SAMPLE PATH PROPERTIES OF GAUSSIAN RANDOM FIELDS AND STOCHASTIC

PARTIAL DIFFERENTIAL EQUATIONS

By

Cheuk Yin Lee

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Statistics—Doctor of Philosophy

2020

ABSTRACT

SAMPLE PATH PROPERTIES OF GAUSSIAN RANDOM FIELDS AND STOCHASTIC

PARTIAL DIFFERENTIAL EQUATIONS

By

Cheuk Yin Lee

Gaussian random fields are studied and applied in a wide range of scientific areas. In

particular, the solutions of stochastic partial differential equations (SPDEs) form an impor-

tant class of random fields and it is of interest to study the properties of their sample paths.

The objective of this dissertation is to develop some methods for studying Gaussian random

fields and to use these methods to investigate the sample path properties of SPDEs. We

study the existence of multiple points for a general class of Gaussian random fields including

fractional Brownian sheets, systems of stochastic heat equations and systems of stochastic

wave equations. We also study the regularity of local times and the Hausdorff measure of

level sets of Gaussian random fields and give an application to the stochastic heat equation.

Moreover, for the stochastic wave equation, we examine further properties including local

nondeterminism, the exact modulus of continuity, and the propagation of singularities.

ACKNOWLEDGMENTS

I wish to express my deep gratitude to my advisor Professor Yimin Xiao, for his continued

support, guidance and encouragement in my research, and his invaluable advice for my future

career.

I would like to thank Professors Shlomo Levental, V. S. Mandrekar and Dapeng

Zhan for serving on my graduate committee. I would like to express my appreciation to the

Department of Statistics and Probability for their support in my PhD program. I would also

like to thank my family and friends for their support. Finally, I would like to give special

thanks to my wife Yuan Chen, for her love and understanding, and being supportive at all

times.

iii

TABLE OF CONTENTS

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Gaussian Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
2.2 Stochastic Partial Differential Equations
2.2.1 Walsh’s Stochastic Integration . . . . . . . . . . . . . . . . . . . . . .
2.2.2
Stochastic Heat Equation . . . . . . . . . . . . . . . . . . . . . . . .
Stochastic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3

Chapter 3 Multiple Points of Gaussian Random Fields

. . . . . . . . . . .
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Assumptions and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Preliminaries
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Proof of Theorem 3.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Examples
. . . . . . . . . . . . . . . . . . . . . . .
System of Stochastic Heat Equations . . . . . . . . . . . . . . . . . .
System of Stochastic Wave Equations . . . . . . . . . . . . . . . . . .

3.5.1 Fractional Brownian Sheets
3.5.2
3.5.3

Chapter 4

Local Times and Level Sets of Gaussian Random Fields . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2 Joint Continuity of Local Times . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 H¨older conditions of Local Times
. . . . . . . . . . . . . . . . . . . . . . . .
4.4 Hausdorff Measure of Level Sets . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Stochastic Heat Equation and Strong Local Nondeterminism . . . . . . . . .

Chapter 5

Local Nondeterminism and the Exact Modulus of Continuity
for the Stochastic Wave Equation . . . . . . . . . . . . . . . . . .
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Local Nondeterminism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 The Exact Uniform Modulus of Continuity . . . . . . . . . . . . . . . . . . .

4
4
7
8
11
15

18
18
19
22
34
41
42
48
50

54
54
57
73
76
79

90
90
92
97

Chapter 6 Propagation of Singularities for the Stochastic Wave Equation 103
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Simultaneous Law of Iterated Logarithm: Upper Bound . . . . . . . . . . . . 106
6.3 Simultaneous Law of Iterated Logarithm: Lower Bound . . . . . . . . . . . . 113
6.4 Singularities and Their Propagation . . . . . . . . . . . . . . . . . . . . . . . 127

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

iv

Chapter 1

Introduction

Gaussian random fields are studied extensively in probability, and have useful applications in

a wide range of scientific areas such as statistics, physics, engineering, biology, economics and

finance. Random fields are generalization of stochastic processes in the sense that they are
indexed not only by a single time variable t ∈ R+, but by multi-dimensional variables such
as spatial position x ∈ Rn (n = 1, 2 or 3) or even time and space variables (t, x) ∈ R+ × Rn.

In particular, a large class of random fields arise naturally as solutions of stochastic partial

differential equations (SPDEs). In this thesis, SPDEs are partial differential equations that

are subject to random perturbations such as a white noise. Some of them are motivated from

physics and can be used to model randomness in physical phenomena. In both mathematic

and scientific point of view, it is interesting and meaningful to investigate properties of

solutions of SPDEs.

For fundamental Gaussian random fields such as the Brownian motion, fractional Brow-

nian motion and Brownian sheet, many sample path properties have been studied in the

literature. Moreover, some unified methods for anisotropic Gaussian random fields with

general assumptions were developed (see e.g. Xiao [70]). These methods allow us to study

many properties including modulus of continuity, small ball probabilities, hitting probabil-

ities, fractal properties of ranges, graphs, and level sets, existence and regularity of local

times, etc.

1

There are different approaches for studying SPDEs. One of them is the random field ap-

proach based on Walsh’s theory of stochastic integration. This approach emphasizes solutions

as real-valued random fields, as opposed to other approaches that consider solutions that

take values in certain infinite dimensional spaces. The links between different approaches

are discussed by Dalang [12]. In this thesis, we focus on the random field approach. As we

will see, the methods of random fields are useful for obtaining precise results on analytic and

geometric properties of the sample paths of SPDEs.

The main purpose of this thesis is to develop some methods of Gaussian random fields

and to use these methods to study sample path properties of the solutions of SPDEs.

This thesis is organized as follows. In Chapter 2, we begin with some preliminaries and

overview of Gaussian random fields and SPDEs. We introduce some important examples

of Gaussian random fields, and then review Walsh’s theory of stochastic integration and

SPDEs. We also introduce two important examples of SPDEs, namely the stochastic heat

equation and wave equation.

In Chapter 3, we study the multiple points (or self-intersections) of the sample paths of

Gaussian random fields. Based on a covering argument, we prove that for a large class of

Gaussian random fields, multiple points do not exist in critical dimensions. We apply this

result to the fractional Brownian sheet, systems of stochastic heat equations and systems of

stochastic wave equations.

Chapter 4 is devoted to the study of the local times and level sets of a class of anisotropic

Gaussian random fields that satisfies the property of strong local nondeterminism. We

prove joint continuity and H¨older conditions for their local times, and discuss the Hausdorff

dimension and Hausdorff measure of their level sets. Our results can be applied to the

solution of stochastic heat equation, which satisfies strong local nondeterminism. We also

2

determine the gauge function for the Hausdorff measure of its level sets.

In Chapters 5 and 6, we examine further properties of the stochastic wave equation driven

by an additive Gaussian noise that is white in time and colored in space. In Chapter 5, we

prove a property of local nondeterminism for the solution of the stochastic wave equation

and apply this property to derive the exact uniform modulus of continuity for the solution.

In Chapter 6, we discuss the notion of singularity for the stochastic wave equation and study

the existence and propagation of singularities based on a simultaneous law of the iterated

logarithm.

Throughout the thesis, we use C and K denote constants whose value may vary in each
appearance, and we use C1, C2, K1, . . . for specific constants. We let R+ denote the set of all
non-negative real numbers. Also, |x| is the absolute value of x if x ∈ R, and the Euclidean
norm of x if x ∈ Rn.

3

Chapter 2

Preliminaries

The purpose of this chapter is to give an overview of Gaussian random fields and stochastic

partial differential equations (SPDEs). We first define Gaussian random fields and give

some important examples. Then we give a self-contained introduction to Walsh’s theory

on stochastic integration and SPDEs. The stochastic heat equation and stochastic wave

equation are the most important examples of SPDEs. We state some existence results and

regularity properties of their solutions. The materials in this chapter are known in the

literature, and they will provide sufficient preliminary knowledge for understanding the rest

of the thesis.

2.1 Gaussian Random Fields

An N -parameter d-dimensional random field, or (N, d)-random field is a stochastic process
u = {u(x) : x ∈ T} that is indexed by a subset T of RN and takes values in Rd, i.e. a family
of random variables u(x) = (u1(x), . . . , ud(x)) : Ω → Rd indexed by x ∈ T . We say that u is

a Gaussian random field if the nd-dimensional random vector (u(x1), . . . , u(xn)) is Gaussian
for all n ≥ 1 and all x1, . . . , xn ∈ T . The probability distributions of the collection of all

these random vectors are called the finite dimensional distributions.

The function m : T → Rd defined by m(x) = E(u(x)) is called the mean function and
the function C : T × T → Rd×d, C = (Cij)1≤i,j≤d, defined by Cij(x, y) = Cov(ui(x), uj(y)),

4

is called the covariance function. We say that u is centered if E(u(x)) = 0 for all x ∈ T .

We can define a Gaussian random field by specifying the mean function and covariance

function because a Gaussian random field is determined by its finite dimensional distribu-

tions, which in turn are determined by the mean function and the covariance function. More
precisely, if we are given a function m : T → Rd and a function C : T × T → Rd×d which is

symmetric i.e. C(x, y) = C(y, x), and nonnegative definite in the sense that

d(cid:88)

n(cid:88)

ai,kaj,l Ci,j(xk, xl) ≥ 0

i,j=1

k,l=1

for all n ≥ 1, for all x1, . . . , xn ∈ T and all ai,k ∈ R (i = 1, . . . , d, k = 1, . . . , n), then
there exists an (N, d)-Gaussian random field {u(x) : x ∈ T} whose mean function is m and

covariance function is C.

The following are two fundamental examples of Gaussian random fields.

Example 2.1.1. Multiparameter fractional Brownian motion.
The (N, d)-fractional Brownian motion with Hurst index H ∈ (0, 1) is defined as a centered
(N, d)-Gaussian random field {B(t) : t ∈ RN} with covariance function

E(Bi(t)Bj(s)) = δij

|t|2H + |s|2H − |t − s|2H

2

for t, s ∈ RN , where δij = 1 if i = j, and δij = 0 otherwise. It follows that the coordinate

components B1, . . . , Bd of B are independent and identically distributed (i.i.d.).

This Gaussian random field has stationary increments in the sense of Yaglom: for any
h ∈ RN , {B(t + h) − B(h) : t ∈ RN} and {B(t) − B(0) : t ∈ RN} are equal in finite

dimensional distributions.

5

When N = 1 and H = 1/2, it is the standard d-dimensional Brownian motion. When

N = 1 and 0 < H < 1, it is the d-dimensional fractional Brownian motion of Hurst index

H. When N > 1 and H = 1/2, it is known as Levy’s multiparameter Brownian motion.

Example 2.1.2. Fractional Brownian sheet.
The (N, d)-fractional Brownian sheet of Hurst indices H1, . . . , HN ∈ (0, 1) is defined as a
centered (N, d)-Gaussian random field {B(t) : t ∈ RN} with covariance function

N(cid:89)

E(Bi(t)Bj(s)) = δij

|t(cid:96)|2H(cid:96) + |s(cid:96)|2H(cid:96) − |t(cid:96) − s(cid:96)|2H(cid:96)

(cid:96)=1

2

for t, s ∈ RN . When H1 = ··· = HN = 1/2, it is called the (N, d)-Brownian sheet.

The fractional Brownian sheet has the property of being anisotropic in the sense that it

can have different regularities and sample properties along different directions.

Also, the fractional Brownian sheet does not have stationary increments and it has subtle

properties that are different from those of the fractional Brownian motion. For example, they

have different form of small ball probabilities and Chung’s law of the iterated logarithm.

For a Gaussian random field u = {u(t) : t ∈ T}, it will be convenient to use the notation

σu(t, s) := (E|u(t) − u(s)|2)1/2

to denote the canonical metric (or pseudo-metric) on T . In many examples, we can find
α1, . . . , αN ∈ (0, 1) and positive finite constants C1, C2 such that

|tj − sj|αj ≤ σu(t, s) ≤ C2

|tj − sj|αj

(2.1)

N(cid:88)

C1

N(cid:88)

j=1

j=1

6

for all t, s ∈ T . The parameters α1, . . . , αN play important roles in characterizing the sample

path properties of u(t) e.g. regularity, fractal properties and hitting probabilities (see [70]).

In Example 2.1.1, the (N, d)-fractional Brownian motion of Hurst index H satisfies (2.1)
with α1 = ··· = αN = H. In Example 2.1.2, the (N, d)-fractional Brownian sheet of Hurst

indices H1, . . . , HN satisfies (2.1) with αi = Hi for i = 1, . . . , N .

In the next section, we will see more examples of Gaussian and non-Gaussian random

fields that arise as solutions of stochastic partial differential equations.

2.2 Stochastic Partial Differential Equations

As explained in the Introduction, solutions of SPDEs form a large class of random fields

and we are interested in studying their sample path properties. We follow [64] and give

an introduction to Walsh’s theory of stochastic integration, which allows us to construct

random field solutions to SPDEs. We will then discuss two of the most important examples

of SPDEs, namely the stochastic heat equation and stochastic wave equation.

Consider a differential operator D with constant coefficients and the SPDE

Du(t, x) = σ(u(t, x)) ˙W (t, x) + b(u(t, x)),

t ≥ 0, x ∈ Rk,

(2.2)

where σ : R → R and b : R → R are Lipschitz functions, and ˙W is a Gaussian noise, whose

definition will be given later.

From the theory of partial differential equations, the differential operator D always has

a fundamental solution G, namely a distribution G that solves DG = δ0, where δ0 is the
Dirac measure at 0 ∈ R1+k, and it follows that u = G ∗ ϕ solves the equation Du = ϕ for

7

any ϕ in S(R+ × Rk), the space of smooth rapidly decreasing functions (Schwartz space),
where G ∗ ϕ is the convolution of G and ϕ in (t, x)-variables. See [55, Ch. 8].

In view of this, a mild solution to the SPDE (2.2) is a jointly measurable real-valued

random field {u(t, x) : t ≥ 0, x ∈ Rk} that satisfies

(cid:17)

u(t, x) = G ∗(cid:16)
(cid:90)(cid:90)

=

[0,t]×Rk

σ(u) ˙W + b(u)

G(t − s, x − y)

(cid:16)

σ(u(s, y)) ˙W (s, y) + b(u(s, y))

(cid:17)

ds dy

and is adapted to a filtration generated by the noise ˙W (defined in (2.5) below). To explain

the meaning of the above stochastic integral, let us introduce Walsh’s approach [64] of

martingale measures and stochastic integration.

2.2.1 Walsh’s Stochastic Integration

We will consider spatially homogeneous (centered) Gaussian noise W that is white in time

and has spatial covariance f , which is a non-negative definite function. The Gaussian noise
is defined as a centered Gaussian process {W (ϕ) : ϕ ∈ C∞
c (R1+k)} that is indexed by ϕ in
C∞
c (R1+k), the space of real-valued smooth functions on R1+k of compact support, and has

covariance

(cid:90)

R+

(cid:90)

Rk

ds

(cid:90)

Rk

dy

E(W (ϕ)W (ψ)) =

for all ϕ, ψ ∈ C∞

c (R1+k). Formally, we write

dy(cid:48)ϕ(s, y)f (y − y(cid:48))ψ(s, y(cid:48))

E( ˙W (s, y) ˙W (s(cid:48), y(cid:48))) = δ0(s − s(cid:48))f (y − y(cid:48)).

(2.3)

8

˙W is a space-time white noise if f = δ0, the Dirac delta function. Another

We say that
spatial covariance that is commonly used is f (y) = |y|−β, where 0 < β < k.

Let Bb(Rk) denote the set of all bounded Borel sets in Rk. The martingale measure
induced by the noise ˙W is the stochastic process {Mt(A) : t ∈ R+, A ∈ Bb(Rk)} defined by

Mt(A) = lim

n→∞ W (ϕn),

(2.4)

where the right-hand side is the limit of a sequence {W (ϕn) : n ≥ 1} in L2(Ω, F , P), and ϕn
is any sequence in C∞
c (R1+k) such that ϕn ↓ 1[0,t]×A. It follows that for each A ∈ Bb(Rk),
the stochastic process {Mt(A) : t ≥ 0} is a martingale with respect to the filtration

Ft = σ{Ms(B) : 0 ≤ s ≤ t, B ∈ Bb(Rk)},

t ≥ 0.

(2.5)

Let us define an elementary process as a stochastic process g(t, x) : Ω → R, with t ≥ 0

and x ∈ Rk, of the form

g(t, x, ω) = X(ω)1(a,b](t)1B(x),

where 0 ≤ a < b, B ∈ Bb(Rk), and X is a bounded, Fa-measurable random variable. For

an elementary process, we can naturally define its stochastic integral as

(cid:90)

[0,t]×Rk

g(s, y)W (ds dy) := X(cid:0)Mt∧b(B) − Mt∧a(B)(cid:1).

By linearity, we can then extend the definition of stochastic integration to the class S of all

linear combinations of elementary processes, which we will call simple processes.

For the martingale measure M , we can define a function QM by QM ((s, t] × B × C) =

9

(cid:104)M·(B), M·(C)(cid:105)t − (cid:104)M·(B), M·(C)(cid:105)s, for any 0 ≤ s < t and B, C ∈ B(Rk).

We say that the martingale measure M is worthy if there exists a random σ-finite measure

KM (A × B × C, ω), where A ∈ B(R+), B, C ∈ B(Rk) and ω ∈ Ω, such that

1. B × C (cid:55)→ KM (A × B × C, ω) is nonnegative definite and symmetric;

2. {KM ((0, t] × B × C) : t ≥ 0} is a σ(S )-measurable process for all B, C ∈ B(Rk);

3. For all t > 0 and compact sets B, C ∈ B(Rk), E[KM ((0, t] × B × C)] < ∞;

4. For all t > 0 and B, C ∈ B(Rk), |QM ((0, t] × B × C)| ≤ KM ((0, t] × B × C) a.s.

Consider t ∈ [0, T ], where T > 0 is fixed. If M is a worthy martingale measure, then for

any t ∈ [0, T ], the stochastic integral defines a linear map

g(s, y)W (ds dy),

(2.6)

(cid:90)

g (cid:55)→

[0,t]×Rk

(cid:90)(cid:90)(cid:90)

from S to L2(Ω, FT , P), which is continuous with respect to the norm (cid:107)·(cid:107)M on S and the
L2-norm on L2(Ω, FT , P), where (cid:107) · (cid:107)M is defined by

(cid:107)g(cid:107)2

M := E

|g(s, y)g(s, y(cid:48))|KM (ds dy dy(cid:48)).

(2.7)

[0,T ]×Rk×Rk

˙W is a space-time white noise i.e. f = δ0 in (2.3). Then
Example 2.2.1. Suppose that
QM ((0, t] × B × C) = tλk(B ∩ C), where λk is the k-dimensional Lebesgue measure. Take
KM (A × B × C) = λ1(A)λk(B ∩ C). It follows that M is a worthy martingale measure and

(cid:107)g(cid:107)2

M = E

|g(s, y)|2 ds dy.

(cid:90) T

(cid:90)

Rk

0

10

Example 2.2.2. Suppose that the Gaussian noise ˙W satisfies (2.3) with f (y) = |y|−β and

0 < β < k. Then

We can take

QM ((0, t] × B × C) = t

KM (A × B × C) = λ1(A)

(cid:90)

(cid:90)

|y − y(cid:48)|−β dy dy(cid:48).

B

(cid:90)

C

(cid:90)

B

C

|y − y(cid:48)|−β dy dy(cid:48),

and the martingale measure M is worthy and

(cid:90) T

(cid:90)

(cid:90)

Rk

Rk

0

(cid:107)g(cid:107)2

M = E

|g(s, y)g(s, y(cid:48))||y − y(cid:48)|−β ds dy dy(cid:48).

Let PM be the set of all σ(S )-measurable processes g such that (cid:107)g(cid:107)M < ∞. Then
(PM ,(cid:107) · (cid:107)M ) is a Banach space and S is dense in PM . It follows that (2.6) extends to a
continuous linear map from PM to L2(Ω, FT , P). Therefore, we are now able to define the

stochastic integral

(cid:90)

g(s, y)W (ds dy)

[0,t]×Rk

as the image of g under this map, for a large class of processes g in PM .

2.2.2 Stochastic Heat Equation

Consider the stochastic heat equation



u(t, x) − ∆u(t, x) = ˙W (t, x),

t ≥ 0, x ∈ Rk,

∂
∂t

(2.8)

u(0, x) = u0(x),

11

where ˙W is a Gaussian noise. The fundamental solution for the heat equation is

G(t, x) =

−|x|2
4t .

e

1

(4πt)1/2

For the moment, suppose that the spatial dimension is 1 (i.e. k = 1) and ˙W is a space-

time white noise. Then the mild solution to (2.8) is the real-valued Gaussian random field
{u(t, x) : t ≥ 0, x ∈ R} defined by

u(t, x) = (G ∗ u0)(t, x) +

G(t − s, x − y)W (ds dy).

However, for k ≥ 2, the stochastic integral above is not well-defined because the norm

(cid:107) · (cid:107)M of the integrand defined in (2.7) is infinite:

(cid:90)

[0,t]×R

(cid:90) t

(cid:90)

Rk

0

(cid:107)G(t − ·, x − ·)(cid:107)2

M =

|G(t − s, x − y)|2 ds dy = ∞.

As a consequence, there is no real-valued process solution for (2.8) when k ≥ 2. The solution

is a random Schwartz distribution, but we will not discuss this kind of solution in this thesis.
To obtain real-valued solutions when k ≥ 2, Dalang’s approach [11] is to replace the

space-time white noise by a Gaussian noise that in white in time but correlated in space.

Consider the nonlinear stochastic heat equation

u(t, x) − ∆u(t, x) = σ(u(t, x)) ˙W (t, x) + b(u(t, x)),

t ≥ 0, x ∈ Rk,

∂
∂t

(2.9)



u(0, x) = u0(x).

12

Suppose that the Gaussian noise ˙W satisfies (2.3):

E[ ˙W (s, y) ˙W (s(cid:48), y(cid:48))] = δ0(s − s(cid:48))f (y − y(cid:48)).

Recall the natural filtration {Ft} of the noise defined in (2.5). By a mild solution to (2.9) we
mean a jointly measurable, {Ft}-adapted, real-valued random field {u(t, x) : t ≥ 0, x ∈ Rk}

that satisfies the integral equation

u(t, x) = (G ∗ u0)(t, x) +

+

(cid:90)
(cid:90)

[0,t]×Rk

[0,t]×Rk

G(t − s, x − y)σ(u(s, y))W (ds dy)

G(t − s, x − y)b(u(s, y)) ds dy.

(2.10)

Suppose that f ≥ 0 and f is a non-negative definite function, i.e.

(cid:90)

Rk

(ϕ ∗ ˜ϕ)(x)f (x)dx ≥ 0

for all ϕ ∈ S(Rk), and ˜ϕ(x) := ϕ(−x). Let µ be a nonnegative measure on Rk whose Fourier
transform is f (x). For example, if f (x) = |x|−β, where 0 < β < k, then µ(dξ) = ck,β|ξ|β−kdξ

for some constant ck,β depending on k and β.

The following is an existence and uniqueness result: if u0 is measurable and bounded, σ

and b are Lipschitz, and µ satisfies Dalang’s condition

(cid:90)

Rk

µ(dξ)

1 + |ξ|2 < ∞,

13

then there exists a unique solution to (2.9) which is L2-continuous and satisfies

E(|u(t, x)|p) < ∞

sup
0≤t≤T

sup
x∈Rk

for any T < ∞ and p ≥ 1. See [11] (cf. [49, 50]).

The solution can be obtained by Picard iteration. Define u0(t, x) = u0(x) and

un+1(t, x) = (G ∗ u0)(t, x) +

+

G(t − s, x − y)σ(un(s, y))W (ds dy)

G(t − s, x − y)b(un(s, y)) ds dy

(cid:90)
(cid:90)

[0,t]×Rk

[0,t]×Rk

for n ≥ 1. One can verify that un(t, x) converges in L2 using Gronwall’s lemma, and show

that the limit u(t, x) satisfies the integral equation (2.10).

Here is a regularity result for the solution: u0 is a bounded, ρ-H¨older continuous function

for some ρ ∈ (0, 1), σ and b are Lipschitz, and

(cid:90)

µ(dξ)

(1 + |ξ|2)η < ∞

(2.11)

Rk

for some η ∈ (0, 1), then the solution u(t, x) of (2.9) is a.s. β1-H¨older continuous in t and
2(ρ∧ (1− η)) and 0 < β2 < ρ∧ (1− η). Indeed,
β2-H¨older continuous in x, for any 0 < β1 < 1
2(ρ ∧ (1 − η)) and 0 < β2 < ρ ∧ (1 − η), there exists C such
for any T > 0, p ≥ 2, 0 < β1 < 1

that

E(cid:0)|u(t, x) − u(s, y)|p(cid:1) ≤ C(cid:0)|t − s|β1p + |x − y|β2p(cid:1)

for all t, s ∈ [0, T ] and x, y ∈ Rk. In particular, if f (y) = |y|−β, then (2.11) is satisfied if
and only if 0 < β < 2η ∧ k. See [58].

14

2.2.3 Stochastic Wave Equation

Consider the stochastic wave equation



u(t, x) − ∆u(t, x) = ˙W (t, x),

t ≥ 0, x ∈ Rk,

∂2
∂2t

(2.12)

u(0, x) = 0,

∂
∂t

u(0, x) = 0,

with an additive Gaussian noise ˙W .

Suppose that if k = 1,

˙W is either a space-time white noise or satisfies

E[ ˙W (s, y) ˙W (s(cid:48), y(cid:48))] = δ0(s − s(cid:48))|y − y(cid:48)|−β,

0 < β < 1;

and if k ≥ 2,

˙W satisfies

E[ ˙W (s, y) ˙W (s(cid:48), y(cid:48))] = δ0(s − s(cid:48))|y − y(cid:48)|−β,

0 < β < 2.

Let G be the fundamental solution of the wave equation. Recall that if k = 1, G(t, x) =

1

21{|x|<t}; if k ≥ 2 and k is even,

G(t, x) = ck

(cid:19)(k−2)/2

(cid:18)1

t

∂
∂t

(t2 − |x|2)

−1/2
+ ;

if k ≥ 3 and k is odd,

(cid:18)1

t

∂
∂t

(cid:19)(k−3)/2 σk

t (dx)

t

,

G(t, x) = ck

where σk

t denotes the uniform surface measure on the sphere {x ∈ Rk : |x| = t}, see [22,

15

Chapter 5].

Since G is a function when k = 1 or 2, the solution of (2.12) can be defined by

(cid:90) t

(cid:90)

G(t − s, x − y)W (ds dy)

(2.13)

u(t, x) =

Rk

0

in the sense of Walsh. For k ≥ 3, G is not a function but a distribution. It is not straight-

forward to define the stochastic integral (2.13) in this case.

However, for all dimensions, the Fourier transform of G in variable x is still a function:

F (G(t,·))(ξ) =

sin(t|ξ|)

|ξ|

t ≥ 0, ξ ∈ Rk.

,

(2.14)

Based on this observation, Dalang [11] extended the definition of Walsh’s stochastic inte-

gration so that the integrand can be taken from certain class of distributions whose Fourier

transform in x is a function. As a result, for all dimensions we can obtain real-valued process

solutions of equation (5.1):

u(t, x) =

(cid:90) t

(cid:90)

Rk

0

G(t − s, x − y) W (ds dy).

The range of β has been chosen so that the stochastic integral exists, and the solution is a

Gaussian random field.

Dalang and Sanz-Sol´e [20, Proposition 4.1] proved that for all 0 < a < a(cid:48) < ∞ and

0 < b < ∞, there exist positive finite constants C1, C2 such that

C1(|t − s|2−β + |x − y|2−β) ≤ E(|u(t, x) − u(s, y)|2) ≤ C2(|t − s|2−β + |x − y|2−β)

16

for all (t, x), (s, y) ∈ [a, a(cid:48)] × [−b, b]k. By the Kolmogorov continuity theorem, the solution
u(t, x) is a.s. H¨older continuous in (t, x) of any exponent < (2 − β)/2.

For the nonlinear stochastic wave equation

∂2
∂t2 u(t, x) − ∆u(t, x) = σ(u(t, x)) ˙W (t, x) + b(u(t, x)),

t ≥ 0, x ∈ Rk,

(2.15)

the case k = 1 was studied by many authors, see e.g. [8, 9, 39, 45, 64]. For k = 2, the

existence and regularity of the solution were studied by Dalang and Frangos [13], and Millet

and Morien [43]. For k = 3, the H¨older-Sobolev regularity of the solution was studied by

Dalang and Sanz-Sol´e [19]. Not much is known for the nonlinear stochastic wave equation
in dimension k ≥ 4.

17

Chapter 3

Multiple Points of Gaussian Random

Fields

3.1

Introduction

In this chapter, we study the non-existence of multiple points of Gaussian random fields.
Let v = {v(x) : x ∈ Rk} be an Rd-valued Gaussian random field. For a set T ⊂ Rk and an
integer m ≥ 2, we say that z ∈ Rd is an m-multiple point of the sample path v(·, ω) on T if
there are m distinct points x1, . . . , xm ∈ T such that z = v(x1, ω) = ··· = v(xm, ω). This

chapter is based on [16].

The existence of multiple points of Gaussian random fields have been studied by several

authors. Sufficient or necessary conditions for the (N, d)-fractional Brownian motion BH =
{BH (t) : t ∈ Rk} with Hurst index H to have multiple points were obtained by Kˆono [32],
Goldman [24] and Rosen [54]. Their results show that if km > (m − 1)Hd, then BH has
m-multiple points on any interval T ⊆ Rk; and if km < (m − 1)Hd, then BH has no m-
multiple points on Rk\{0}. The multiple points of the Brownian sheet was also studied by

Rosen [54] via self-intersection local times.

For BH , the critical dimension is km = (m − 1)Hd. In general, the problem for proving

the non-existence of multiple points of a random field in the critical dimensions is more

18

difficult than the non-critical case. The critical case for the fractional Brownian motion

and the Brownian sheet has been solved by different methods. The former was solved by

Talagrand [61] and the latter was solved by Dalang et al. [15] and Dalang and Mueller [17].

Our study is motivated by the interest in the intersection problems for the solutions of

linear systems of stochastic heat and wave equations, where the method in [15, 17] fails in

general. Based on the framework in [18], we extend Talagrand’s approach in [61] to a large

class of Gaussian random fields including fractional Brownian sheets and the solutions of

systems of stochastic heat and wave equations with constant coefficients. Moreover, our

theorem provides an alternative proof for the results in [15, 17] with the use of general

Gaussian principles and the harmonizable representation of the Brownian sheet.

The chapter is organized as follows. In Section 3.2, we state our assumptions and main

result (Theorem 3.2.4). In Section 3.3, we establish some necessary lemmas and the main

estimate Proposition 3.3.6 for proving the main theorem and, in Section 3.4, we prove the

theorem. In Section 3.5, we provide some examples of Gaussian random fields to which the

theorem can be applied, including the Brownian sheet, fractional Brownian sheets, and the

solutions of systems of stochastic heat and wave equations.

3.2 Assumptions and Main Result

Throughout this chapter, we assume that v = {v(x) : x ∈ Rk} is a centered, continuous Rd-
valued Gaussian random field defined on a probability space (Ω, F , P) with i.i.d. components.
Write v(x) = (v1(x), . . . , vd(x)) for x ∈ Rk. We will study the existence problem of multiple
points of v(x) on a set T ⊂ Rk.

By a closed interval in Rk we mean a set I of the form (cid:81)k

j=1[cj, dj], where cj < dj.

19

We assume that T ⊂ Rk is a fixed index set that can be written as a countable union of
compact intervals. To avoid trivial multiple points, we will take, for example, T = Rk\{0}
or T = (0,∞)k.

We consider the following two assumptions, which are slight modification of Assumptions

2.1 and 2.4 in [18].

Assumption 3.2.1. There exists a centered Gaussian random field {v(A, x), A ∈ B(R+), x ∈
T}, where B(R+) is the Borel σ-algebra on R+ = [0,∞), such that the following hold:

(a) For all x ∈ T , A (cid:55)→ v(A, x) is an Rd-valued white noise (or, more generally, an

independently scattered Gaussian noise with a control measure µ) with i.i.d. components,
v(R+, x) = v(x), and v(A,·) and v(B,·) are independent whenever A and B are disjoint.

(b) There exist constants γj > 0, j = 1, . . . , k with the following properties: For every
compact interval F ⊂ T , there exist constants c0 > 0 and a0 ≥ 0 such that for all a0 ≤ a ≤
b ≤ ∞ and x, y ∈ F ,

(cid:107)v([a, b), x) − v(x) − v([a, b), y) + v(y)(cid:107)L2 ≤ c0

aγj|xj − yj| + b−1

,

(3.1)

(cid:18) k(cid:88)

j=1

(cid:19)

and

k(cid:88)
In the above, (cid:107)X(cid:107)L2 =(cid:0)E|X|2(cid:1)1/2 for a random vector X.

(cid:107)v([0, a0), x) − v([0, a0), y)(cid:107)L2 ≤ c0

j=1

|xj − yj|.

(3.2)

Notice that in Assumption 3.2.1 the constants a0 and c0 may depend on F , but γj

(j = 1, . . . , k) do not. As shown by Dalang et al. [18], the parameters γj (j = 1, . . . , k) play

important roles in characterizing sample path properties (e.g., regularity, fractal properties,

20

hitting probabilities) of the random field {v(x), x ∈ T}.

Let αj = (γj + 1)−1 and Q =(cid:80)k

. Define the metric ∆ on Rk by

j

j=1 α−1
k(cid:88)

∆(x, y) =

|xj − yj|αj .

(3.3)

j=1

Assumption 3.2.2. For every compact interval F ⊂ T , there are positive constants ε0, C
and δj ∈ (αj, 1], j = 1, . . . , k, such that the following holds:

For all closed intervals I ⊂ F , x ∈ I and 0 < ρ ≤ ε0, there is x(cid:48) ∈ I(ρ) (here and below,
I(ρ) denotes the ρ-neighbourhood of I in the Euclidean norm) such that for all y, ¯y ∈ I(ρ)
with ∆(x, y) ≤ 2ρ and ∆(x, ¯y) ≤ 2ρ,

(cid:12)(cid:12)E((vi(y) − vi(¯y))vi(x(cid:48)))(cid:12)(cid:12) ≤ C

k(cid:88)

|yj − ¯yj|δj ,

i = 1, . . . , d.

(3.4)

j=1

The constants ε0 and C may depend on F .

In addition, we impose a non-degeneracy assumption.

Assumption 3.2.3. For any m distinct points x1, . . . , xm in T , the random variables

v1(x1), . . . , v1(xm) are linearly independent, or equivalently, the Gaussian distribution of

(v1(x1), . . . , v1(xm)) is non-degenerate.

The main result of this chapter is the following.

Theorem 3.2.4. Let m ≥ 2. Suppose that Assumptions 3.2.1, 3.2.2 and 3.2.3 hold. If
mQ ≤ (m − 1)d, then {v(x), x ∈ T} has no m-multiple points almost surely.

21

3.3 Preliminaries

In this section, we provide some lemmas and a main estimate that will be used for proving
Theorem 3.2.4. It suffices to prove that if mQ ≤ (m − 1)d then, for every compact interval
F ⊂ T , {v(x), x ∈ F} has no m-multiple points. Therefore, from now on, we will assume

that T is a compact interval.

center x and radius r in the metric ∆ in (3.3) and let Br(x) =(cid:81)k

For x ∈ T and r > 0, denote by S(x, r) = {y ∈ Rk : ∆(x, y) ≤ r} the closed ball with
j=1[xj − r1/αj , xj + r1/αj ].

Notice that S(x, r) ⊆ Br(x) and Br/k(x) ⊆ S(x, r).

Fix m ≥ 2. Given any m distinct points t1, . . . , tm ∈ T , we can find an integer n ≥ 1

such that ∆(ti, tj) ≥ 1/n for i (cid:54)= j. For ρ > 0, let Bi

ρ = Bρ(ti) (i = 1, . . . , m).

Consider the random set

Mt1,...,tm;ρ =

(cid:110)
z ∈ Rd :∃ (x1, . . . , xm) ∈ m(cid:89)

Bi
ρ

i=1

such that z = v(x1) = ··· = v(xm)

(cid:111)

,

(3.5)

which is the intersection of the images v(Bi
ρ) for i = 1, . . . , m. By the continuity of the
process v(x), the set of m-multiple points of {v(x) : x ∈ T} can be written as a countable

union

(cid:91)

n≥1

(cid:91)

(cid:91)

(cid:91)

(t1,...,tm)∈An

ρ0∈(0,1/n)∩Q

ρ∈(0,ρ0)∩Q

Mt1,...,tm;ρ

(3.6)

where An = {(t1, . . . , tm) : ti ∈ T ∩ Qk, ∆(ti, tj) ≥ 1/n for i (cid:54)= j}.

For the rest of this section, we fix n and (t1, . . . , tm) ∈ An. Let ρ0 ∈ (0, 1/n) be a small
number which may depend on t1, . . . , tm and will be determined in Lemma 3.3.8 below. For

22

simplicity of notation, we assume that Bρ0(ti) ⊆ T for i = 1, . . . , m (otherwise we take the
intersection with T ), and we omit the subscripts t1, . . . , tm in (3.5) and write Mρ.

Recall from [18] that, under Assumption 3.2.1, ∆ provides an upper bound for the L2-
norm of the increments of {v(x), x ∈ T} and in particular v(x) is continuous in L2(Ω, F , P).

Lemma 3.3.1. [18, Proposition 2.2] Under Assumption 3.2.1, for all x, y ∈ T with ∆(x, y) ≤
min{a−1

0 , 1}, we have (cid:107)v(x) − v(y)(cid:107)L2 ≤ 4c0∆(x, y).

Assumption 3.2.1 suggests that for any s ∈ T and x that is close to s, the increment
v(x)−v(s) can be approximated well by v([a, b), x)−v([a, b), s) if we choose a and b carefully.

The following lemma from [18] quantifies the approximation error on S(s, cr).

Lemma 3.3.2. Let c > 0 be a constant. Consider b > a > 1, ε0 > r > 0 and set

α−1
j −1

α−1
j + b−1.

r

a

A =

k(cid:88)

j=1

There are constants A0, ˜K and ˜c (depending on c0 in Assumption 3.2.1 and c) such that if
A ≤ A0r and

u ≥ ˜KA log1/2(cid:16) r

(cid:17)

A

,

(3.7)

then for any s ∈ T ,

(cid:26)

P

sup

x∈S(s, cr)

|v(x) − v(s) − (v([a, b), x) − v([a, b), s))| ≥ u

(cid:27)

(cid:18)

− u2
˜cA2

(cid:19)

.

≤ exp

Remark 3.3.3. The constant c in Lemma 3.3.2 and Proposition 3.3.6 below is not important.

It merely helps to simplify the presentation in Section 3.4, where sometimes we switch back

and forth between a ball S(s, r) and an interval Br(x).

23

For describing the contribution of the main part v([a, b], x) − v([a, b], s), we will apply

the small ball probability estimate given in Lemma 3.3.5 below. We refer to Lemma 2.2 of

[60] for a general lower bound on the small ball probability of Gaussian processes. However,

it was pointed out by Slobodan Krstic (personal communication) that the condition of that

lemma is not correctly stated. Indeed, the lemma fails if we consider S consisting of two

points and independent standard normal random variables indexed by the two points. We

will make use of the following reformulation of the presentation of Talagrand’s lower bound

given by Ledoux [34, (7.11)–(7.13) on p. 257].

Lemma 3.3.4. Let {X(t), t ∈ S} be a separable, vector-valued, centered Gaussian process
indexed by a bounded set S with the canonical metric dX (s, t) = (E|X(s) − X(t)|2)1/2. Let

Nε(S) denote the smallest number of dX -balls of radius ε needed to cover S. If there is a
decreasing function ψ : (0, δ] → (0,∞) such that Nε(S) ≤ ψ(ε) for all ε ∈ (0, δ] and there
are constants c2 ≥ c1 > 1 such that

c1ψ(ε) ≤ ψ(ε/2) ≤ c2ψ(ε)

(3.8)

for all ε ∈ (0, δ], then there is a constant K depending only on c1 and c2 such that for all
u ∈ (0, δ),

(cid:19)

≥ exp(cid:0) − Kψ(u)(cid:1).

(3.9)

(cid:18)

P

|X(s) − X(t)| ≤ u

sup
s,t∈S

Let ρ ∈ (0, ρ0/3), recall that B1

2ρ, . . . , Bm

2ρ are the rectangles centered at t1, . . . , tm. By

applying Assumption 3.2.1 and Lemma 3.3.4, we derive the following lemma.

Lemma 3.3.5. Suppose that Assumption 3.2.1 holds and ρ ∈ (0, ρ0/3) is a constant. Then

there exist constants K and 0 < η0 < ρ0/3, depending on c0 in Assumption 3.2.1, such that

24

for all (s1, . . . , sm) ∈ B1

2ρ × ··· × Bm

2ρ, for all 0 < a < b and 0 < u < r < η0, we have

(cid:18)

P

sup
1≤i≤m

sup

xi∈S(si,r)

|v([a, b), xi) − v([a, b), si)| ≤ u

≥ exp

.

(3.10)

(cid:19)

(cid:33)

(cid:32)

−K

rQ
uQ

Proof. As suggested by the proof of (3.3) in Talagrand [61], (3.10) can be derived from

Lemma 3.3.4. However, there was a typo in the exponent in (3.3) in [61] (the ratio

r

u1/α

there should be raised to the power N ) and the suggested proof by introducing the auxiliary

process Z does not give the correct power for

r

u1/α in (3.3) in [61], which is needed for

proving Proposition 3.4 in [61]. Hence we give a proof of (3.10).

2ρ and r < ρ0/3, define S =(cid:83)m

For (s1, . . . , sm) ∈ B1

2ρ × ··· × Bm

i=1 S(si, r). Under our
assumption, we have S(si, r) ⊆ T for i = 1, . . . , m. Thus, S ⊆ T . It follows from Assumption
3.2.1 that for all x, y ∈ S,

(cid:107)v([a, b), x) − v([a, b), y)(cid:107)2

L2 = (cid:107)v(x) − v(y)(cid:107)2
L2 − (cid:107)v(R+ \ [a, b), x) − v(R+ \ [a, b), y)(cid:107)2
≤ (cid:107)v(x) − v(y)(cid:107)2
L2.

L2

By Lemma 3.3.1, we have that the canonical metric for {v([a, b), x), x ∈ S} satisfies

dv(s, t) := (cid:107)v([a, b), x) − v([a, b), y)(cid:107)L2 ≤ 4c0∆(x, y)

for all x, y ∈ S with ∆(x, y) small. Hence there is a constant η0 ∈ (0, ρ0/3) such that for all
r ∈ (0, η0) and ε ≤ r, the minimal number of dv-balls of radius ε needed to cover S is

Nε(S) ≤ ψ(ε) := CN,Q(r/ε)Q.

25

Note that this function ψ(ε) satisfies (3.8) with the constants c1 = c2 = 2Q which are greater

than 1. It follows from Lemma 3.3.4 that there is a constant K such that (3.10) holds. This

proves Lemma 3.3.5.

The following is the main estimate. It is an extension of Proposition 3.4 in [61].

Proposition 3.3.6. Let c > 0 be a constant and suppose that Assumption 3.2.1 holds. Then
there are constants K1 and 0 < η1 < 1 such that for all 0 < r0 < η1, ρ ∈ (0, ρ0/3), and
(s1, . . . , sm) ∈ B1

2ρ × ··· × Bm

2ρ, we have

(cid:32)

P

∃ r ∈ [r2

0, r0],

sup
1≤i≤m

sup

xi∈S(si, cr)

(cid:18)
|v(xi) − v(si)| ≤ K1r
(cid:32)

≥ 1 − exp

−

(cid:19)−1/Q(cid:33)
(cid:19)1/2(cid:33)

1
r

.

log log

(cid:18)

log

1
r0

Proof. The method of proof is similar to that of Proposition 3.4 in Talagrand [61]. But

the latter contains several typos. For completeness we provide a proof of Proposition 3.3.6

here. The main ingredients are the small ball probability estimate in Lemma 3.3.5 and the

estimate of the approximation error in Lemma 3.3.2,

As in [60, 61] and [18], let U > 1 be fixed for now and its value will be chosen later. Set

r(cid:96) = r0U−2(cid:96) and a(cid:96) = U 2(cid:96)−1/r0. Consider the largest integer (cid:96)0 such that

(cid:96)0 ≤ log(1/r0)
2 log U

.

(3.11)

26

Then for (cid:96) ≤ (cid:96)0, we have r(cid:96) ≥ r2

0. It suffices to show that, for some large constant K1,

P

∃1 ≤ (cid:96) ≤ (cid:96)0,
(cid:32)

≥ 1 − exp

sup
1≤i≤m

(cid:18)

−

log

sup

(cid:19)1/2(cid:33)

xi∈S(si, cr(cid:96))
1
r0

.

|v(xi) − v(si)| ≤ K1

r(cid:96)
(log log 1
r(cid:96)

)1/Q



It follows from Lemma 3.3.5 that, for K1 large enough so that K/KQ

1 ≤ 1/4,

|v([a(cid:96), a(cid:96)+1), xi) − v([a(cid:96), a(cid:96)+1), si)| ≤ K1

r(cid:96)
(log log 1
r(cid:96)

)1/Q



(3.12)



P

1≤i≤m

 sup
(cid:18)

≥ exp

≥

(cid:33)

sup

xi∈S(si, cr(cid:96))

(cid:32)

− K
KQ
1

(cid:19)−1/4

.

log

1
r(cid:96)

log log

1
r(cid:96)

P

∃(cid:96) ≤ (cid:96)0,
1 − P
(cid:96)0(cid:89)

sup
1≤i≤m

(cid:96)=1

= 1 −

(cid:18)

Thus, by the independence of the Gaussian processes v([a(cid:96), a(cid:96)+1),·) ((cid:96) = 1, . . . , (cid:96)0), we have

sup

xi∈S(si, cr(cid:96))

|v([a(cid:96), a(cid:96)+1), xi) − v([a(cid:96), a(cid:96)+1), si)| ≤ K1

r(cid:96)
(log log 1
r(cid:96)

)1/Q

sup
1≤i≤m

sup

xi∈S(si, cr(cid:96))

|v([a(cid:96), a(cid:96)+1, xi) − v([a(cid:96), a(cid:96)+1), si)|

≤ K1

r(cid:96)
(log log 1
r(cid:96)

)1/Q

(cid:19) .

27

By (3.12), we see that the last expression is greater than or equal to

(cid:40)

(cid:96)0(cid:89)

(cid:96)=1

1 −

(cid:18)

(cid:19)−1/4(cid:41)

1 −

log

1
r(cid:96)

1 −
(cid:32)
−(cid:96)0

log

1
r2
0

(cid:32)

log

(cid:33)−1/4(cid:96)0
(cid:33)−1/4 .

1
r2
0

≥ 1 −

≥ 1 − exp

(3.13)

Set

k(cid:88)

j=1

α−1
j −1

(cid:96)

a

A(cid:96) =

α−1

j

r
(cid:96)

+ a−1
(cid:96)+1.

Notice that r(cid:96)a(cid:96) = U−1 and r(cid:96)a(cid:96)+1 = U . Then
k(cid:88)

k(cid:88)

α−1
j −1

+ (a(cid:96)+1r(cid:96))−1 =

A(cid:96)r−1

(cid:96) =

(a(cid:96)r(cid:96))

j=1

j=1

−(α−1

j −1)

U

+ U−1 ≤ (k + 1)U−β,

(3.14)

with β = min{1, minj=1,...,k(α−1
large enough, A(cid:96) ≤ A0r(cid:96), and for u ≥ ˜Kr(cid:96)U−β√

j − 1)} > 0 since αj < 1 for j = 1, . . . , k. Therefore, for U

log U , (3.7) is satisfied. Hence, by Lemma

3.3.2 and (3.14),

(cid:32)

P

(cid:33)
(cid:12)(cid:12)v(xi) − v(si) − v([a(cid:96), a(cid:96)+1, xi) + v([a(cid:96), a(cid:96)+1, si)(cid:12)(cid:12) ≥ u

sup
1≤i≤m

sup

xi∈S(si, cr(cid:96))

(cid:19)

− u2
˜cA2
(cid:96)

(cid:18)
(cid:18)

≤ exp

≤ exp

−

u2

˜c(k + 1)2r2
(cid:96)

U 2β

(cid:19)

.

Now we take u = K1r(cid:96)(log log 1
r0

(cid:18)

)−1/Q, which is allowed provided

(cid:19)−1/Q ≥ ˜Kr(cid:96)U−β(cid:112)log U .

K1r(cid:96)

log log

1
r0

28

This is equivalent to

U β(log U )−1/2 ≥ ˜K
K1

(cid:18)

log log

1
r0

(cid:19)1/Q

,

which holds if U is large enough. It follows from the above that

(cid:12)(cid:12)v(xi) − v(si) − v([a(cid:96), a(cid:96)+1), xi) + v([a(cid:96), a(cid:96)+1), si)(cid:12)(cid:12) ≥

sup
1≤i≤m

(cid:32)

sup

xi∈S(si, cr(cid:96))
−

U 2β

˜c(k + 1)2(log log 1
r0

.

)2/Q

(cid:33)

(3.15)

(cid:33)

K1r(cid:96)
(log log 1
r0

)1/Q

(3.16)

(cid:32)

P

≤ exp

Let

F(cid:96) =

G(cid:96) =

Then

P

 ,
 .

 sup
 sup

1≤i≤m

1≤i≤m

|v([a(cid:96), a(cid:96)+1), xi) − v([a(cid:96), a(cid:96)+1, si)| ≤ K1
2

r(cid:96)
(log log 1
r(cid:96)

|v(xi) − v(si) − v([a(cid:96), a(cid:96)+1, xi) + v([a(cid:96), a(cid:96)+1, si)|

sup

xi∈S(si, cr(cid:96))

sup

xi∈S(si, cr(cid:96))

)1/Q

≥ K1
2

r(cid:96)
(log log 1
r(cid:96)

)1/Q

|v(xi) − v(si)| ≤ K1

r(cid:96)
(log log 1
r(cid:96)

)1/Q




(cid:19)c
 .

G(cid:96)

G(cid:96)

(3.17)

∃1 ≤ (cid:96) ≤ (cid:96)0,
(cid:32) (cid:96)0(cid:91)
(cid:18) (cid:96)0(cid:91)
 (cid:96)0(cid:91)

≥ P

≥ P

≥ P

(cid:96)=1

(cid:96)=1

sup
1≤i≤m

sup

xi∈S(si, cr(cid:96))

(cid:33)
(cid:18) (cid:96)0(cid:91)
 (cid:96)0(cid:91)

(cid:96)=1

(F(cid:96) ∩ Gc
(cid:96))
(cid:19)
 − P

F(cid:96)

∩

F(cid:96)

(cid:96)=1

(cid:96)=1

29

By (3.13), we have

and by (3.16),

(cid:96)=1

P

P

 (cid:96)0(cid:91)
 (cid:96)0(cid:91)
∃1 ≤ (cid:96) ≤ (cid:96)0,
−(cid:96)0

≥ 1 − exp

(cid:32)

(cid:96)=1

G(cid:96)

sup
1≤i≤m

P

Combining this with (3.17), we get

F(cid:96)

 ≥ 1 − exp
−
 ≤ (cid:96)0 exp

−(cid:96)0

(cid:32)

log

1
r2
0

U 2β

(cid:33)−1/4 ,
 .

)2/Q

˜c(k + 1)2(log log 1
r0

|v(xi) − v(si)| ≤ K1

sup

(cid:33)−1/4 − (cid:96)0 exp

xi∈S(si, cr(cid:96))
1
r2
0

−

log



r(cid:96)
(log log 1
r(cid:96)

)1/Q

U 2β

 .

˜c(k + 1)2(log log 1
r0

)2/Q

Therefore, the proof will be completed provided

−(cid:96)0

(cid:32)
(cid:32)

exp

log

(cid:33)−1/4 + (cid:96)0 exp
(cid:19)1/2(cid:33)

1
r2
0

(cid:18)

.

≤ exp

−

log

1
r0

−



U 2β

˜c(k + 1)2(log log 1
r0

)2/Q

(3.18)

Recall the condition (3.15), and the definition of (cid:96)0 in (3.11). If we set

(cid:18)

(cid:19)1/(2β)

1
r0

U =

log

,

then for r0 small enough, by (3.11),

(cid:18)

(cid:96)0 >

β
2

log

(cid:19)(cid:18)

log log

(cid:19)−1

1
r0

> 1.

1
r0

30

Therefore, the left-hand side of (3.18) is bounded above by

(cid:19)

−



log 1
r0

˜c(k + 1)2(log log 1
r0

)2/Q

1 + log

1
r0

exp

−

exp

(cid:18)

 +
(cid:19)1/2(cid:33)

(log 1
r0

)3/4

(cid:32)

(cid:18)

˜c(k + 1)2 log log 1
r0

≤ exp

−

log

1
r0

provided r0 is small enough. This completes the proof of Proposition 3.3.6.

For each small ρ > 0, by Assumption 3.2.2, there are (ˆt1, . . . , ˆtm) ∈ B1

3ρ ×···× Bm

3ρ such

that for all i = 1, . . . , m and all x, y ∈ Bi
2ρ,

(cid:12)(cid:12)E(cid:0)(v(x) − v(y)) · v(ˆti)(cid:1)(cid:12)(cid:12) ≤ C

k(cid:88)

j=1

The points ˆt1, . . . , ˆtm are fixed.

|xj − yj|δj .

(3.19)

Let Σ2 denote the σ-algebra generated by v(ˆt1), . . . , v(ˆtm). Define

v2(x) = E(cid:0)v(x)|Σ2

(cid:1),

v1(x) = v(x) − v2(x).

(3.20)

The Gaussian random fields v1 = {v1(x), x ∈ T} and v2 = {v2(x), x ∈ T} are independent.

Lemma 3.3.7. There is a constant K2 depending on ˆt1, . . . , ˆtm and the constant C in
Assumption 3.2.2 such that for all i = 1, . . . , m and all x, y ∈ Bi
2ρ,

(cid:12)(cid:12)v2(x) − v2(y)(cid:12)(cid:12) ≤ K2

k(cid:88)

j=1

(cid:12)(cid:12)v(ˆti)(cid:12)(cid:12).

|xj − yj|δj max
1≤i≤m

by v(ˆt1), . . . , v(ˆtm), has dimension m ≥ 2. Let {(cid:80)m

Proof. By Assumption 3.2.3, the subspace in L2(Ω; Rd) of random vectors Ω → Rd spanned
i=1 ai,jv(ˆti) : j = 1, . . . , m} be an

31

orthonormal basis of this subspace, where ai,j are constants that depend on ˆt1, . . . , ˆtm.

Then

m(cid:88)

(cid:20) m(cid:88)

E

j=1

i=1

v2(x) =

ai,jv(ˆti) · v(x)

(cid:19)

a(cid:96),jv(ˆt(cid:96))

.

(cid:21)(cid:18) m(cid:88)

(cid:96)=1

By (3.19), we have

(cid:12)(cid:12)v2(x) − v2(y)(cid:12)(cid:12) =

ai,ja(cid:96),jE(cid:104)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m(cid:88)
(cid:18) m(cid:88)
k(cid:88)

(cid:96)=1

i=1

m(cid:88)

j=1

≤ K

|xj − yj|δj max
1≤(cid:96)≤m

(cid:12)(cid:12)v(ˆt(cid:96))(cid:12)(cid:12).

(v(x) − v(y)) · v(ˆti)

v(ˆt(cid:96))

(cid:105)(cid:19)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

j=1

This completes the proof.

Lemma 3.3.8. Suppose Assumptions 3.2.1, 3.2.2 and 3.2.3 are satisfied. Then there exist
constants K and ρ0 > 0 depending on t1, . . . , tm such that for all ρ ∈ (0, ρ0), a2, . . . , am ∈
Rd, r > 0, and all (x1, . . . , xm) ∈ B1

ρ × ··· × Bm
ρ ,

(cid:32)

(cid:33)

P

|v2(x1) − v2(xi) − ai| ≤ r

sup
2≤i≤m

≤ Kr(m−1)d.

Proof. We first assume d = 1. We claim that if ρ0 is small then v2(x1), . . . , v2(xm) are
linearly independent for all ρ ∈ (0, ρ0) and (x1, . . . , xm) ∈ B1

Assumption 3.2.3, we can find K > 0 such that Var((cid:80)m

Indeed, by
i=1 biv(ti)) ≥ K|b|2 for all b ∈ Rm.

ρ × ··· × Bm
ρ .

32

By the Cauchy–Schwarz inequality, we have

(cid:19)2(cid:35)1/2

(cid:34)
(cid:17)2(cid:21)1/2

≤ |b|

(cid:32) m(cid:88)
(cid:16)
(cid:17)2
(cid:20)
E(cid:16)E(v(ˆti) − v(xi)|Σ2)

v(ti) − v2(xi)

(cid:33)(cid:35)1/2
(cid:17)2(cid:21)1/2(cid:33)

(cid:34)

E

i=1

(cid:18) m(cid:88)
≤ |b| m(cid:88)
≤ |b| m(cid:88)

i=1

i=1

bi(v(ti) − v2(xi))

E

(cid:32)(cid:20)
E(cid:16)
(cid:16)(cid:107)v(ti) − v(ˆti)(cid:107)L2 + (cid:107)v(ˆti) − v(xi)(cid:107)L2

v(ti) − v(ˆti)

i=1

+

(cid:17)

.

It follows that

(cid:34)

(cid:18) m(cid:88)

E

i=1

biv2(xi)

(cid:19)(cid:35)1/2

≥

≥

(cid:34)
(cid:32)

E

(cid:18) m(cid:88)
K1/2 − m(cid:88)

i=1

biv(ti)

(cid:34)

(cid:18) m(cid:88)

(cid:19)2(cid:35)1/2
(cid:16)(cid:107)v(ti) − v(ˆti)(cid:107)L2 + (cid:107)v(ˆti) − v(xi)(cid:107)L2

bi(v(ti) − v2(xi))

−

i=1

E

(cid:19)2(cid:35)1/2
(cid:17)(cid:33)

|b|.

i=1

Notice that, Assumption 3.2.1 implies the L2(P)-continuity of v(x) [cf. Lemma 3.3.1], we
can find a small constant ρ0 > 0 depending on t1, . . . , tm so that the above is ≥ C|b| for all
ρ ∈ (0, ρ0) and (x1, . . . , xm) ∈ B1
ρ , where C > 0. It follows that v2(x1), . . . , v2(xm)
are linearly independent, and so are v2(x1) − v2(x2), v2(x1) − v2(x3), . . . , v2(x1) − v2(xm).

ρ×···×Bm

Denote the determinant of the covariance matrix of the last random vector by

det Cov(v2(y1) − v2(y2), v2(y1) − v2(y3), . . . , v2(y1) − v2(ym)).

Then the map (y1, . . . , ym) (cid:55)→ det Cov(v2(y1)− v2(y2), v2(y1)− v2(y3), . . . , v2(y1)− v2(ym))

is continuous and positive on the compact set B1
ρ0

× ··· × Bm
ρ0

, so it is bounded from below

33

by a positive constant depending on t1, . . . , tm. This and Anderson’s theorem [3] imply that

(cid:33)

(cid:32)

(cid:33)

(cid:32)

P

|v2(x1) − v2(xi) − ai| ≤ r

≤ P

sup
2≤i≤m

|v2(x1) − v2(xi)| ≤ r

sup
2≤i≤m

≤ Krm−1.

Since v(x) has i.i.d. components, the case d > 1 follows readily.

We end this section with the following lemma which is obtained by applying Theorem

2.1 and Remark 2.2 of [28] to the metric space (T, ∆). It provides nested families of “cubes”

sharing most of the good properties of dyadic cubes in the Euclidean spaces. For this reason,

we call the sets in Qq generalized dyadic cubes of order q. Their nested property will help

us to construct an efficient covering for Mρ.

q=1

Qq, Qq = {Iq,(cid:96) : (cid:96) = 1, . . . , nq}, such that the following hold.

Lemma 3.3.9. There exist constants c1, c2, and a family Q of Borel subsets of T , where

Q =(cid:83)∞
(i) T =(cid:83)nq
(ii) Either Iq,(cid:96) ∩ Iq(cid:48),(cid:96)(cid:48) = ∅ or Iq,(cid:96) ⊂ Iq(cid:48),(cid:96)(cid:48) whenever q ≥ q(cid:48), 1 ≤ (cid:96) ≤ nq, 1 ≤ (cid:96)(cid:48) ≤ nq(cid:48).
(iii) For each q, (cid:96), there exists xq,(cid:96) ∈ T such that S(xq,(cid:96), c12−q) ⊂ Iq,(cid:96) ⊂ S(xq,(cid:96), c22−q) and

(cid:96)=1 Iq,(cid:96) for each q ≥ 1.

{xq,(cid:96) : 1, . . . , nq} ⊂ {xq+1,(cid:96) : (cid:96) = 1, . . . , nq+1} for all q ≥ 1.

3.4 Proof of Theorem 3.2.4

Recall that, by (3.6), it suffices to show that for all integers n and all points t1, . . . , tm ∈ T
such that ∆(ti, tj) ≥ 1/n for i (cid:54)= j, we can find a small ρ0 > 0 depending on t1, . . . , tm so
that for all ρ ∈ (0, ρ0), Mρ is empty with probability 1. When mQ < (m − 1)d (we refer

this as the sub-critical case), the last statement can be proved easily by using a standard

34

covering argument based on the uniform modulus of continuity of v = {v(x), x ∈ T} on

compact intervals.

In the following we provide a unified proof for both the critical and

subcritical cases.

Now let t1, . . . , tm ∈ T be m distinct points such that ∆(ti, tj) ≥ 1/n for i (cid:54)= j and some
integer n ≥ 1. They are fixed in the rest of the proof. We choose a constant ρ0 > 0 such
that both Lemma 3.3.8 and Assumption 3.2.2 hold for all ρ ≤ ρ0 (e.g., we take ρ0 ≤ ε0).
Hence we can find (ˆt1, . . . , ˆtm) ∈ B1
assume that there is a compact interval F ⊂ T such that the Bj

3ρ such that (3.19) holds. Furthermore, we
⊂ F for all 1 ≤ j ≤ m.

3ρ × ··· × Bm

3ρ0
Fix ρ ∈ (0, ρ0). For each integer p ≥ 1, consider the random set

(cid:40)

Rp =

(s1, . . . , sm) ∈ B1

2ρ × ··· × Bm

2ρ : ∃ r ∈ [2−2p, 2−p] such that

sup
1≤i≤m

sup

xi∈S(si,4c2r)

|v(xi) − v(si)| ≤ K1r

log log

1
r

(cid:18)

(cid:19)−1/Q(cid:41)

,

where c2 is the constant given by Lemma 3.3.9. Let β = min{β∗, 1}/2, where β∗ =
min{δj/αj − 1 : j = 1, . . . , k}. Let λ denote the Lebesgue measure on Rmk. Consider

the events

Ωp,1 =

λ(Rp) ≥ λ(B1

2ρ × ··· × Bm

2ρ)(1 − exp(−√

(cid:111)

p/4))

,

Ωp,2 =

max
1≤i≤m

|v(ˆti)| ≤ 2βp

.

By applying Proposition 3.3.6 with c = 4c2 and Fubini’s theorem, we derive that for p

sufficiently large,

(cid:17) ≥ 1 − exp(−√

p/2)

(s1, . . . , sm) ∈ Rp

(cid:110)
(cid:26)

P(cid:16)

(cid:27)

35

2ρ. Then by Fubini’s theorem, (cid:80)∞

p=1

P(Ωc

p,1) < ∞.

for all (s1, . . . , sm) ∈ B1

Moreover, it is clear that(cid:80)∞
Denote by Q =(cid:83)∞

p=1

2ρ × ··· × Bm

P(Ωc

p,2) < ∞.

p=1

Qp the family of generalized dyadic cubes given by Lemma 3.3.9

that intersect the compact interval F . Consider the event

(cid:26)
∀ I ∈ Q2p, sup
x,y∈I

Ωp,3 =

|v(x) − v(y)| ≤ K32−2pp1/2

(cid:27)

.

For every I ∈ Q2p, Lemma 3.3.1 implies that the diameter of I under the canonical metric
dv(x, y) = (cid:107)v(x) − v(y)(cid:107)L2 is at most c3 2−2p. By applying Lemma 2.1 in Talagrand [60]
(see also Lemma 3.1 in [18]) we see that for any positive constant K3 and p large,

(cid:18)

P

|v(x) − v(y)| ≥ K32−2pp1/2

sup
x,y∈I

≤ exp

(cid:19)

(cid:18)

−(cid:16) K3

(cid:17)2

c3

(cid:19)

p

.

most K22pQ. We can verify directly that (cid:80)∞

Notice that the cardinality of the family Q2p of generalized dyadic cubes of order 2p is at
p,3) < ∞ provided K3 is chosen to

P(Ωc

p=1

satisfy K3 > 2c3Q ln 2.

Let Ωp = Ωp,1 ∩ Ωp,2 ∩ Ωp,3 and

(cid:91)

(cid:92)

(cid:96)≥1

p≥(cid:96)

Ωp.

Ω∗ =

It follows that the event Ω∗ occurs with probability 1. We will show that, for every ω ∈ Ω∗,
we can construct families of balls in Rd that cover Mρ.

For each p ≥ 1, we first construct a family Gp of subsets in Rmk (depending on ω).
× ··· × Iq,(cid:96)m for some
∈ Qq are the generalized dyadic cubes of order q in Lemma

Denote by Cp the family of subsets of T m of the form C = Iq,(cid:96)1
integer q ∈ [p, 2p], where Iq,(cid:96)i

36

3.3.9.

We say that a dyadic cube C = I1 × ··· × Im of order q is good if it has the property

that

where

sup
1≤i≤m

sup
x,y∈Ii

|v1(x) − v1(y)| ≤ dq,

k(cid:88)

dq = 2(K1 + K2

(2c2)δj /αj )2−q(log log 2q)−1/Q.

j=1

(3.21)

(3.22)

2ρ × ··· × Bm

For each x ∈ B1
2ρ, consider the good dyadic cube C containing x (if any) of
smallest order q, where p ≤ q ≤ 2p. By property (ii) of Lemma 3.3.9, we obtain in this way
a family of disjoint good dyadic cubes of order q ∈ [p, 2p] that meet the set B1
2ρ × ··· × Bm
2ρ.

We denote this family by G 1
p .

Let G 2

p be the family of dyadic cubes in T m of order 2p that meet B1

p . Let Gp = G 1

p ∪ G 2

ρ × ··· × Bm

ρ but
p . Notice that for each C ∈ Cp, the

are not contained in any cube of G 1
events {C ∈ G 1

p } and {C ∈ G 2

p } are in the σ-algebra Σ1 := σ(v1(x) : x ∈ T ).

Now we construct a family Fp of balls in Rd (depending on ω) as follows. For each C ∈
2ρ×···×Bm
2ρ).

Cp, we choose a distinguished (non-random) point xC = (x1

C ) in C∩(B1

C , . . . , xm

If C is a cube of order q, then we define the ball Bp,C as follows.

(i) If C ∈ G 1

p , take Bp,C as the Euclidean ball of center v(x1

C ) of radius rp,C = 4dq. Recall

that dq is defined in (3.22).

(ii) If C ∈ G 2

p , take Bp,C as the Euclidean ball of center v(x1

C ) of radius rp,C = 2K32−2pp1/2.

(iii) Otherwise, take Bp,C = ∅ and rp,C = 0.

Note that for each p ≥ 1, C ∈ Cp, the random variable rp,C is Σ1-measurable. Consider the

37

event

(cid:40)

ω ∈ Ω :

Ωp,C =

sup
2≤i≤m

(cid:12)(cid:12)v(x1

(cid:41)
C , ω)(cid:12)(cid:12) ≤ rp,C (ω)

.

C , ω) − v(xi

If ω ∈ Ω∗ ∩ Ωp,C , define Fp(ω) = {Bp,C : C ∈ Gp(ω)}. Otherwise, define Fp(ω) = ∅.
Choose an integer p0 such that 2c22−p ≤ ρ and

pmQ/2(log p)m exp(−√

p/4) ≤ ρmQ

(3.23)

for all p ≥ p0. We now show that Fp(ω) covers Mρ(ω) whenever p ≥ p0 and ω ∈ Ωp.
Let ω ∈ Ωp and z ∈ Mρ(ω). By definition, we can find a point (y1, . . . , ym) ∈ B1
ρ such that z = v(y1, ω) = ··· = v(ym, ω). By the definitions of G 1
Bm
ρ , thus the point (y1, . . . , ym) is contained in some
Gp(ω) of dyadic cubes covers B1
C = I1 × ··· × Im ∈ Gp(ω). We will show that z ∈ Bp,C and ω ∈ Ωp,C . To this end, we

ρ ×···× Bm

ρ ×···×

p and G 2

p , the family

distinguish two cases.
Case 1. If C ∈ G 1

p (ω), then it is a good dyadic cube of order q ∈ [p, 2p] such that

|v1(xi

C , ω) − v1(yi, ω)| ≤ dq.

sup
1≤i≤m

By Lemma 3.3.9, xi

C , yi ∈ Ii ⊂ S(x∗, c22−q) for some x∗ ∈ T , so we have

(2c2)δj /αj 2−q(1+β∗),

(3.24)

k(cid:88)

j|δj ≤ k(cid:88)

|xi
C,j − yi

j=1

j=1

38

αj

(cid:8) δj
− 1(cid:9). Since ω ∈ Ωp,2, Lemma 3.3.7 and (3.24) imply that
C ) − v2(yi)(cid:12)(cid:12) ≤ K2
(cid:12)(cid:12)v2(xi

(2c2)δj /αj 2−q(1+β∗−β) ≤ dq.

k(cid:88)

(3.25)

j=1

recall that β∗ = min
1≤j≤k

sup
1≤i≤m

It follows that

(cid:12)(cid:12)v(xi
C , ω) − z(cid:12)(cid:12) = sup

1≤i≤m

(cid:12)(cid:12)v(xi
C , ω) − v(yi, ω)(cid:12)(cid:12) ≤ 2dq,

sup
1≤i≤m

which implies that z ∈ Bp,C and ω ∈ Ωp,C .

Case 2. Now we assume C ∈ G 2

p (ω). Since ω ∈ Ωp,3, we have

|v(xi

C , ω) − z| = sup

i

sup

i

|v(xi

C , ω) − v(yi, ω)| ≤ K32−2pp1/2,

hence z ∈ Bp,C and ω ∈ Ωp,C .

Therefore, for every ω ∈ Ω∗, Fp(ω) covers Mρ(ω) when p is large enough. We claim
that, with probability 1, the family Fp is empty for infinitely many p. This will imply that

Mρ is empty with probability 1 and the proof will then be complete.

We prove the aforementioned claim by contradiction. Suppose the claim is not true.
Then the event Ω(cid:48) that Fp is nonempty for all large p has positive probability and the event

Ω(cid:48) ∩ Ω∗ =(cid:83)

(cid:84)

(cid:96)≥1

p≥(cid:96)(Ω(cid:48) ∩ Ωp) also has positive probability. Denote

φ(r) = rmQ−(m−1)d(log log(1/r))m,

f (r) = rmQ(log log(1/r))m,

39

and consider the random variables Xp defined by

(cid:88)

Bp,C∈Fp

(cid:88)

C∈Cp

Xp := 1Ω(cid:48)∩Ωp

φ(rp,C ) = 1Ω(cid:48)∩Ωp

f (rp,C )r

−(m−1)d
p,C

1{C∈Gp}1Ωp,C

.

(3.26)

Let X := lim infp Xp. Since mQ ≤ (m − 1)d, we have φ(r) → ∞ as r → 0+. Moreover,
for every ω ∈ Ω(cid:48) ∩ Ω∗, Fp(ω) is not empty for all large p. This and the definition of Xp in
(3.26) imply that X(ω) = ∞ on Ω(cid:48) ∩ Ω∗. In particular, E(X) = ∞.

On the other hand, notice that G 1

p covers Rp on the event Ωp for all p ≥ p0. Indeed, if
ω ∈ Ωp, s = (s1, . . . , sm) ∈ Rp(ω), and C = I1×···× Im is the dyadic cube of order q in G 1
containing s, then there exists r ∈ [2−2p, 2−p] that satisfies the condition in the definition of
Rp and we can find q such that 2−q−1 < r ≤ 2−q, p ≤ q ≤ 2p, and

p

sup
1≤i≤m

sup

xi∈S(si,2c22−q)

|v(xi) − v(si)| ≤ K12−q(log log 2q)−1/Q.

(3.27)

By the property that Ii ⊂ S(x(cid:48), c22−q) for some x(cid:48) and by Lemma 3.3.7, it follows from
(3.25) and (3.27) that (3.21) holds. Thus C is a good dyadic cube. This proves that G 1

p (ω)

covers Rp(ω).

2ρ × ··· × Bm

2ρ, thus in B1

By the choice of p0, the cubes in G 2

··· × Bm
I1 × ··· × Im ∈ G 2

p are contained in B1
2ρ \ Rp, whose Lebesgue measure is at most exp(−√

2ρ ×
p/4) on Ωp. For any C =
p of order 2p, each Ii contains a set S(xi, c12−2p) for some xi and the set
has Lebesgue measure K2−2pQ, so Ωp is contained in the event (cid:101)Ωp that the cardinality of
p is at most K22pmQ exp(−√
p depend on Σ1. We see that (cid:101)Ωp belongs to the σ-algebra Σ1.
p and G 2

Recall that both G 1

p/4).

G 2

40

Hence for p ≥ p0,

E(Xp) ≤ E

C∈Cp

(cid:18)
(cid:88)
1(cid:101)Ωp
(cid:18)
(cid:88)
1(cid:101)Ωp
(cid:18)
(cid:88)
C∈Cp
1(cid:101)Ωp
≤ KE

= E

C∈Cp

(cid:19)

f (rp,C )r

−(m−1)d
p,C

f (rp,C )r

−(m−1)d
p,C

1{C∈Gp}1Ωp,C

(cid:19)
1{C∈Gp}P(Ωp,C|Σ1)
(cid:19)

f (rp,C )1{C∈Gp}

,

(3.28)

where the last inequality follows from Lemma 3.3.8 and independence of v1 and v2.

Now consider any dyadic cube C ∈ Cp of order q. If C ∈ G 1

p , then f (rp,C ) ≤ K2−qmQ ≤

p , then f (rp,C ) ≤ K2−2pmQpmQ/2(log p)m.

Kλ(C) (where λ(·) denotes Lebesgue measure); if C ∈ G 2
Moreover, for p ≥ p0 the dyadic cubes in G 1
p are disjoint and contained in B1
These observations, together with (3.28) and (3.23), imply that for all p ≥ p0,
(cid:19)

(cid:18) (cid:88)

p } + pmQ/2(log p)m exp(−√

E(Xp) ≤ KE

λ(C)1{C∈G 1

p/4)

≤ KρmQ.

3ρ × ··· × Bm
3ρ.

C∈Cp

By Fatou’s lemma, we derive E(X) ≤ KρmQ < ∞. This is a contradiction. The proof of

Theorem 3.2.4 is complete.

3.5 Examples

In this section we provide some examples where Theorem 3.2.4 is applicable. These in-

clude fractional Brownian sheets, and the solutions to systems of stochastic heat and wave

equations.

41

3.5.1 Fractional Brownian Sheets

The (N, d)-fractional Brownian sheet with Hurst parameter H = (H1, . . . , HN ) ∈ (0, 1)N
is an Rd-valued continuous Gaussian random field {v(x), x ∈ RN
+} with mean zero and

covariance

E(vj(x)v(cid:96)(y)) = δj,(cid:96)

N(cid:89)

i=1

1
2

(cid:16)|xi|2Hi + |yi|2Hi − |xi − yi|2Hi

(cid:17)

.

When N = 1, it is the fractional Brownian motion and the non-existence of multiple points
in the critical dimension was proved by Talagrand [61]. So we focus on the case N ≥ 2.

Let α ∈ (0, 1) be a constant. We start with the identity that any x ∈ R,

(cid:90)

|x|2α = c2

α

1 − cos xξ
|ξ|2α+1 dξ,

where cα =

R

1 − cos ξ
|ξ|2α+1 dξ

R

(cid:18)(cid:90)

(cid:19)−1/2

,

which can be obtained by a change of variable in the integral. It implies that for any x, y ∈ R,

(cid:16)|x|2α + |y|2α − |x − y|2α(cid:17)

1
2

(cid:90)

(cid:20)(1 − cos xξ)(1 − cos yξ)

= c2
α

R

|ξ|2α+1

(cid:21)

+

sin xξ sin yξ

|ξ|2α+1

dξ.

It follows that for H ∈ (0, 1)N and x, y ∈ RN , we can write

(cid:16)|xi|2Hi + |yi|2Hi − |xi − yi|2Hi

(cid:17)

N(cid:89)

i=1

1
2

(cid:88)

(cid:90)

= c2
H

p∈{0,1}N

RN

N(cid:89)

i=1

fpi(xiξi)fpi(yiξi)

|ξi|2Hi+1

dξ,

(3.29)
where f0(t) = 1−cos t and f1(t) = sin t. It gives a representation for the fractional Brownian
sheet: If Wp, p ∈ {0, 1}N , are independent Rd-valued Gaussian white noises on RN and

v(x) := cH

fpi(xiξi)
|ξi|Hi+1/2

Wp(dξ),

(3.30)

(cid:88)

(cid:90)

N(cid:89)

p∈{0,1}N

RN

i=1

42

then (a continuous modification of) {v(x), x ∈ RN

+} is an (N, d)-fractional Brownian sheet

with Hurst index H. In particular, when Hi = 1
2 for i = 1, . . . , k, the Gaussian random field
{v(x), x ∈ RN} is the Brownian sheet and (3.30) provides a harminozable representation for

it.

We take T = (0,∞)N [since v(x) = 0 for all x ∈ ∂RN

+ a.s., the existence of multiple points

is trivial on ∂RN

+ ]. We use the representation (3.30) to show that the fractional Brownian

sheet satisfies the assumptions of Theorem 3.2.4 on T .

Define the random field {v(A, x), A ∈ B(R+), x ∈ T} by

(cid:88)

(cid:90)

v(A, x) = cH

p∈{0,1}N

{maxi |ξi|Hi∈A}

N(cid:89)

i=1

fpi(xiξi)
|ξi|Hi+1/2

Wp(dξ).

Lemma 3.5.1. For any n ≥ 1, let Fn = [1/n, n]N , ε0 = (2n)−1, a0 = 0 and γi = H−1
i − 1.
There is a constant c0 > 0 depending on n such that for all 0 ≤ a < b ≤ ∞ and x, y ∈ Fn,

(cid:13)(cid:13)(v(x) − v([a, b), x)) − (v(y) − v([a, b), y))(cid:13)(cid:13)L2 ≤ c0

(cid:18) N(cid:88)

i=1

(cid:19)

aγi|xi − yi| + b−1

.

(3.31)

Proof. Without loss of generality, we may assume d = 1. For any 0 ≤ a < b ≤ ∞, let
B = {ξ ∈ RN : maxi |ξi|Hi ∈ [a, b)}. Then we can express its complement as

RN \ B =(cid:8)|ξk| < ak,∀1 ≤ k ≤ N(cid:9) ∪ N(cid:91)

(cid:8)|ξk| ≥ bk

(cid:9),

where ai = a1/Hi and bi = b1/Hi.

k=1

43

Note that

− N(cid:89)

N(cid:89)
fpi(xiξi)
|ξi|Hi+1/2
N(cid:88)

i=1

=

(cid:18) fpi(xiξi) − fpi(yiξi)

fpi(yiξi)
|ξi|Hi+1/2
(cid:89)

i=1

|ξi|Hi+1/2

1≤j<i

(cid:89)

i<j≤N

fpj (yjξj)
|ξj|Hj +1/2

fpj (xjξj)
|ξj|Hj +1/2

(cid:19)

.

i=1

It follows that

+ c

(cid:34)(cid:90)
N(cid:88)

k=1

(cid:88)
(cid:88)
(cid:34)(cid:90)
N(cid:88)
N(cid:88)
N(cid:88)

i=1

p∈{0,1}N

(cid:88)
(cid:88)

p

(cid:34)(cid:90)

≤ c

+ c

(cid:107)(v(x) − v([a, b), x)) − (v(y) − v([a, b), y))(cid:107)L2
fpi(xiξi)
≤ c
|ξi|Hi+1/2

{|ξk|<ak,∀k}

p∈{0,1}N

(cid:18) N(cid:89)
(cid:18) N(cid:89)

i=1

(cid:34)(cid:90)

{|ξk|≥bk}

fpi(xiξi)
|ξi|Hi+1/2

i=1

(cid:18) fpi(xiξi) − fpi(yiξi)
(cid:18) fpi(xiξi) − fpi(yiξi)

|ξi|Hi+1/2

|ξi|Hi+1/2

{|ξk|<ak,∀k}

p

k=1

i=1

{|ξk|≥bk}

(cid:19)2

fpi(yiξi)
|ξi|Hi+1/2

fpi(yiξi)
|ξi|Hi+1/2

fpj (yjξj)
|ξj|Hj +1/2

i=1

− N(cid:89)
− N(cid:89)
(cid:89)
(cid:89)

1≤j<i

i=1

fpj (yjξj)
|ξj|Hj +1/2

1≤j<i

dξ

(cid:35)1/2
(cid:35)1/2
(cid:19)2
(cid:89)
(cid:89)

i<j≤N

dξ

i<j≤N

(cid:19)2
(cid:19)2

dξ

2

(cid:35) 1
(cid:35) 1

2

dξ

fpj (xjξj)
|ξj|Hj +1/2

fpj (xjξj)
|ξj|Hj +1/2

1, we see that the above is at most

Using the bounds |fpi(xξ) − fpi(yξ)| ≤ |x − y||ξ| and |fpi(xξ) − fpi(yξ)| ≤ 2 for pi = 0 and
(cid:88)

(cid:18) (cid:89)

(cid:89)

(cid:19)2

|xi − yi|2
|ξi|2Hi−1

1≤j<i

{|ξk|≥bk}

4

|ξi|2Hi+1

fpj (yjξj)
|ξj|Hj +1/2

(cid:18) (cid:89)

fpj (xjξj)
|ξj|Hj +1/2
(cid:89)

i<j≤N

i<j≤N
fpj (yjξj)
|ξi|Hi+1/2

(cid:35)1/2
(cid:19)2

dξ

fpj (xjξj)
|ξj|Hj +1/2

(cid:35)1/2

dξ

.

(cid:34)(cid:90)
N(cid:88)
N(cid:88)
(cid:88)

i=1

c

p

+ c

(cid:34)(cid:90)

{|ξi|<ai}

N(cid:88)

p

k=1

i=1

1≤j<i

44

Then by (3.29) the above is bounded from above by

(cid:88)

p

c

(cid:34)
N(cid:88)
(cid:88)

i=1

+ c

2−2Hi

a

i

N(cid:88)

|xi − yi|2 (cid:89)
(cid:34)
(cid:89)
N(cid:88)

−2Hk

1≤j<i

|yj|2Hj (cid:89)
|yj|2Hj (cid:89)

i<j≤N

b

k

1≤j<i

i<j≤N

p

k=1

i=1

(cid:35)1/2
|xj|2Hj
(cid:35)1/2

|xj|2Hj

.

Since |xj|,|yj| ≤ n + (2n)−1, we obtain (3.31) for some c0 depending on n.

Lemma 3.5.2. For any n ≥ 1, there is ˜c > 0 such that for all x ∈ [1/n, n]N , (cid:107)vj(x)(cid:107)L2 ≥ ˜c
for all j. There is C > 0 such that for all x ∈ [1/n, n]N and y, ¯y with |xi − yi| ≤ 1/2n and
|xi − ¯yi| ≤ 1/2n,

(cid:12)(cid:12)δi

i=1

N(cid:88)

for all j, where δi = min{2Hi, 1}.

(cid:12)(cid:12)E((vj(y) − vj(¯y))vj(x))(cid:12)(cid:12) ≤ C

(cid:12)(cid:12)yi − ¯yi
Proof. The first statement is obvious because (cid:107)vj(x)(cid:107)L2 ≥ ((cid:81)N
(cid:12)(cid:12)(cid:12)(cid:12) N(cid:89)
(|xi|2Hi +|yi|2Hi −|xi− yi|2Hi)− N(cid:89)

statement, it suffices to show that

i=1

i=1

(|xi|2Hi +|¯yi|2Hi −|xi− ¯yi|2Hi)

i=1 |xi|2Hi)1/2. For the second

(cid:12)(cid:12)(cid:12)(cid:12) ≤ K

(cid:12)(cid:12)yi− ¯yi

(cid:12)(cid:12)δi.

N(cid:88)

i=1

For 1 ≤ (cid:96) ≤ N , let A(cid:96) = U(cid:96) − V(cid:96), where

(cid:96)(cid:89)

U(cid:96) =

(cid:0)|xi|2Hi + |yi|2Hi − |xi − yi|2Hi(cid:1),

(cid:0)|xi|2Hi + |¯yi|2Hi − |xi − ¯yi|2Hi(cid:1).
When (cid:96) = 1, we have |A1| ≤(cid:12)(cid:12)|y1|2H1 −|¯y1|2H1(cid:12)(cid:12) +(cid:12)(cid:12)|x1 − y1|2H1 −|x1 − ¯y1|2H1(cid:12)(cid:12). If 2H1 ≤ 1,

V(cid:96) =

i=1

i=1

(cid:96)(cid:89)

then by the triangle inequality, |A1| ≤ 2|y1 − ¯y1|2H1; if 2H1 > 1, then we can use the mean

45

value theorem to get |A1| ≤ K|y1 − ¯y1|. Thus |A1| ≤ K|y1 − ¯y1|δ1. For 2 ≤ (cid:96) ≤ N ,

A(cid:96) = U(cid:96)−1(|x(cid:96)|2H(cid:96) + |y(cid:96)|2H(cid:96) − |x(cid:96) − y(cid:96)|2H(cid:96)) − V(cid:96)−1(|x(cid:96)|2H(cid:96) + |¯y(cid:96)|2H(cid:96) − |x(cid:96) − ¯y(cid:96)|2H(cid:96))

= A(cid:96)−1(|x(cid:96)|2H(cid:96) + |y(cid:96)|2H(cid:96) − |x(cid:96) − y(cid:96)|2H(cid:96))
+ V(cid:96)−1(|y(cid:96)|2H(cid:96) − |¯y(cid:96)|2H(cid:96) + |x(cid:96) − ¯y(cid:96)|2H(cid:96) − |x(cid:96) − y(cid:96)|2H(cid:96)).

Then |A(cid:96)| ≤ K(|A(cid:96)−1+|y(cid:96)−¯y(cid:96)|δ(cid:96)) and by induction we obtain |AN| ≤ K(cid:80)N

(cid:96)=1 |y(cid:96)−¯y(cid:96)|δ(cid:96).

The following lemma verifies Assumption 3.2.3 for fractional Brownian sheets. The sec-

torial local nondeterminism in Theorem 1 of Wu and Xiao [65] provides more information

on the conditional variances among v(x1), . . . , v(xm).

Lemma 3.5.3. If x1, . . . , xm ∈ (0,∞)N are distinct points, then the random variables

v(x1), . . . , v(xm) are linearly independent.

Proof. Suppose that a1, . . . , am are real numbers such that(cid:80)m
N(cid:89)

the representation (3.30) for v(x), we have

(cid:32) m(cid:88)

(cid:88)

(cid:90)

0 = E

(cid:18) m(cid:88)

(cid:19)2
Then for each p ∈ {0, 1}N ,(cid:80)m

a(cid:96)v(x(cid:96))

(cid:96)=1

p∈{0,1}N

RN

a(cid:96)

(cid:96)=1

j=1

(cid:96)=1 a(cid:96)

j=1 fpj (x(cid:96)

jξj) = 0 and, equivalently,

(cid:96)=1 a(cid:96)v(x(cid:96)) = 0 a.s. Recalling

(cid:33)2

dξ.

fpj (x(cid:96)
jξj)
|ξj|Hj +1/2

= c2
H

(cid:81)N
m(cid:88)

a(cid:96)

N(cid:89)

(cid:96)=1

j=1

46

˜fpj (x(cid:96)

jξj) = 0

for all ξ ∈ RN , where ˜f0(t) = 1 − cos t and ˜f1(t) = −i sin t. It follows that

m(cid:88)

a(cid:96)

N(cid:89)

(cid:16)

(cid:96)=1

j=1

(cid:17)

(cid:88)

m(cid:88)

N(cid:89)

p∈{0,1}N

(cid:96)=1

j=1

1 − exp(ix(cid:96)

jξj)

=

a(cid:96)

˜fpj (x(cid:96)

jξj) = 0

(3.32)

for all ξ ∈ RN . We claim that a1 = 0. Let L1,1, . . . , L1,k1
be partitions of {1, . . . , m}
obtained from the equivalence classes of the equivalence relation ∼1 defined by (cid:96) ∼1 k if and
only if x(cid:96)
1 for all
(cid:96) ∈ L1,k, k = 1, . . . , m1. Let ξ2, . . . , ξN ∈ R be arbitrary and define c1,1, c1,2, . . . , c1,m1 by

1. We may assume 1 ∈ L1,1. Let ˆx1

be such that x(cid:96)

1 = xk

1 = ˆxk

1, . . . , ˆx

m1
1

(cid:88)

N(cid:89)

(cid:16)

j=2

c1,k =

a(cid:96)

(cid:96)∈L1,k

(cid:17)

1 − exp(ix(cid:96)

jξj)

.

Then by (3.32), we have, for all ξ1 ∈ R,

c1,1 exp(iˆx1

1ξ1) + ··· + c1,m1 exp(iˆx

1 ξ1) + (c1,1 + ··· + c1,m1) = 0.
m1

m1
1

1, . . . , ˆx

m1
Since ˆx1
1 ξ), 1 are
linearly independent over C, we have c1,1 = ··· = c1,m1 = 0. In particular, we have

are non-zero and distinct, the functions exp(iˆx1

1ξ), . . . , exp(iˆx

(cid:88)

a(cid:96)

(cid:96)∈L1,1

N(cid:89)

(cid:16)

j=2

(cid:17)

1 − exp(ix(cid:96)

jξj)

= 0

for all ξ2, . . . , ξN ∈ R. Next we consider the partitions L2,1, . . . , L2,m2 of {1, . . . , m} obtained
2 (with 1 ∈ L2,1). Then the
from equivalence classes of ∼2 defined by (cid:96) ∼2 k iff x(cid:96)

2 = xk

47

argument above yields

By induction, we obtain

(cid:88)

(cid:96)∈L1,1∩L2,1

a(cid:96)

N(cid:89)

(cid:16)

j=3

(cid:88)

(cid:17)

= 0.

1 − exp(ix(cid:96)

jξj)

(cid:96)∈L1,1∩···∩LN,1

a(cid:96) = 0.

Note that L1,1 ∩ ··· ∩ LN,1 = {1} because x1, . . . , xm are distinct. Hence a1 = 0. Similarly,

we can show that a(cid:96) = 0 for (cid:96) = 2, . . . , m.

Proposition 3.5.4. Let v = {v(x), x ∈ RN

Hurst parameter H ∈ (0, 1)N . If mQ ≤ (m − 1)d where Q = (cid:80)N

+} be an (N, d)-fractional Brownian sheet with

, then v has no

i=1 H−1

i

m-multiple points on (0,∞)N almost surely.

orem 3.2.4 with Q =(cid:80)N

Proof. By the three lemmas above, {v(x), x ∈ [1/n, n]N} satisfies the assumptions of The-
for every n ≥ 1. Hence the result follows immediately from

i=1 H−1

i

the theorem.

We remark that for the case of Brownian sheet i.e. Hi = 1/2 for all i, the above result

provides an alternative proof for the main results in [15, 17] .

3.5.2 System of Stochastic Heat Equations

Let k ≥ 1 and β ∈ (0, k ∧ 2), or k = 1 = β. Consider the Rd-valued random field
{v(t, x), (t, x) ∈ R+ × Rk} defined by

(cid:90)

(cid:90)

v(t, x) =

R

Rk

e−iξ·x e−iτ t − e−t|ξ|2

|ξ|2 − iτ

|ξ|−(k−β)/2 W (dτ, dξ),

48

where W is a Cd-valued space-time Gaussian white noise on R1+k i.e. W = W1 + iW2 and
W1, W2 are independent Rd-valued space-time Gaussian white noises on R1+k. According
to Proposition 7.2 of [18], the process ˆv(t, x) := Re v(t, x), (t, x) ∈ R+ × Rk, has the same

law as the mild solution to the system of stochastic heat equations



ˆv(0, x) = 0,

ˆvj(t, x) = ∆ˆvj(t, x) +

˙ˆWj(t, x),

j = 1, . . . , d,

∂
∂t

(3.33)

where ˆW is an Rd-valued spatially homogeneous Gaussian noise that is white in time with
spatial covariance |x−y|−β if k ≥ 1 and β ∈ (0, k∧2); it is an Rd-valued space-time Gaussian
white noise when k = 1 = β. Note that, in this case, we take T = (0,∞) × Rk.

The H¨older exponents of v(t, x) are α1 = (2 − β)/4 in time and α2 = ··· = α1+k =

(2 − β)/2 in space. See [18, §7] or [14]. In this case, we have Q = (4 + 2k)/(2 − β).

The following lemma can also be found in [52, Lemma A.5.3].

Lemma 3.5.5. Let (t1, x1), . . . , (tm, xm) be distinct points in (0,∞)×Rk. Then the random

variables ˆv1(t1, x1), . . . , ˆv1(tm, xm) are linearly independent.

Proof. Suppose that a1, . . . , am are real numbers such that(cid:80)m

j=1 aj ˆv1(tj, xj) = 0 a.s. Then

0 = E

(cid:18) m(cid:88)
and thus (cid:80)m

j=1

(cid:19)2

(cid:90)

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12) m(cid:88)

j=1

aj ˆv1(tj, xj)

=

R

Rk

aje−iξ·xj

(e−iτ tj − e−tj|ξ|2

)

dτ dξ

(|ξ|4 + τ 2)|ξ|k−β

(cid:12)(cid:12)(cid:12)(cid:12)2

j=1 aje−iξ·xj

(e−iτ tj − e−tj|ξ|2

) = 0 for all τ ∈ R and ξ ∈ Rk. We claim that

aj = 0 for all j = 1, . . . , m. Let ˆt1, . . . , ˆtp be all distinct values of the tj’s. Fix an arbitrary

49

ξ ∈ Rk. Then for all τ ∈ R, we have

p(cid:88)

(cid:18) (cid:88)

aje−iξ·xj(cid:19)

e−iτ ˆt(cid:96) − m(cid:88)

aje−iξ·xj−tj|ξ|2

= 0.

(cid:96)=1

j:tj =ˆt(cid:96)

j=1

Since the functions e−iτ ˆt1

, . . . , e−iτ ˆtp

, 1 are linearly independent over C, it follows that for

all ξ ∈ Rk, for all (cid:96) = 1, . . . , p, (cid:88)

j:tj =ˆt(cid:96)

aje−iξ·xj

= 0.

(3.34)

Since (t1, x1), . . . , (tn, xn) are distinct, the xj’s in the sum in (3.34) are distinct for any fixed
(cid:96). By linear independence of the functions e−iξ·xj

, we conclude that aj = 0 for all j.

The following result solves the existence problem of m-multiple points for (3.33).

Proposition 3.5.6. If m(4 + 2k)/(2− β) ≤ (m− 1)d, then {ˆv(t, x), t ∈ (0,∞), x ∈ Rk} has

no m-multiple points a.s.

Proof. Assumptions 3.2.1 and 3.2.2 are satisfied with Q = (4 + 2k)/(2 − β) by Lemma 7.3

and 7.5 of [18]. Assumption 3.2.3 is also satisfied by Lemma 3.5.5 above. The result follows

from Theorem 3.2.4.

3.5.3 System of Stochastic Wave Equations

Let k ≥ 1 and β ∈ [1, k ∧ 2), or k = 1 = β. Consider the Rd-valued random field
{v(t, x), (t, x) ∈ R+ × Rk} defined by

(cid:90)

(cid:90)

v(t, x) =

R

Rk

F (t, x, τ, ξ)|ξ|−(k−β)/2 W (dτ, dξ),

50

where W is a Cd-valued space-time Gaussian white noise on R1+k and

F (t, x, τ, ξ) =

e−iξ·x−iτ t

2|ξ|

(cid:20)1 − eit(τ +|ξ|)

τ + |ξ|

(cid:21)

.

− 1 − eit(τ−|ξ|)

τ − |ξ|

By Proposition 9.2 of [18], the process ˆv(t, x) = Re v(t, x), (t, x) ∈ R+ × Rk, has the same

law as the mild solution to the system of stochastic wave equations



∂2
∂t2 ˆvj(t, x) = ∆ˆvj(t, x) +

˙ˆWj(t, x),

j = 1, . . . , d,

ˆv(0, x) = 0,

∂
∂t

ˆv(0, x) = 0,

where ˆW is the spatially homogeneous Rd-valued Gaussian noise as in (3.33).

The H¨older exponents of v(t, x) are α1 = α2 = ··· = α1+k = (2 − β)/2 in both time and

space. See [18, §9] or [20]. In this case, we have Q = (2 + 2k)/(2 − β).

Lemma 3.5.7. Let (t1, x1), . . . , (tm, xm) be distinct points in T = (0,∞) × Rk. Then the

random variables ˆv1(t1, x1), . . . , ˆv1(tm, xm) are linearly independent.

Proof. Suppose that a1, . . . , am are real numbers such that(cid:80)m

j=1 aj ˆv1(tj, xj) = 0 a.s. Then

0 = E

aj ˆv1(tj, xj)

(cid:18) m(cid:88)

(cid:19)2
It follows that τ ∈ R and ξ ∈ Rk,(cid:80)m
m(cid:88)

j=1

(cid:90)

(cid:90)

=

R

Rk

(cid:12)(cid:12)(cid:12)(cid:12) m(cid:88)

j=1

(cid:12)(cid:12)(cid:12)(cid:12)2 dτ dξ

|ξ|k−β

.

ajF (tj, xj, τ, ξ)

j=1 ajF (tj, xj, τ, ξ) = 0 and thus

bje−iτ tj

+ c1τ + c2 = 0,

j=1

51

where bj = −2aj|ξ|e−iξ·xj

,

and

c1 = − m(cid:88)
m(cid:88)

j=1

c2 =

j=1

aje−iξ·xj

(eitj|ξ| − e−itj|ξ|)

aj|ξ|e−iξ·xj

(eitj|ξ| + e−itj|ξ|).

We claim that aj = 0 for all j = 1, . . . , m. Let ˆt1, . . . , ˆtp be all distinct values of the tj’s. If
we take arbitrary ξ ∈ Rk and take derivative with respect to τ , we see that

(cid:18)

p(cid:88)

− iˆt(cid:96) (cid:88)

(cid:19)
e−iτ ˆt(cid:96)

bj

+ c1 = 0

(cid:96)=1

j:tj =ˆt(cid:96)

for all τ ∈ R. Since the functions e−iτ ˆt1

, . . . , e−iτ ˆtp

, 1 are linearly independent over C, we

have

−iˆt1 (cid:88)

j:tj =ˆt(cid:96)

bj = 0

for all (cid:96) = 1, . . . , p. It implies that for all ξ ∈ Rk, for all (cid:96) = 1, . . . , p,

(cid:88)

aje−iξ·xj

= 0.

(3.35)

j:tj =ˆt(cid:96)

Since (t1, x1), . . . , (tm, xm) are distinct, the xj’s that appear in the sum in (3.35) are distinct
for any fixed (cid:96). By linear independence of the functions e−iξ·xj

, we conclude that aj = 0 for

all j.

52

Proposition 3.5.8. If m(2 + 2k)/(2− β) ≤ (m− 1)d, then {ˆv(t, x), t ∈ (0,∞), x ∈ Rk} has

no m-multiple points a.s.

Proof. Assumptions 3.2.1 and 3.2.2 are satisfied with Q = (2 + 2k)/(2 − β) by Lemmas 9.3

and 9.6 of [18]. Assumption 3.2.3 is also satisfied by Lemma 3.5.7. Hence the result follows

from Theorem 3.2.4.

53

Chapter 4

Local Times and Level Sets of

Gaussian Random Fields

4.1

Introduction

The purpose of this chapter is to study the local times and level sets of anisotropic Gaussian

random fields satisfying strong local nondeterminism with respect to an anisotropic metric.

We will prove joint continuity for the local times in Section 4.2 and H¨older condition in

Section 4.3. Then we discuss the Hausdorff dimension and Hausdorff measure of the level

sets in Section 4.4. As an example, we apply these results to the stochastic heat equation in

Section 4.5.

Let Y = {Y (t) : t ∈ RN} be a real-valued centered Gaussian random field. Let us

consider the (N, d)-Gaussian random field X = {X(t) : t ∈ RN} defined by

X(t) = (X1(t), . . . , Xd(t)),

where X1, . . . , Xd are i.i.d. copies of Y . We will study the regularities of the local times of
X and the Hausdorff measure of the level sets {t ∈ RN : X(t) = x}.

Consider a fixed closed bounded cube T ⊂ RN . Suppose there is a constant vector

54

H = (H1, . . . , HN ) ∈ (0, 1)N (not depending on T ) and two positive finite constants C1 and

C2 such that

C1ρ(t, s)2 ≤ E[(Y (t) − Y (s))2] ≤ C2ρ(t, s)2

(4.1)

for all t, s ∈ T , where ρ is the metric defined by

N(cid:88)

ρ(t, s) =

|tj − sj|Hj .

j=1

Suppose that Y satisfies strong local nondeterminism in the following sense: there is a

positive finite constant C3 such that for all integers n ≥ 1, for all t, t1, . . . , tn ∈ T ,

Var(Y (t)|Y (t1), . . . , Y (tn)) ≥ C3 min
0≤k≤n

ρ(t, tk)2,

(4.2)

where t0 = 0.

The property of local nondeterminism (LND) is useful for investigating sample paths of

Gaussian random fields. This terminology was first introduced by Berman [6] for Gaussian

processes and extended by Pitt [51] for Gaussian random fields to study their local times.

Later, the property of strong local nondeterminism was developed to study exact regularity

of local times, small ball probability and other sample paths properties for Gaussian random

fields (see, e.g., [68, 69]).

For example, the multiparameter fractional Brownian motion satisfies strong local non-
determinism with ρ(t, s) = |t − s|H (see Pitt [51]). Sufficient conditions in terms of spectral

measures for Gaussian random fields with stationary increments to satisfy strong LND can

be found in [70, 37]. In Section 4.5, we will show that the stochastic heat equation satisfies

strong local nondeterminism.

55

The H¨older conditions for the local times and the Hausdorff measure of the level sets of

strongly locally nondeterministic Gaussian random fields with stationary increments were

studied by Xiao [67]. The case of anisotropic Gaussian random fields satisfying a weaker

form of LND called sectorial local nondeterminism was considered by Wu and Xiao [66].

An example of Gaussian random field that satisfies sectorial local nondeterminism is the

fractional Brownian sheet. The Gaussian random field X that we consider here satisfies

strong local nondeterminism (4.2) with respect to an anisotropic metric, but it does not

necessarily have stationary increments.

Let S ⊂ RN be a Borel set. We say that an Rd-valued random field X = {X(t) : t ∈ RN}
has a local time on S if the occupation measure µS(A) = λN{t ∈ S : X(t) ∈ A}, A ∈ B(Rd),
is absolutely continuous with respect to the Lebesgue measure λd on Rd. In this case, the

local time is defined as (a version of) the Radon–Nikodym derivative

L(x, S) =

dµS
dλd

(x),

x ∈ Rd.

Note that if X has local time on S, then it also has local time on any Borel set B ⊆ S.

By Theorem 6.4 of [23], the local time satisfies the following occupation density formula:

for any Borel set B ⊂ S and any nonnegative measurable function f : Rd → R,

(cid:90)

f (X(t)) dt =

f (x)L(x, B) dx.

(4.3)

By Theorem 8.1 of [70], if condition (4.1) holds on T and if d <(cid:80)N

B

j=1 H−1

j

, then X has

(cid:90)

Rd

56

a local time L(·, S) ∈ L2(Rd) on any Borel set S ⊆ T , with the representation

du e−i(cid:104)u,x(cid:105)(cid:90)

(cid:90)

Rd

S

dt ei(cid:104)u,X(t)(cid:105)

(4.4)

L(x, S) = (2π)−d

for almost every x ∈ Rd.

4.2 Joint Continuity of Local Times

Let T =(cid:81)N

j=1[τj, τj + hj] ⊂ RN be a closed bounded cube, where hj > 0 for j = 1, . . . , N .
Suppose X has a local time L(x,·) on T . We say that the local time is jointly continuous

on T if we can find a version of the process

(cid:40)

(cid:16)

N(cid:89)

(cid:17)

: x ∈ Rd, s ∈ N(cid:89)

[0, hj]

(cid:41)

L

x,

[τj, τj + sj]

j=1

j=1

such that with probability 1, the sample function

(cid:16)

N(cid:89)

(cid:17)

(x, s) (cid:55)→ L

x,

[τj, τj + sj]

is continuous in all variables on the domain Rd ×(cid:81)N

j=1

j=1[0, hj]. If the local time is jointly
continuous on T , then for each x, L(x,·) is a well-defined measure on the Borel sets in T ,
supported on the level set X−1(x) ∩ T = {t ∈ T : X(t) = x} (see [1], Theorem 8.6.1).

The goal of this section is to prove the following:

Theorem 4.2.1. Suppose (4.1) and (4.2) hold on the closed bounded cube T ⊂ RN . Suppose

d < Q, where Q =(cid:80)N

j=1 H−1

j

. Then the Gaussian random field X has a jointly continuous

local time on T almost surely.

57

The key of the proof is to derive moment bounds for the increments of the local time.

We follow Lemma 2.5 of Xiao [67].

For a ∈ RN and r > 0, let Bρ(a, r) = {t ∈ RN : ρ(t, a) ≤ r} denote the anisotropic ball

at a of radius r under the metric ρ.

Q =(cid:80)N

j=1 H−1

j

Lemma 4.2.2. Let T be a closed bounded cube in RN . Suppose 0 < d ≤ β ≤ β0 < Q, where

. Then there exists a positive finite constant C depending on N, d, H and β0

only such that for all subset S of T , for all integers j ≥ 1, for all t1, . . . , tj ∈ T , we have

(cid:90)

(cid:20)

(cid:21)−β

min

0≤k≤j−1

S

ρ(t, tk)

dt ≤ Cjβ/QλN (S)1−β/Q.

(4.5)

In particular, for all a ∈ RN , 0 < r < 1, with D := Bρ(a, r) ⊆ T , for all integers j ≥ 1, for
all t1, . . . , tj ∈ T , we have

(cid:90)

(cid:20)

min

0≤k≤j−1

D

ρ(t, tk)

(cid:21)−β

dt ≤ Cjβ/QrQ−β.

(4.6)

Proof. Let I denote the integral in (4.6). For l = 0, . . . , j − 1, define

(cid:26)

Γl =

t ∈ S : ρ(t, tl) = min

ρ(t, tk)

.

0≤k≤j−1

(cid:27)

Then S =(cid:83)j−1

l=0 Γl and

(cid:90)
(cid:90)

Γl

j−1(cid:88)
j−1(cid:88)

l=0

l=0

I =

=

ρ(t, tl)−βdt

(cid:18) N(cid:88)

Γl

m=1

|tm − tl

m|Hm

58

(cid:19)−β

dt.

Fix l ∈ {0, 1, . . . , j − 1}. Let us consider a change of variables on Γl:

t1 = tl

1 + h1/H1[cos(θ1)]2/H1,

2 + h1/H2[sin(θ1) cos(θ2)]2/H2,

t2 = tl
...

tN−1 = tl

N−1 + h1/HN−1[sin(θ1) . . . sin(θN−2) cos(θN−1)]2/HN−1,

tN = tl

N + h1/HN [sin(θ1) . . . sin(θN2

) sin(θN−1)]2/HN ,

for θ = (θ1, . . . , θN−1) ∈ A := [0, 2π]× [0, π]N−2 and h ∈ [0, hl(θ)], where [x]p := sgn(x)|x|p.

We may write the integral

(cid:90)

(cid:18) N(cid:88)

Γl

m=1

(cid:19)−β

dt

|tm − tl

m|Hm

into a sum of 2N terms, each of which is an integral over the intersection of Γl with one of

the 2N open quadrants centered at tl. Then the Jacobian exists on each open quadrant and
the absolute value of its determinant is hQ−1ϕ(θ) for some bounded function ϕ. We can use

the change of variables formula for each term and then recombine the terms to get

(cid:90) hl(θ)

0

hl(θ)Q−β ϕ(θ) dθ.

59

I = N−β

dθ ϕ(θ)

hQ−1−βdh

(cid:90)
j−1(cid:88)
j−1(cid:88)

l=0

A

(cid:90)

A

l=0

N−β
Q − β

=

where CN :=(cid:82)

Then by Jensen’s inequality and (4.7),

A ϕ(θ)dθ. Since 0 < β < Q, the function x (cid:55)→ x1−β/Q is concave on [0,∞).

(4.7)

Note that the Lebesgue measure of Γl is

(cid:90)

(cid:90) hl(θ)
hl(θ)Q C−1

0

dθ ϕ(θ)

(cid:90)

A

A
CN
Q

hQ−1dh

N ϕ(θ) dθ,

λ(Γl) =

=

A

l=0

(cid:16)

(cid:90)
j−1(cid:88)
(cid:18)(cid:90)
j−1(cid:88)
(cid:18) Q
j−1(cid:88)
(cid:18) Q

l=0

l=0

A

CN

∨ 1

CN

I ≤ CN N−d
Q − β0
≤ CN N−d
Q − β0
CN N−d
Q − β0
≤ CN N−d
Q − β0

=

hl(θ)Q(cid:17)1−β/Q

N ϕ(θ) dθ

hl(θ)QC−1
(cid:19)1−β/Q
(cid:19)1−d/Q j−1(cid:88)

λ(Γl)

C−1
(cid:19)1−β/Q
N ϕ(θ) dθ

λ(Γl)1−β/Q.

l=0

Then by Jensen’s inequality again,

I ≤ C j

≤ C j

(cid:18) 1
(cid:18) 1

j

j

j−1(cid:88)
j−1(cid:88)

l=0

l=0

(cid:19)
λ(Γl)1−β/Q
(cid:19)1−β/Q

λ(Γl)

= C jβ/Qλ(S)1−β/Q,

where C depends on N, d, Q and β0. Hence we obtain (4.5). This implies (4.6) immediately,
since λN (Bρ(a, r)) ≤ CrQ.

60

The following proposition gives a moment estimate for the local time on anisotropic balls.

Proposition 4.2.3. Suppose (4.1) and (4.2) hold on the closed bounded cube T ⊂ RN .

Suppose d < Q, where Q =(cid:80)N

j=1 H−1

j

. Then there exists a positive finite constant C such

that for all subset S of T , for all x ∈ Rd and all integers n ≥ 1, we have

E[L(x, S)n] ≤ Cn(n!)d/QλN (S)n(1−d/Q).

In particular, for all a ∈ RN , r ∈ (0, 1) with D := Bρ(a, r) ⊆ T , for all x ∈ Rd and all
integers n ≥ 1, we have

E[L(x, D)n] ≤ Cn(n!)d/Qrn(Q−d).

Proof. By (4.4), we have

E[L(x, S)n] = (2π)−nd

(cid:90)

Rnd

(cid:90)

Sn

d¯u

−i(cid:80)n

d¯t e

j=1(cid:104)uj ,x(cid:105) E

(cid:20)

j=1(cid:104)uj ,X(tj )(cid:105)(cid:21)
i(cid:80)n

e

,

where ¯u = (u1, . . . , un) and ¯t = (t1, . . . , tn). Since X1, . . . , Xd are i.i.d. copies of Y , we have

(cid:90)

(cid:90)

d(cid:89)
(cid:104)

k=1

d¯t

Sn

(cid:90)

Sn

2 Var((cid:80)n

− 1

d¯uk e

Rn

j
j=1 u
k

Y (tj ))

(cid:105)−d/2

det Cov(Y (t1), . . . , Y (tn))

d¯t

E[L(x, S)n] ≤ (2π)−nd

= (2π)−nd/2

where ¯uk = (u1

k, . . . , un

k ). Since

n(cid:89)

j=2

Var(Y (tj)|Y (t1), . . . , Y (tj−1)),

det Cov(Y (t1), . . . , Y (tn)) = Var(Y (t1))

61

it follows from assumption (4.2) that

E[L(x, S)n] ≤ C(2π)−nd/2

(cid:90)

(cid:20)

n(cid:89)

Sn

j=1

min

0≤k≤j−1

(cid:21)−d

ρ(tj, tk)

d¯t.

(4.8)

If we integrate (4.8) in the order of dtn, dtn−1, . . . , dt1, and apply Lemma 4.2.2 (with β = d)

repeatedly, we deduce that

E[L(x, S)n] ≤ Cn(n!)d/QλN (S)n(1−d/Q).

This yields the first statement of the proposition. The last statement follows immediately
since λN (D) ≤ CrQ.

Next, we would like to extend the moment estimate in the above proposition to moment

estimates for the increments of the local time. To this end, we need some lemmas. The

following lemma is taken from [10, Lemma 2].

Lemma 4.2.4. Let Y1, . . . , Yn be mean zero Gaussian random variables that are linearly

independent and assume that(cid:82)R g(v)e−εv2
(cid:90)

(cid:19)(cid:21)

(cid:18) n(cid:88)

(cid:20)

g(v1) exp

Rn

−1
2

Var

l=1

dv < ∞ for all ε > 0. Then

(cid:90)

vlYl

dv1 . . . dvn =

det Cov(Y1, . . . , Yn)1/2

R

(2π)(n−1)/2

g(v/σ1)e−v2/2dv

where σ1 = Var(Y1|Y2, . . . , Yn).

Let us recall the following version of Besicovitch’s covering theorem for cubes in RN . See

[26], Theorem 1.1.

Lemma 4.2.5. There exists a positive integer M = M (N ) depending only on N with the
following property. For any bounded subset A of RN and any family B = {Q(x) : x ∈ A}

62

of closed cubes such that Q(x) is centered at x for every x ∈ A, there exists a sequence {Qi}
in B such that:

(i) A ⊂(cid:83)

i Qi;

(ii) the cubes of {Qi} can be distributed in M families of disjoint cubes.

We will use the covering theorem to prove Lemma 4.2.6 below. Before that, let us

introduce the notation:

Note that

˜ρ(t, s) = max
1≤j≤N

|tj − sj|Hj .

˜ρ(t, s) ≤ ρ(t, s) ≤ N ˜ρ(t, s)

(4.9)

for all t, s ∈ RN . Then under the assumption (4.2), for any closed bounded cube T in RN ,
there exists a positive finite constant C3 (depending on T ) such that for all integers n ≥ 1
and all t, t1, . . . , tn ∈ T ,

Var(Y (t)|Y (t1), . . . , Y (tn)) ≥ C3 min
0≤l≤n

˜ρ(t, tl)2.

(4.10)

Lemma 4.2.6. There exists a positive integer K = K(N ) depending only on N such that
for any integer n ≥ 1, for any distinct points s0, s1, . . . , sn ∈ RN , the cardinality of the set
of all j ∈ {1, . . . , n} such that

˜ρ(sj, s0) = min{˜ρ(sj, si) : 0 ≤ i ≤ n, i (cid:54)= j}

(4.11)

is at most K.

Proof. Without loss of generality, we may assume that (4.11) is satisfied for j = 1, . . . , k.

63

Note that for s ∈ RN and r > 0, the ball B˜ρ(s, r) := {t ∈ RN : ˜ρ(t, s) ≤ r} under the metric
˜ρ is the closed cube centered at s with side lengths 2r1/H1, . . . , 2r1/HN . We will use Lemma
4.2.5 to show that k ≤ K for some positive integer K = K(N ) that depends on N only.

To this end, let

δ0 = min

(cid:40)

(cid:41)

.

˜ρ(si, s0)
˜ρ(sj, s0)

: i, j ∈ {1, . . . , k}

Note that 0 < δ0 ≤ 1. Take a small 0 < ε0 < 1 such that (1− ε0)1/Hp(1 + δ
) ≥ 1 for all
p ∈ {1, . . . , N}. Let ε = ε0 min{˜ρ(si, s0) : 1 ≤ i ≤ k}. Let A = {s1, . . . , sk} and consider the
family B = {B˜ρ(s1, r1), . . . , B˜ρ(sk, rk)} of closed cubes, where ri = ˜ρ(si, s0)− ε. By Lemma
4.2.5, we can find B1, . . . , BM ⊂ B, with Bi = {B˜ρ(si,1, ri,1), . . . , B˜ρ(si,J(i), ri,J(i))} such

1/Hp
0

that

A = {s1, . . . , sk} ⊂ M(cid:91)

J(i)(cid:91)

i=1

j=1

B˜ρ(si,j, ri,j)

and for each i, the cubes of Bi are pairwise disjoint, where M = M (N ) is a positive integer
which depends on N only. For each 1 ≤ j ≤ k, by the assumption (4.11), if (cid:96) (cid:54)= j, then
˜ρ(sj, s(cid:96)) ≥ ˜ρ(sj, s0) > rj. In other words, for each 1 ≤ j ≤ k, the cube B˜ρ(sj, rj) does not
contain any other s(cid:96), where (cid:96) (cid:54)= j. It means that we need at least k cubes to cover the set
A, and hence k ≤ J(1) + ··· + J(M ).

Let us fix i and estimate the cardinality J(i) of the family Bi. Let us consider the family
B∗
i = {B˜ρ(si,1, r∗
are pairwise disjoint, for any pair si,(cid:96) (cid:54)= si,j we can find p ∈ {1, . . . , N} such that

i,j = ˜ρ(si,j, s0). Since the cubes of Bi

i,1), . . . , B˜ρ(si,J(i), r∗

i,J(i))}, where r∗

|si,(cid:96)
p − si,j

p | > r

1/Hp
i,(cid:96) + r

1/Hp
i,j

.

64

Then by the definition of ε, ε0 and δ0,

|si,(cid:96)
p − si,j

p | > (˜ρ(si,(cid:96), s0) − ε)1/Hp + (˜ρ(si,j, s0) − ε)1/Hp

≥ (1 − ε0)1/Hp(cid:16)

˜ρ(si,(cid:96), s0)1/Hp + ˜ρ(si,j, s0)1/Hp(cid:17)

1/Hp
0

)˜ρ(si,j, s0)1/Hp

= (1 − ε0)1/Hp(1 + δ
≥ ˜ρ(si,j, s0)1/Hp
= (r∗

i,j)1/Hp.

i,j, which means that the cube B(si,j, r∗

It follows that ˜ρ(si,(cid:96), si,j) > r∗
i,j) does not contain any
other si,(cid:96) where (cid:96) (cid:54)= j. On the other hand, every cube of B∗
i contains s0, so these cubes are
not pairwise disjoint. Then another application of Lemma 4.2.5 to the set {si,1, . . . , si,J(i)}
and the family B∗
implies that J(i) ≤ M . Therefore, we have k ≤ M 2 and we may take

i

K = M 2.

We use the previous lemma to prove the lemma below, which provide a correction to the

estimate (2.20) in Xiao [67].

Lemma 4.2.7. Let 0 < γ < 1. Let t0 = 0 and t1, . . . , tn be distinct points in {t ∈ RN : |t| ≤
R}\{0}, where R > 0. Then there exist a positive integer K = K(N ) depending on N only,
a positive finite constant C depending only on R, H, N , and a permutation π of {0, 1, . . . , n}

with π(0) = 0 such that

n(cid:89)

j=1

min{˜ρ(tj, ti)γ : 0 ≤ i ≤ n, i (cid:54)= j} ≤ Cn

1

n(cid:89)

j=1

1

˜ρ(tπ(j), tπ(j−1))2Kγ

.

Proof. Since (minj c

−Hj )˜ρ(t, s) ≤ ˜ρ(c−1t, c−1s) ≤ (maxj c

−Hj )˜ρ(t, s), it suffices to prove the

65

result for R = 1/2. In this case, ˜ρ(ti, tj) ≤ 1 for all i, j. Let π(0) = 0. Define π(1), . . . , π(n)

inductively such that

and for j ≥ 2,

˜ρ(tπ(1), 0) = min{˜ρ(ti, 0) : i = 1, . . . , n}

˜ρ(tπ(j), tπ(j−1)) = min

(cid:110)

˜ρ(ti, tπ(j−1)) : i ∈ {1, . . . , n} \ {π(1), . . . , π(j − 1)}(cid:111)

.

Then π is a permutation of {0, 1, . . . , n} with π(0) = 0 and we have

n(cid:89)

n(cid:89)

min{˜ρ(tj, ti)

γ

: 0 ≤ i ≤ n, i (cid:54)= j} =

min{˜ρ(tπ(j), ti)

γ

: 0 ≤ i ≤ n, i (cid:54)= π(j)}.

j=1

j=1

For 1 ≤ j ≤ n and 0 ≤ m ≤ n with m (cid:54)= π(j), let us define

1 if ˜ρ(tπ(j), tm) = min{˜ρ(tπ(j), ti) : 0 ≤ i ≤ n, i (cid:54)= π(j)},

0 otherwise.

Ij,m =

For each j, by the definition of π, ˜ρ(tπ(j), tm) ≥ ˜ρ(tπ(j), tπ(j+1)) for all m such that π−1(m) >

j + 1. This implies

min{˜ρ(tπ(j), ti)

γ

: 0 ≤ i ≤ n, i (cid:54)= π(j)} ≥ (cid:89)

0≤m≤n,
m(cid:54)=π(j),

π−1(m)≤j+1

˜ρ(tπ(j), tm)Ij,mγ.

Indeed, if there is a unique m such that ˜ρ(tπ(j), tm) = min{˜ρ(tπ(j), ti) : 0 ≤ i ≤ n, i (cid:54)= π(j)},

then the equality holds; if there are more than one m such that the minimum is attained

(i.e. Ij,m = 1 for more than one m), then we get the above inequality by the condition that

66

˜ρ(ti, tj) ≤ 1 for all i, j. It follows that

n(cid:89)

j=1

n(cid:89)

j=1

min{˜ρ(tj, ti)

γ

˜ρ(tπ(j), tm)Ij,mγ

: 0 ≤ i ≤ n, i (cid:54)= j} ≥ (cid:89)
(cid:89)

1≤j≤n

(cid:89)
(cid:89)

0≤m≤n,
m(cid:54)=π(j),

π−1(m)≤j+1

=

0≤m≤n

1≤j≤n,
π(j)(cid:54)=m,

π−1(m)≤j+1

˜ρ(tπ(j), tm)Ij,mγ.

Putting m = π((cid:96)) with 0 ≤ (cid:96) ≤ n, we have

: 0 ≤ i ≤ n, i (cid:54)= j} ≥ (cid:89)

min{˜ρ(tj, ti)

γ

(cid:89)

1≤j≤n,
j(cid:54)=(cid:96), j+1≥(cid:96)

˜ρ(tπ(j), tπ((cid:96)))

Ij,π((cid:96))γ

.

0≤(cid:96)≤n

By the definition of π, for 0 ≤ (cid:96) ≤ n and 1 ≤ j ≤ n with j (cid:54)= (cid:96), j ≥ (cid:96) − 1, we have

˜ρ(tπ(j), tπ((cid:96))) ≥ min(cid:8)˜ρ(tπ((cid:96)−1), tπ((cid:96))), ˜ρ(tπ((cid:96)+1), tπ((cid:96)))(cid:9),

with the notation π(−1) := π(1) and π(n + 1) := π(n − 1). Then

j=1

n(cid:89)
≥ (cid:89)
(cid:89)

=

0≤(cid:96)≤n

0≤(cid:96)≤n

min{˜ρ(tj, ti)

γ

: 0 ≤ i ≤ n, i (cid:54)= j}

(cid:16)

min(cid:8)˜ρ(tπ((cid:96)−1), tπ((cid:96))), ˜ρ(tπ((cid:96)+1), tπ((cid:96)))(cid:9)(cid:17)Ij,π((cid:96))γ

(cid:89)
min(cid:8)˜ρ(tπ((cid:96)−1), tπ((cid:96))), ˜ρ(tπ((cid:96)+1), tπ((cid:96)))(cid:9)(cid:17)K(cid:96)γ

1≤j≤n,
j(cid:54)=(cid:96), j+1≥(cid:96)

(cid:16)

,

67

where

(cid:88)

K(cid:96) =

Ij,π((cid:96)).

1≤j≤n,
j(cid:54)=(cid:96), j+1≥(cid:96)

For each fixed (cid:96), we have(cid:80)
min(cid:8)˜ρ(tπ((cid:96)−1), tπ((cid:96))), ˜ρ(tπ((cid:96)+1), tπ((cid:96)))(cid:9)(cid:17)K(cid:96)γ ≥ ˜ρ(tπ((cid:96)−1), tπ((cid:96)))Kγ ˜ρ(tπ((cid:96)+1), tπ((cid:96)))Kγ.
(cid:16)

j: j(cid:54)=(cid:96) Ij,π((cid:96)) ≤ K(N ) = K by Lemma 4.2.6. Hence K(cid:96) ≤ K and

Therefore, we get that

n(cid:89)

min(cid:8)˜ρ(tj, ti)

γ

j=1

This proves the lemma.

: 0 ≤ i ≤ n, i (cid:54)= j(cid:9) ≥ (cid:89)

1≤(cid:96)≤n

˜ρ(tπ((cid:96)), tπ((cid:96)−1))2Kγ.

Now, we prove a moment estimate for the increments of the local time.

Proposition 4.2.8. Suppose (4.1) and (4.2) hold on the closed bounded cube T ⊂ RN .

Suppose d < Q, where Q =(cid:80)N

j=1 H−1

j

. Then there is a positive integer K depending only

on N , and there is a positive finite constant C depending only on T, N, d and H such that
for all small 0 < γ < 1, for all Borel set S ⊆ T , for all x, y ∈ Rd, for all even integers
n ≥ 2, we have

E[(L(x, S) − L(y, S))n] ≤ Cn|x − y|nγ(n!)d/Q+(1+2K/Q)γλN (S)n(1−(d+2Kγ)/Q).

In particular, for all small 0 < γ < 1, for all a ∈ RN and 0 < r < 1 with D := Bρ(a, r) ⊆ T ,
for all x, y ∈ Rd, for all even integers n ≥ 2, we have

E[(L(x, D) − L(y, D))n] ≤ Cn|x − y|nγ(n!)d/Q+(1+2K/Q)γrn(Q−d−2Kγ).

68

Proof. Recall that for any even integer n ≥ 2 and x, y ∈ Rd,

E[(L(x, S) − L(y, S))n]
= (2π)−nd

d¯t

(cid:90)

(cid:90)

Sn

Rnd

e−i(cid:104)uj ,x(cid:105) − e−i(cid:104)uj ,y(cid:105)(cid:21)
(cid:20)

E

(cid:20)

l=1(cid:104)ul,X(tl)(cid:105)(cid:21)
ei(cid:80)n

,

(4.12)

n(cid:89)

j=1

d¯u

where ¯u = (u1, . . . , un) ∈ Rnd and ¯t = (t1, . . . , tn) ∈ Sn. Note that

(cid:20)

l=1(cid:104)ul,X(tl)(cid:105)(cid:21)
ei(cid:80)n

E

d(cid:88)

(cid:20)

−1
2

(cid:18) n(cid:88)

= exp

Var

ul
kY (tl)

k=1

l=1

(cid:19)(cid:21)

(4.13)

for all u1, . . . , un ∈ Rd and t1, . . . , tn ∈ S. For any 0 < γ < 1, we have |eiu − 1| ≤ 21−γ|u|γ
and |u + v|γ ≤ |u|γ + |v|γ for all u, v ∈ R. It follows that

n(cid:89)

j=1

(cid:12)(cid:12)(cid:12)(cid:12)e−i(cid:104)uj ,x(cid:105) − e−i(cid:104)uj ,y(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) ≤ 2(1−γ)n|x − y|nγ(cid:88)

n(cid:89)

¯k

j=1

|γ

|uj
kj

(4.14)

for all u1, . . . , un, x, y ∈ Rd, where the summation is taken over all ¯k = (k1, . . . , kn) ∈
{1, . . . , d}n. Then (4.12), (4.13) and (4.14) imply that

(cid:90)

Sn

E[(L(x, S) − L(y, S))n] ≤ (2π)−nd 2n|x − y|nγ(cid:88)
(cid:18) n(cid:88)

(cid:18) n(cid:89)

d(cid:88)

(cid:90)

(cid:20)

¯k

|γ(cid:19)

J(¯t, ¯k) =

|uj
kj

exp

−1
2

Var

Rnd

j=1

k=1

l=1

(cid:19)(cid:21)

d¯u

kY (tl)
ul

J(¯t, ¯k) d¯t,

(4.15)

where

for ¯t = (t1, . . . , tn) ∈ Sn and ¯k = (k1, . . . , kn) ∈ {1, . . . , d}n. By the generalized H¨older

69

inequality,

J(¯t, ¯k) ≤ n(cid:89)

j=1

(cid:40)(cid:90)

Rnd

(cid:20)

−1
2

d(cid:88)

k=1

(cid:18) n(cid:88)

l=1

Var

|nγ

|uj
kj

exp

(cid:19)(cid:21)

(cid:41)1/n

ul
kY (tl)

d¯u

.

If we fix ¯t, ¯k and j, then by Lemma 4.2.4, we have

(cid:90)

=

(cid:20)

d(cid:88)

k=1

Var

(cid:18) n(cid:88)
(cid:90)

l=1

|uj
kj

Rnd

−1
2

|nγ
exp
(2π)(nd−1)/2

det Cov(Y (t1), . . . , Y (tn))d/2

R

(cid:19)(cid:21)

d¯u

ul
kY (tl)

(cid:12)(cid:12)(cid:12) v

σj

(cid:12)(cid:12)(cid:12)nγ

e−v2/2dv,

where σj = Var(Y (tj)|Y (tl) : 1 ≤ l ≤ n, l (cid:54)= j). By Jensen’s inequality and the moments of

the standard Gaussian,

(cid:90)

|v|nγe−v2/2dv ≤

R

It follows that

√

2π(n!)γ.

(cid:18)(cid:90)

R

√

2π

|v|ne−v2/2dv

1√
2π

Cn(n!)γ

(cid:19)γ ≤
n(cid:89)

det Cov(Y (t1), . . . , Y (tn))d/2

j=1

1
σγ
j

(4.16)

for some C depending on d. By (4.10),

J(¯t, ¯k) ≤

≤ n(cid:89)

n(cid:89)

j=1

1
σγ
j

j=1

C

γ/2

3 min{˜ρ(tj, tl)γ : 0 ≤ l ≤ n, l (cid:54)= j}

1

≤ (C3 ∧ 1)−n/2

n(cid:89)

j=1

min{˜ρ(tj, tl)γ : 0 ≤ l ≤ n, l (cid:54)= j}.

1

70

Then by Lemma 4.2.7 and (4.9),

n(cid:89)

j=1

n(cid:89)

j=1

1

ρ(tπ(j), tπ(j−1))2Kγ

≤ Cn

1
σγ
j

where K is a positive integer that depends only on N , and C is a positive finite constant

that depends on T, N and H. By assumption (4.1),

Var(Y (tπ(j))|Y (tπ(l)) : l = 1, . . . , j − 1) ≤ C2ρ(tπ(j), tπ(j−1))2.

Hence

n(cid:89)

j=1

≤ Cn

1
σγ
j

= Cn

n(cid:89)

j=1

Var(Y (tπ(j))|Y (tπ(l)) : l = 1, . . . , j − 1)Kγ

CKγ

2

(C2 ∨ 1)Kn

det Cov(Y (t1), . . . , Y (tn))Kγ

.

Then (4.16), (4.17) and assumption (4.2) imply that

J(¯t, ¯k) ≤

Cn(n!)γ

(cid:21)−d−2Kγ

det Cov(Y (t1), . . . , Y (tn))d/2+Kγ

≤ Cn(n!)γ

min

0≤l≤j−1

ρ(tj, tl)

(cid:20)

n(cid:89)

j=1

(4.17)

(4.18)

where C depends on T, N, H and d. Since d < Q, we can choose and fix some d < β0 < Q.

If 0 < γ < 1 is chosen small enough such that

d < d + 2Kγ ≤ β0 < Q,

71

then by Lemma 4.2.2 with β = d + 2Kγ, we have

(cid:21)−d−2Kγ

(cid:90)

(cid:20)

n(cid:89)

Sn

j=1

which gives

min

0≤l≤j−1

ρ(tj, tl)

d¯t ≤ Cn(n!)(d+2Kγ)/QλN (S)n(1−(d−2Kγ)/Q),

(cid:90)

Sn

J(¯t, ¯k) d¯t ≤ Cn(n!)d/Q+(1+2K/Q)γλN (S)n(1−(d+2Kγ)/Q).

(4.19)

Note that this bound does not depend on ¯k. Combining (4.15), (4.18) and (4.19), we have

E[(L(x, S) − L(y, S))n] ≤ Cn|x − y|nγ(n!)d/Q+(1+2K/Q)γλN (S)n(1−(d+2Kγ)/Q).

This completes the proof of Proposition 4.2.8.

Now, we turn to the proof of Theorem 4.2.1.

Proof of Theorem 4.2.1. By Proposition 4.2.8 and the multiparameter version of Kolmogorov’s
continuity theorem (see e.g. [29], Theorem 4.3), for any anisotropic ball Bρ(a, r) ⊂ T , X has

a local time L(x, Bρ(a, r)) that is continuous in x.

Next, we prove the joint continuity. Let T =(cid:81)N
[τ, τ + s] =(cid:81)N

j=1[τj, τj + sj] for any s1, . . . , sN ≥ 0. For all x, y ∈ Rd, s, t ∈(cid:81)N

j=1[τj, τj + hj]. For simplicity, denote

j=1[0, hj]

and all even integers n ≥ 2, we have

E[(L(x, [τ, τ + s]) − L(y, [τ, τ + t]))n]

≤ 2n−1(cid:8)E[(L(x, [τ, τ + s]) − L(y, [τ, τ + s]))n] + E[(L(y, [τ, τ + s]) − L(y, [τ, τ + t]))n](cid:9).

72

By Proposition 4.2.8, we can find some small γ > 0 such that the first term is

E[(L(x, [τ, τ + s]) − L(y, [τ, τ + s]))n] ≤ C|x − y|nγ.

For the second term, by considering the symmetric difference of the cubes [τ, τ + s] and
[τ, τ + t], we see that L(y, [τ, τ + s]) − L(y, [τ, τ + t]) can be written as a sum of M terms
of the form ±L(y, Ti), where M is a finite number depending only on N and Ti is a closed
bounded cube in T = [τ, τ + h] with at least one edge length ≤ |s − t|. Then by Proposition

4.2.3, the second term is

E[(L(y, [τ, τ + s]) − L(y, [τ, τ + t]))n] ≤ M n−1

M(cid:88)

i=1

E[L(y, Ti)n]

≤ M nCn(n!)d/QλN (Ti)n(1−d/Q)
≤ C|s − t|n(1−d/Q).

Combining the two terms, we have

E[(L(x, [τ, τ + s]) − L(y, [τ, τ + t]))n] ≤ C(|x − y|nβ + |s − t|nβ)

for some small β > 0. Therefore, by Kolmogorov’s continuity theorem, X has a jointly

continuous local time on T .

4.3 H¨older conditions of Local Times

In the previous section, we derive joint continuity for the local time. In fact, we can also

derive a H¨older condition for the local time.

73

j=1 H−1

j

Suppose d < Q where Q =(cid:80)N

Theorem 4.3.1. Suppose X satisfies (4.1) and (4.2) on a closed bounded cube T ⊂ RN .
. For any x ∈ Rd, let L(x,·) be a (joint continuous)
local time of X, a random measure on T that is supported on the level set X−1(x) ∩ T =
{t ∈ T : X(t) = x}. Then there exists a positive finite constant C such that for any x ∈ Rd,
with probability 1, for L(x,·)-almost every t ∈ T ,

L(x, Bρ(t, r) ∩ T )

ϕ(r)

≤ C,

lim sup
r→0+

(4.20)

where ϕ(r) = rQ−d(log log(1/r))d/Q.

Proof. For any x ∈ Rd and any integer k ≥ 1, consider the random measure Lk(x,·) on

Borel subsets C of T defined by

(cid:90)
(cid:90)

C

(cid:18) k
(cid:90)

2π

(cid:19)d/2

1

Rd

(2π)d

C

(cid:19)

(cid:18)
−k|X(t) − x|2
(cid:18)
−|u|2

2k

exp

exp

dt

2
+ i(cid:104)u, X(t) − x(cid:105)

(cid:19)

(4.21)

du dt.

Lk(x, C) =

=

By the occupation density formula (4.3) and the continuity of y (cid:55)→ L(y, C) for all rectangles
C in T , one can verify that a.s. for all C, Lk(x, C) → L(x, C) as k → ∞. It follows that
a.s., Lk(x,·) converges weakly to L(x,·).

For each m ≥ 1, define fm(t) = L(x, Bρ(t, 2−m)). By Proposition 4.2.3, 4.2.8 and the

multiparameter version of Kolmogorov’s continuity theorem [29], fm(t) is a.s. bounded and
continuous on T . Then by the a.s. weak convergence of Lk(x,·), for all m, n ≥ 1,

(cid:90)

[fm(t)]nL(x, dt) = lim
k→∞

T

T

[fm(t)]nLk(x, dt) a.s.

(cid:90)

74

Hence, by the dominated convergence theorem, (4.21) and (4.4), we have

(cid:90)

E

(2π)d

T
1

1

(cid:90)

(cid:90)

[L(x, Bρ(t, 2−m))]nL(x, dt)
(cid:90)

lim
k→∞

E

T

dt

Rd

du exp

+ i(cid:104)u, X(t) − x(cid:105)(cid:17)
du ei(cid:104)u,X(t)−x(cid:105)E[L(x, B(t, 2−m))n]
(cid:104)x,u(cid:96)(cid:105)E(cid:16)
−i(cid:80)n+1

(cid:16) − |u|2
(cid:90)

2k

(cid:96)=1

(cid:90)
(cid:90)

Rd

(cid:90)

dt

d¯u e

(2π)d
1

T

(2π)(n+1)d

T

Bρ(t,2−m)n

d¯s

R(n+1)d

=

=

=

[L(x, B(t, 2−m))]n

ei(cid:80)n+1

(cid:96)=1

(cid:104)u(cid:96),X(s(cid:96))(cid:105)(cid:17)

,

where ¯u = (u1, . . . , un+1) ∈ R(n+1)d, ¯s = (t, s2, . . . , sn+1) and s1 = t. Similar to the proof

of Proposition 4.2.3, we can deduce that

(cid:90)

E

≤ Cn

[L(x, Bρ(t, 2−m))]nL(x, dt)
(cid:90)

T

d¯s

T×Bρ(t,2−m)n

[det Cov(Y (t), Y (s2), . . . , Y (sn+1))]d/2

(4.22)

≤ Cn(n!)d/Q2−nm(Q−d).

Let A > 0 be a constant to be determined. Consider the random set

Bm = {t ∈ T : L(x, Bρ(t, 2−m)) > Aϕ(2−m)}.

Consider the random measure µ on T defined by µ(B) = L(x, B) for any B ∈ B(T ). Take
n = (cid:98)log m(cid:99), the integer part of log m. Then by (4.22) and Stirling’s formula,

E µ(Bm) ≤ E(cid:82)

T [L(x, Bρ(t, 2−m))]nL(x, dt)

[Aϕ(2−m)]n
≤ Cn(n!)d/Q2−nm(Q−d)
An2−nm(Q−d)(log m)nd/Q

≤ m−2

75

provided A > 0 is chosen large enough. This implies that

∞(cid:88)

E

µ(Bm) < ∞.

By the Borel–Cantelli lemma, with probability 1, for µ-a.e. t ∈ T , we have

m=1

L(x, Bρ(t, 2−m))

ϕ(2−m)

≤ A.

lim sup
m→∞

(4.23)

For any r > 0 small enough, there exists an integer m such that 2−m ≤ r < 2−m+1 and

(4.23) can be applied. Since ϕ(r) is increasing near r = 0, we can use a monotonicity

argument to obtain (4.33).

4.4 Hausdorff Measure of Level Sets

Let us consider the class C of functions ϕ : [0, δ0] → R+ such that ϕ is nondecreasing,

continuous, ϕ(0) = 0, and satisfies the doubling condition, i.e. there exists a positive finite

constant c0 such that

for all s ∈ (0, δ0/2).

ϕ(2s)
ϕ(s)

≤ c0

(4.24)

For any Borel set A in RN , the Hausdorff measure of A with respect to the function

ϕ ∈ C in the metric ρ is defined by

Hϕ
ρ (A) = lim
δ→0+

inf

ϕ(2rn) : A ⊆

(cid:40) ∞(cid:88)

n=1

Bρ(tn, rn) where tn ∈ RN , rn ≤ δ for all n

(cid:41)

.

∞(cid:91)

n=1

When ϕ(s) = sβ, where β is a positive real number, Hϕ

ρ (A) is called the β-dimensional

76

(cid:26) k(cid:88)

j=1

(cid:27)

Hausdorff measure of A in the metric ρ, and the Hausdorff dimension of A in the metric ρ

is defined as

dimρ

H (A) = inf{β > 0 : Hϕ

ρ (A) = 0}.

(cid:80)N
Suppose X satisfies (4.1) and (4.2) on a closed bounded cube T ⊂ RN . Let Q =
j=1 H−1
and consider the level set X−1(x) ∩ T = {t ∈ T : X(t) = x}. By Theorem 7.1
of [70], if d < Q, then X−1(x) ∩ T = ∅ a.s.; if Q > d, then the Hausdorff dimension of
X−1(x) ∩ T in the Euclidean metric is (assuming that 0 < H1 ≤ H2 ≤ ··· ≤ HN < 1)

j

dimH (X−1(x) ∩ T ) = min

+ N − k − Hkd : 1 ≤ k ≤ N

Hk
Hj

(4.25)

which is also equal to

τ(cid:88)

where τ is the unique integer between 1 and N such that(cid:80)τ−1

j=1

j ≤ d <(cid:80)τ

j=1 H−1

j=1 H−1

j

. On

+ N − τ − Hτ d

Hτ
Hj

the other hand, Theorem 4.2 of [66] implies that if d < Q, then the Hausdorff dimension of
X−1(x) ∩ T in the metric ρ is

dimρ

H (X−1(x) ∩ T ) = Q − d.

It would be interesting to determine the exact gauge function for the Hausdorff measure

of the level set, that is, to find a function ϕ such that

0 < Hϕ

ρ (X−1(x) ∩ T ) < ∞ a.s.

The following theorem is a partial result that gives a lower bound for the Hausdorff measure.

77

Theorem 4.4.1. Suppose X satisfies (4.1) and (4.2) on a closed bounded cube T ⊂ RN .

Suppose d < Q where Q =(cid:80)N

j=1 H−1

j

. Then there is a positive finite constant C such that

CL(x, T ) ≤ Hϕ

ρ (X−1(x) ∩ T ) a.s.,

where ϕ(r) = rQ−d(log log(1/r))d/Q.

Proof. Recall that there exists a positive constant c ≥ 1 depending only on c0 in (4.24) such
that for any finite Borel measure µ on RN and any Borel set E ⊂ RN ,

c−1Hϕ

ρ (E) inf
t∈E

D

where

ϕ,ρ

µ (t) ≤ µ(E) ≤ cHϕ

ρ (E) sup
t∈E

D

ϕ,ρ
µ (t)

(4.26)

D

ϕ,ρ
µ (t) := lim sup
r→0+

µ(Bρ(t, r))

ϕ(r)

is called the ρ-upper ϕ-density of µ at the point t (see Theorem 4.1 of [66]). We can take
µ = L(x,· ∩ T ), which is a.s. a finite Borel measure on RN supported on X−1(x) ∩ T . Then

by Theorem 4.3.1, there exists a positive finite constant C such that

ϕ,ρ

µ (t) ≤ C a.s.

D

sup
t∈E

This and the upper bound of (4.26) with E = X−1(x) ∩ T yields the desired result.

78

4.5 Stochastic Heat Equation and Strong Local Non-

determinism

Consider the system of stochastic heat equations



∂

∂tuj(t, x) − ∆uj(t, x) = ˙Wj(t, x),

t ≥ 0, x ∈ RN ,

uj(0, x) = 0,

j = 1, . . . , d,

where ˙Wj, j = 1, . . . , d, are i.i.d. Gaussian noises that are white in time and colored in space

with covariance

E[ ˙Wj(t, x) ˙Wj(s, y)] = δ0(t − s)|x − y|−β

where 0 < β < 2 ∧ N . Let u(t, x) = (u1(t, x), . . . , ud(t, x)). Then {u(t, x) : t ≥ 0, x ∈ RN}

is a (1 + N, d)-Gaussian random field and u1, . . . , ud are i.i.d.

Recall that for any 0 < a < b < ∞, there exist positive finite constants C1, C2 such that

C1ρ((t, x), (s, y)) ≤ E[(u1(t, x) − u1(s, y))2]1/2 ≤ C2ρ((t, x), (s, y))

(4.27)

for all (t, x), (s, y) ∈ [a, b] × [−b, b]N , where ρ((t, x), (s, y)) = |t − s|(2−β)/4 + |x − y|(2−β)/2.

See e.g. Lemma 4.2 of [14]. It shows that u satisfies condition (4.1) on any closed bounded
cube in (0,∞) × RN .

Recall Section 3.4.2. We may assume that u1(t, x) has the following representation:

(cid:90)

(cid:90)

u1(t, x) =

R

RN

e−i(cid:104)ξ,x(cid:105) e−iτ t − e−t|ξ|2

|ξ|2 − iτ

|ξ|−(N−β)/2 ˜W (dτ dξ),

(4.28)

79

where ˜W is a C-valued space-time Gaussian white noise on R1+N . The following proposition

shows that u also satisfies the condition (4.2) of strong local nondeterminism.

Proposition 4.5.1. For any 0 < a < b < ∞, there exists a positive finite constant C such
that for all integers n ≥ 1, for all (t, x), (t1, x1), . . . , (tn, xn) ∈ [a, b] × [−b, b]N ,

Var(u1(t, x)|u1(t1, x1), . . . , u1(tn, xn)) ≥ C min
1≤i≤n

ρ((t, x), (ti, xi))2.

(4.29)

Proof. Since u is Gaussian, the conditional variance in (4.29) is the squared L2-distance of
u1(t, x) from the linear subspace of L2(P) spanned by u1(t1, x1), . . . , u1(tn, xn), that is,

Var(u1(t, x)|u1(t1, x1), . . . , u1(tn, xn)) =

inf

a1,...,an∈R

E

(cid:34)(cid:18)
u1(t, x) − n(cid:88)

j=1

(cid:19)2(cid:35)

aju1(tj, xj)

.

Therefore, it suffices to show that there exists a positive constant C such that

(cid:34)(cid:18)
u1(t, x) − n(cid:88)

E

(cid:19)2(cid:35)

aju1(tj, xj)

≥ Cr2−β,

j=1

for any n ≥ 1, any (t, x), (t1, x1), . . . , (tn, xn) ∈ [a, a(cid:48)] × [−b, b]N , and any a1, . . . , an ∈ R,

where

r = min
1≤j≤n

(|t − tj|1/2 ∨ |x − xj|).

From (4.28), we have

(cid:34)(cid:18)
u1(t, x) − n(cid:88)
(cid:90)
(cid:90)

j=1

dτ

R

dξ

RN

aju1(tj, xj)

(cid:19)2(cid:35)
(cid:12)(cid:12)(cid:12)(cid:12)e−i(cid:104)ξ,x(cid:105)(e−iτ t − e−t|ξ|2

E

≥ C

aje−i(cid:104)ξ,xj(cid:105)(e−iτ tj − e−tj|ξ|2

) − n(cid:88)

j=1

80

(4.30)

(cid:12)(cid:12)(cid:12)(cid:12)2 |ξ|β−N

|ξ|4 + |τ|2 .

)

Let M be a positive finite constant depending on a(cid:48) and b such that |t−t(cid:48)|1/2∨|x−x(cid:48)| ≤ M
for all (t, x), (t(cid:48), x(cid:48)) ∈ [a, a(cid:48)] × [−b, b]N . Let ρ = min{a/M 2, 1}. Let ϕ : R → R and
ψ : RN → R be nonnegative smooth bump functions that vanish outside [−ρ, ρ] and the
unit ball respectively and satisfy ϕ(0) = ψ(0) = 1. Let ϕr(τ ) = r−2ϕ(r−2τ ) and φr(ξ) =
r−N ψ(r−1ξ). Let us consider the integral

(cid:90)

(cid:90)

(cid:20)
e−i(cid:104)ξ,x(cid:105)(e−iτ t − e−t|ξ|2

) − n(cid:88)

I :=

dτ

R

dξ

RN

(cid:21)
aje−i(cid:104)ξ,xj(cid:105)(e−iτ tj − e−tj|ξ|2
×ei(cid:104)ξ,x(cid:105)eiτ t(cid:98)ϕr(τ )(cid:98)ψr(ξ).

)

j=1

By inverse Fourier transform, we have

I = (2π)1+N

(cid:20)
ϕr(0)ψr(0)−ϕr(t)(pt ∗ ψr)(0)

− n(cid:88)

(cid:16)

j=1

where pt(x) is the heat kernel

aj

ϕr(t − tj)ψr(x − xj) − ϕr(t)(p

tj ∗ ψr)(x − xj)

(cid:17)(cid:21)

,

pt(x) =

1

(4πt)N/2

e−|x|2/(4t).

By the definition of r, for every j, we have either |t − tj| ≥ r2 or |x − xj| ≥ r, thus
ϕr(t − tj)ψr(x − xj) = 0. Moreover, since t/r2 ≥ a/M 2 ≥ ρ, we have ϕr(t) = 0 and hence

I = (2π)1+N r−2−N .

(4.31)

81

(cid:19)2(cid:35)(cid:90)

On the other hand, by the Cauchy–Schwarz inequality and (4.30),

(cid:34)(cid:18)
u1(t, x) − n(cid:88)

I2 ≤ C E

(cid:12)(cid:12)(cid:98)ϕr(τ )(cid:98)ψr(ξ)(cid:12)(cid:12)2(cid:0)|ξ|4 + |τ|2(cid:1)|ξ|N−βdτ dξ.
Note that (cid:98)ϕr(τ ) = (cid:98)ϕ(r2τ ) and (cid:98)ψr(ξ) = (cid:98)ψ(rξ). Then by a scaling of variables, we have

aju1(tj, xj)

R

RN

(cid:90)

j=1

(cid:90)

(cid:90)

RN

R
= r−6+β−2N

(cid:90)

(cid:90)

R

RN

(cid:12)(cid:12)(cid:98)ϕr(τ )(cid:98)ψr(ξ)(cid:12)(cid:12)2(cid:0)|ξ|4 + |τ|2(cid:1)|ξ|N−βdτ dξ

(cid:12)(cid:12)(cid:98)ϕ(τ )(cid:98)ψ(ξ)(cid:12)(cid:12)2(cid:0)|ξ|4 + |τ|2(cid:1)|ξ|N−βdτ dξ,

where the last integral is finite since (cid:98)ϕ and (cid:98)ψ are Schwartz functions. It follows that

(cid:34)(cid:18)
u(t, x) − n(cid:88)
I2 ≤ C0r−6+β−2N E

(cid:19)2(cid:35)

aju(tj, xj)

j=1

(4.32)

for some finite constant C0 (depending on a, a(cid:48) and b). Combining (4.31) and (4.32), we

obtain

(cid:34)(cid:18)

E

u(t, x) − n(cid:88)

j=1

(cid:19)2(cid:35)

aju(tj, xj)

≥ (2π)2+2N C−1

0 r2−β.

The proof is complete.

With (4.27) and Proposition 4.5.1, the following result is a direct consequence of Theorem

4.2.1 and 4.3.1.

Theorem 4.5.2. Suppose d < Q, where Q = (4 + 2N )/(2 − β). Let T be any closed
bounded cube in (0,∞) × RN . Then {u(t, x) : t ≥ 0, x ∈ RN} has a jointly continuous local
time L(·, T ) on T that satisfies the following H¨older condition: there exists a positive finite

82

constant C such that for any x ∈ Rd, with probability 1,

lim sup
r→0+

L(x, Bρ(t, r))

rQ−d(log log(1/r))d/Q

≤ C

(4.33)

for L(x,·)-almost every t ∈ T .

The theorem below identifies the correct gauge function for the Hausdorff measure (in
the metric ρ) of the level sets u−1(z) = {(t, x) ∈ (0,∞)× RN : u(t, x) = z} of the stochastic

heat equation.

Theorem 4.5.3. Suppose d < Q := (4 + 2N )/(2 − β). Then for any z ∈ Rd and any closed
bounded cube T ⊂ (0,∞) × RN , there exists a positive finite constant C such that

CL(z, T ) ≤ Hϕ

ρ (u−1(z) ∩ T ) < ∞ a.s.,

(4.34)

where ϕ(r) = rQ−d(log log(1/r))d/Q.

Remark 4.5.4. We conjecture that there exist positive finite constants C1 and C2 such that

C1L(z, T ) ≤ Hϕ

ρ (u−1(z) ∩ T ) ≤ C2L(x, T ) a.s.

We also conjecture that the gauge function for the Hausdorff measure in the Euclidean metric

of the level set is ϕ(r) = rβ(log log(1/r))d/Q, where β is the Hausdorff dimension in (4.25).

Proof of Theorem 4.5.3. The lower bound of (4.34) follows immediately from Theorem 4.4.1.

To prove that the Hausdorff measure is finite, we use the method in [67], which is similar

to Talagrand’s covering argument in [61] and Chapter 3 of this thesis. To this end, note that

83

we may assume T = B((t0, x0), η0), where η0 > 0 small and (t0, x0) ∈ T are fixed. Let

u1(t, x) = u(t, x) − u2(t, x),
u2(t, x) = E(u(t, x)|u(t0, x0))

Note that u1 and u2 are independent.

By Proposition 3.3.6, there exists η1 > 0 small such that for all 0 < r0 < η1, and all

(t, x) ∈ T , we have

(cid:32)

P

∃ r ∈ [r2

0, r0],

sup

(s,y)∈Bρ((t,x),2c2r)

(cid:16)
|u(t, x) − u(s, y)| ≤ K1r
(cid:18)

≥ 1 − exp

log log

−(cid:16)

log

(cid:33)
(cid:17)−1/Q
(cid:17)1/2(cid:19)

.

1
r

1
r0

(4.35)

Moreover, recall that Assumption 3.2.2 is satisfied with δj = 1 (see [18], Lemma 7.5). Then
by Lemma 3.3.7, for all (t, x), (s, y) ∈ T ,

|u2(t, x) − u2(s, y)| ≤ K2

(cid:0)|t − s| +

N(cid:88)

|xj − yj|(cid:1)|u(t0, x0)|.

(4.36)

j=1

Let

(cid:40)
(t, x) ∈ T :∃ r ∈ [2−2p, 2−p] such that

Rp =

sup

(s,y)∈Bρ((t,x),2c2r)

|u(t, x) − u(s, y)| ≤ K1r

(cid:16)

log log

1
r

(cid:17)−1/Q

(cid:41)

.

84

(cid:111)

p/4))

,

Consider the events

Ωp,1 =

(cid:110)
ω : λN (Rp) ≥ λN (T )(1 − exp(−√
ω : |u(t0, x0)| ≤ 2pb(cid:111)
(cid:110)
1 − exp(−(cid:112)p/2). Then by Fubini’s theorem, (cid:80)∞
see that(cid:80)∞

where b > 0 is chosen and fixed such that

Ωp,2 =

P(Ωc

,

p=1

in T given by Lemma 3.3.9. Consider the event

p,2) < ∞. Let Q =(cid:83)∞
(cid:26)

ω : ∀ I ∈ Q2p,

sup

p=1

(t,x),(s,y)∈I

2

2−β − b > 1. By (4.35), P((t, x) ∈ Rp) ≥
p,1) < ∞. Moreover, it is easy to
Qp be the family of (generalized) dyadic cubes

P(Ωc

p=1

(cid:27)

.

Ωp,3 =

|u(t, x) − u(s, y)| ≤ K32−2pp1/2

It is shown in the proof of Theorem 3.2.4 (Section 3.4) that (cid:80)∞

P(Ωc

p,3) < ∞ provided

p=1

K3 is a large enough constant. Let Ωp = Ωp,1 ∩ Ωp,2 ∩ Ωp,3 and

(cid:91)

(cid:92)

(cid:96)≥1

p≥(cid:96)

Ωp.

Ω∗ =

Then Ω∗ is an event of probability 1.

We are going to construct a random covering of the level set u−1(z) ∩ T . For any p ≥ 1
and (t, x) ∈ T , let Ip(t, x) ∈ Qp be the unique dyadic cube of order p containing (t, x). We

say that Iq(t, x) is a good dyadic cube of order q if it satisfies the following property:

sup

(s,y),(s(cid:48),y(cid:48))∈Iq(t,x)

|u1(s, y) − u1(s(cid:48), y(cid:48))| ≤ K12−q(log log 2q)−1/Q.

(4.37)

For each (t, x) ∈ Rp, since Iq is contained in some ball Bρ(c22−q) by Lemma 3.3.9 (iii), there

85

is a good dyadic cube I ∈ Q containing (t, x) of smallest order q, where p ≤ q ≤ 2p. By

property (ii) of Lemma 3.3.9, we obtain in this way a family G 1
cover Rp. On the other hand, we cover T \ Rp by a family G 2

order 2p that are not contained in any cube of G 1
only on the random field {u1(t, x) : (t, x) ∈ T}.

p . Let Gp = G 1

p of disjoint dyadic cubes that

p of dyadic cubes in Q2p of
p ∪ G 2

p . Note that Gp depends

For any dyadic cube I ∈ Q, choose a distinguished point (tI , xI ) ∈ I ∩ T . Fix p ≥ 1. For

any I ∈ Qq of order q, where p ≤ q ≤ 2p, consider the event

Ωp,I = {ω : |u(tI , xI ) − z| ≤ 2rp,I}

where

rp,I =

K12−q(log log 2q)−1/Q if I ∈ G 1

p ,
if I ∈ G 2
p .

K12−2pp1/2

Let Fp be the subcover of Gp (depending on ω) defined by

Fp(ω) = {I ∈ Gp(ω) : ω ∈ Ωp,I}.

We claim that for p large, on the event Ωp, Fp covers the set u−1(z) ∩ T . Suppose Ωp
occurs and (t, x) ∈ u−1(z) ∩ T . Since Gp covers T , the point (t, x) is contained in some
dyadic cube I and either I ∈ G 1

p or I ∈ G 2
p .

Case 1: if I ∈ G 1

p , then I = Iq(t, x) is a good dyadic cube of order q, where p ≤ q ≤ 2p,
and (4.37) holds. Note that I is contained in some ball Bρ(c22−q) by Lemma 3.3.9 (iii).

86

Since Ωp,2 occurs, it follows that from (4.36) and (4.37) that

|u(tI , xI ) − z| ≤ |u1(tI , xI ) − u1(t, x)| + |u2(tI , xI ) − u2(t, x)|

≤ K12−q(log log 2q)−1/Q + K2

4
2−β
2

(2c

+ 2N c

2
2−β
2

(cid:16)

2−β(cid:17)

−q 2

)2

2pb.

This is ≤ 2rp,I for p large because b is chosen such that

2

2−β − b > 1. Hence I ∈ Fp.

Case 2: if I ∈ G 2

p , since Ωp,3 occurs, we have

|u(tI , xI ) − z| = |u(tI , xI ) − u(t, x)| ≤ K32−2pp1/2.

In this case, I ∈ Fp. Hence the claim is proved.

Let Σ1 be the σ-field generated by {u1(t, x) : (t, x) ∈ T}. To estimate the conditional

probability P(Ωp,I|Σ1), note that by (4.27), for all (t, x) ∈ T = Bρ((t0, x0), η0),

Var(E(u(t, x)|u(t0, x0))) = Var(u(t, x)) − E[Var(u(t, x)|u(t0, x0))]

Var(u(t, x)) − C2

ρ((t, x), (t0, x0))2

sup
(t,x)∈T

≥ inf
(t,x)∈T
≥ K > 0

provided η0 > 0 is chosen small enough. Then P(|u2(t, x) − v| ≤ r) ≤ Krd for all (t, x) ∈ T ,
v ∈ Rd and r > 0. It follows from the independence of u1 and u2 that P(Ωp,I|Σ1) ≤ Krd
p,I .

Let ˜Ωp denote the event that the cardinality of G 2

that ˜Ωp ∈ Σ1. Since T \ Rp has Lebesgue measure ≤ exp(−√
p/4) on Ωp,1 and each I of
order 2p has Lebesgue measure ∼ K2−2pQ by Lemma 3.3.9 (iii), it follows that the event

p is at most K22pQ exp(−√

p/4). Note

87

Ωp,1 is contained in ˜Ωp. Let q[I] denote the order of I ∈ Fp. Then

1Ωp

E

(cid:88)

I∈Fp

ϕ(2c22−q[I])

 ≤ E

= E

= E

≤ KE



ϕ(2c22−q)1{I∈Gp}1Ωp,I

1 ˜Ωp

q=p

I∈Qq

q=p

I∈Qq

2p(cid:88)
(cid:88)
ϕ(2c22−q)1{I∈Gp}1Ωp,I
(cid:88)
2p(cid:88)

1 ˜Ωp
E
(cid:18)
1 ˜Ωp
ϕ(2c22−q)1{I∈Gp}E(cid:16)
2p(cid:88)
(cid:88)
1 ˜Ωp
(cid:88)
2p(cid:88)

ϕ(2c22−q)rd

p,I 1{I∈Gp}

q=p

I∈Qq

 .

|Σ1

1Ωp,I

q=p

I∈Qq

(cid:19)
(cid:12)(cid:12)(cid:12)(cid:12)Σ1
(cid:17)

If I ∈ G 1

p is of order q, then

ϕ(2c22−q)rd

p,I ≤ K2−q(Q−d)(log log 2q)d/Q2−qd(log log 2q)−d/Q ≤ KλN (I),

and these I’s are disjoint sets contained in T . If I ∈ G 2

p , then

ϕ(2c22−2p)rd

p,I ≤ K2−2pQ(log log 22p)d/Qpd/2

88

and there are at most K22pQ exp(−√

p/4) many such I’s on ˜Ωp. It follows that

1Ωp

E

(cid:88)

I∈Fp

ϕ(2−q[I])

(cid:88)
 ≤ KE
(cid:16)
≤ K(cid:0)λN (T ) + 1(cid:1)

≤ K

(cid:88)

2−pQ(log log 22p)d/Qpd/2

λN (I) + 1 ˜Ωp

I∈G 1
p
λN (T ) + (log log 22p)d/Qpd/2 exp(−√

I∈G 2
p

p/4)

(cid:17)



provided p is large. Recall that Fp is a cover for u−1(z) ∩ T on Ωp for large p and each I is
contained in a ρ-ball of radius c22−q[I]. Therefore, by Fatou’s lemma,

E(cid:104)Hϕ

(cid:105)
ρ (u−1(z) ∩ T )

≤ E

1Ω∗ Hϕ

(cid:105)
ρ (u−1(z) ∩ T )

= E(cid:104)
lim inf
1Ωp

(cid:88)
(cid:88)
≤ K(cid:0)λN (T ) + 1(cid:1) < ∞.

≤ lim inf
p→∞ E

p→∞ 1Ωp

I∈Fp

I∈Fp




ϕ(2c22−q[I])

ϕ(2c22−q[I])

This completes the proof of Theorem 4.5.3.

89

Chapter 5

Local Nondeterminism and the Exact

Modulus of Continuity for the

Stochastic Wave Equation

5.1

Introduction

Let k ≥ 1 and 0 < β < k ∧ 2, or k = 1 = β. Let us consider the linear stochastic wave
equation in arbitrary spatial dimension k ≥ 1:



∂2
∂t2 u(t, x) = ∆u(t, x) + ˙W (t, x),

t ≥ 0, x ∈ Rk,

u(0, x) =

∂
∂t

u(0, x) = 0.

(5.1)

Here,

˙W is the space-time Gaussian white noise if k = 1 = β; and is a Gaussian noise that

is white in time and colored in space with covariance

E( ˙W (t, x) ˙W (s, y)) = δ0(t − s)|x − y|−β

if k ≥ 1 and 0 < β < k ∧ 2. The purpose of this chapter is to study the exact modulus of

continuity for the solution. This part is based on [35].

90

In Chapter 4, we considered Gaussian random fields with the property of strong local

nondeterminism and we showed that the stochastic heat equation satisfies this property. In

this chapter, we will show that the stochastic wave equation satisfies a different form of local

nondeterminism. As an application, we use this property to determine the exact uniform

modulus of continuity of the solution in time and space variables jointly.

It is well known that the Brownian sheet {B(t) : t ∈ RN

+} does not satisfy strong local

nondeterminism, but it satisfies sectorial local nondeterminism (see [31], Proposition 4.2).
Namely, for all ε > 0, there exists C > 0 such that for all n ≥ 1, for all t, t1, . . . , tn ∈ [ε,∞)N ,

Var(B(t)|B(t1), . . . , B(tn)) ≥ C

|tj − ti
j|.

min
1≤i≤n

N(cid:88)

j=1

Recall from Theorem 3.1 of [64] that if

˙W is the space-time white noise, the solution u(t, x)

of (5.1) has the representation

(cid:18) t − x√

2

,

t + x√
2

(cid:19)

,

u(t, x) =

ˆW

1
2

(5.2)

where ˆW is a modified Brownian sheet (cf. [64, p.281]). In this special case, u(t, x) shares

many properties with the Brownian sheet.

It is therefore natural to study whether the

stochastic wave equation satisfies local nondeterminism.

In this chapter, we investigate the property of local nondeterminism for the solution

of (5.1) and use this property to study the uniform modulus of continuity of its sample

functions. The main results are Proposition 5.2.1 and Theorem 5.3.1. Proposition 5.2.1

shows that for any spatial dimension k, the solution u(t, x) satisfies an integral form of local

nondeterminism. When k = 1 and β = 1, this property (see (5.6) below) can also be derived

91

from the sectorial local nondeterminism for the Brownian sheet in [31, Proposition 4.2] after
a change of coordinates. While for k = 1 and β ∈ (0, 1), property (5.6) is similar to the

sectorial local nondeterminism in [65, Theorem 1] for a fractional Brownian sheet, which

suggests that the sample function u(t, x) may have subtle properties that are different from

those of Gaussian random fields with stationary increments (an important example of the

latter is fractional Brownian motion). We believe that Proposition 5.2.1 is useful for studying

precise regularity and other sample path properties of u(t, x). In Theorem 5.3.1, we apply

Proposition 5.2.1 to derive the exact uniform modulus of continuity of u(t, x).

The exact modulus of continuity provides precise information about the regularity and

oscillation of sample paths. General conditions for uniform and local exact moduli of con-

tinuity of Gaussian processes were studied by Marcus and Rosen [40]. The exact moduli of

continuity for anisotropic Gaussian random fields were studied by Meerschaert, Wang and

Xiao [42], with applications to fractional Brownian sheets and one-dimensional stochastic

heat equation driven by the space-time white noise. Similar results for the stochastic heat

equation driven by fractional-colored noise can be found in [62, 27].

5.2 Local Nondeterminism

Let G be the fundamental solution of the wave equation. Recall from Section 2.2.3 that for
k ≥ 3, G is not a function but a distribution. Also recall that for any dimension k ≥ 1, the

Fourier transform of G in variable x is given by

F (G(t,·))(ξ) =

sin(t|ξ|)

|ξ|

t ≥ 0, ξ ∈ Rk.

,

(5.3)

92

In [11], Dalang extended Walsh’s stochastic integration and proved that the real-valued

process solution of equation (5.1) is given by

u(t, x) =

G(t − s, x − y) W (ds dy),

(cid:90) t

(cid:90)

Rk

0

(cid:19)2(cid:21)

where W is the martingale measure induced by the noise ˙W . The range of β has been chosen

so that the stochastic integral is well-defined. Recall from Theorem 2 of [11] that

(cid:20)(cid:18)(cid:90) t

(cid:90)

E

Rk

0

(cid:90) t

0

(cid:90)

Rk

H(s, y)W (ds dy)

= ck,β

ds

dξ |ξ|β−k|F (H(s,·))(ξ)|2

(5.4)

provided that s (cid:55)→ H(s,·) is a deterministic function with values in the space of nonnegative

distributions with rapid decrease and

(cid:90) t

0

(cid:90)

ds

Rk

dξ |ξ|β−k|F (H(s,·)(ξ)|2 < ∞.

The following result shows that the solution u(t, x) satisfies an integral form of local nonde-

terminism.

Proposition 5.2.1. Let 0 < a < a(cid:48) < ∞ and 0 < b < ∞. There exist constants
C > 0 and δ > 0 depending on a, a(cid:48) and b such that for all integers n ≥ 1 and all
(t, x), (t1, x1), . . . , (tn, xn) in [a, a(cid:48)] × [−b, b]k with |t − tj| + |x − xj| ≤ δ, we have

Var (u(t, x)|u(t1, x1), . . . , u(tn, xn)) ≥ C

|(t − tj) + (x − xj) · w|2−β dw, (5.5)

min
1≤j≤n

Sk−1

where dw is the surface measure on the unit sphere Sk−1.

Remark 5.2.2. When k = 1, the surface measure dw in (5.5) is supported on {−1, 1}. It

93

(cid:90)

follows that u(t, x) satisfies sectorial local nondeterminism:

(cid:18)

Var(u(t, x)|u(t1, x1), . . . , u(tn, xn))
≥ C

|(t − tj) + (x − xj)|2−β + min
1≤j≤n

min
1≤j≤n

|(t − tj) − (x − xj)|2−β

(cid:19)

.

(5.6)

As we pointed out in the introduction in Section 5.1, property (5.6) is similar to the sectorial

local nondeterminism but different from the strong local nondeterminism of Gaussian ran-

dom fields with stationary increments. It indicates that u(t, x) may have properties that are

different from those of Gaussian random fields with stationary increments such as fractional

Brownian motion.

Proof of Proposition 5.2.1. Take δ = a/2. For each w ∈ Sk−1, let

r(w) = min
1≤j≤n

|(tj − t) − (xj − x) · w|.

Since u is a centered Gaussian random field, the conditional variance Var(u(t, x)|u(t1, x1), . . . ,

u(tn, xn)) is the squared distance of u(t, x) from the linear subspace spanned by u(t1, x1), . . . ,
u(tn, xn) in L2(P). Thus, it suffices to show that there exist constants C > 0 and δ > 0 such
that for all (t, x), (t1, x1), . . . , (tn, xn) in [a, a(cid:48)]× [−b, b]k with |t− tj| +|x− xj| ≤ δ, we have

≥ C

Sk−1

r(w)2−β dw

(5.7)

(cid:20)(cid:18)
u(t, x) − n(cid:88)

E

αju(tj, xj)

j=1

(cid:19)2(cid:21)

(cid:90)

for any choice of real numbers α1, . . . , αn. Using (5.3), (5.4) and spherical coordinate ξ = ρ w,

94

(cid:19)2(cid:21)

αju(tj, xj)

we have

E

(cid:20)(cid:18)
u(t, x) − n(cid:88)
(cid:90)
(cid:90) ∞

j=1

= ck,β

ds

dξ

Rk

×

|ξ|2+k−β

(cid:12)(cid:12)(cid:12)(cid:12) sin((t − s)|ξ|)1[0,t](s) − n(cid:88)
(cid:90) ∞
(cid:90) ∞

dρ
ρ3−β

Sk−1

(cid:90)

(cid:90)

dw

0

dρ
|ρ|3−β

−∞

dw

Sk−1

(s)

j=1

[0,tj ]

αje−i(xj−x)·ξ sin((tj − s)|ξ|)1

(cid:12)(cid:12)(cid:12)(cid:12) sin((t − s)ρ) − n(cid:88)
αje−iρ(xj−x)·w sin((tj − s)ρ)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)
ei(t−s)ρ − e−i(t−s)ρ(cid:17)
(cid:18)
− n(cid:88)
ei(tj−s)ρ − e−i(tj−s)ρ

αje−iρ(xj−x)·w

j=1

(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)2

j=1

A(w) dw.

0

(cid:90) a/2
(cid:90) a/2

0

ds

ds

0

(cid:90)

Sk−1

≥ ck,β

=

ck,β
8

=:

ck,β
8

Let λ = min{1, a/[2(a(cid:48) + 2

√

kb)]} and consider the bump function ϕ : R → R defined by

(cid:19)(cid:21)
e−i(t−s)ρ(cid:98)ϕr(w)(ρ).

ei(tj−s)ρ − e−i(tj−s)ρ

95

|y| < λ,

,

|y| ≥ λ.

ϕ(y) =

(cid:19)

1

0,

1−|λ−1y|2

1 −

(cid:18)

exp
ei(t−s)ρ − e−i(t−s)ρ(cid:17)

I(w) :=

ds

(cid:90) a/2

0

(cid:90) ∞
− n(cid:88)
−∞ dρ

(cid:20)(cid:16)
αje−iρ(xj−x)·w

(cid:18)

j=1

Let ϕr(y) = r−1ϕ(y/r). For each w ∈ Sk−1 such that r(w) > 0, consider the integral

By the inverse Fourier transform (or one can apply the Plancherel theorem), we have

(cid:0)2(t − s)(cid:1)

(cid:20)

(cid:90) a/2
(cid:16)

0

− n(cid:88)

I(w) = 2π

ds

ϕr(w)(0) − ϕr(w)

(cid:0)(xj − x) · w − (tj − t)(cid:1) − ϕr(w)

(cid:0)(xj − x) · w − (tj − t) + 2(tj − s)(cid:1)(cid:17)(cid:21)

.

αj

ϕr(w)

j=1

√
Note that r(w) ≤ |tj −t|+|xj −x| ≤ a(cid:48) +2
a/[(a(cid:48) + 2

√
kb)] and |(xj − x) · w − (tj − t)|/r(w) ≥ 1, thus

kb. For any s ∈ [0, a/2], we have 2(t−s)/r(w) ≥

(cid:0)2(t − s)(cid:1) = 0 and ϕr(w)

(cid:0)(xj − x) · w − (tj − t)(cid:1) = 0 for j = 1, . . . , n.

ϕr(w)

Also, [(xj − x) · w − (tj − t) + 2(tj − s)]/r(w) ≥ (−δ + a)/[(a(cid:48) + 2

√
kb)] ≥ λ, thus

(cid:0)(xj − x) · w − (tj − t) + 2(tj − s)(cid:1) = 0.

ϕr(w)

It follows that

I(w) = aπ r(w)−1.

On the other hand, by the Cauchy–Schwarz inequality and scaling, we obtain

(aπ)2r(w)−2 = |I(w)|2 ≤ A(w) ×

(cid:90) a/2

0

ds

(cid:90) ∞
−∞ dρ|(cid:98)ϕ(r(w)ρ)|2|ρ|3−β
(cid:90) ∞
−∞ dρ|(cid:98)ϕ(ρ)|2|ρ|3−β

= (a/2)A(w)r(w)β−4

= CA(w)r(w)β−4

96

for some finite constant C. Hence we have

A(w) ≥ C(cid:48)r(w)2−β

(5.8)

and this remains true if r(w) = 0. Integrating both sides of (5.8) over Sk−1 yields (5.7).

5.3 The Exact Uniform Modulus of Continuity

Let f : I → R be a function with I ⊂ RN . Let φ : [0,∞) → [0,∞) be a function such that

limε→0+ φ(ε) = φ(0) = 0. Recall that φ is called a modulus of continuity for f on I if there

exists a finite constant C such that

|f (x) − f (y)| ≤ Cφ(|x − y|)

for all x, y ∈ I.

In order to identify the optimal modulus function, Marcus and Rosen [40] introduced the

following definition. Let σ be a metric on I. We say that φ is an exact modulus of continuity

for f on (I, σ) if there exists a positive finite constant C such that

|f (x) − f (y)|
φ(σ(x, y))

= C.

lim
ε→0+

sup
x,y∈I:

0<σ(x,y)≤ε

For example, L´evy’s theorem of modulus of continuity shows that the exact modulus of

continuity for the Brownian motion is φ(ε) =(cid:112)ε log(1/ε) with C =

√
2 (and σ being the

Euclidean metric).

It is known that sectorial local nondeterminism is useful for proving the exact uniform

97

modulus of continuity for Gaussian random fields [42].

In this section we show that the

integral form of local nondeterminism in Proposition 5.2.1 can serve the same purpose for

deriving the exact uniform modulus of continuity of the solution u(t, x) to (5.1).

Let us denote

σ(cid:2)(t, x), (t(cid:48), x(cid:48))(cid:3) = E[(u(t, x) − u(t(cid:48), x(cid:48)))2]1/2.

Recall from [20, Proposition 4.1] that for any 0 < a < a(cid:48) < ∞ and 0 < b < ∞, there are

(cid:18)

k(cid:88)

(cid:19)2−β ≤ σ[(t, x), (t(cid:48), x(cid:48))]2 ≤ C2

(cid:18)
|t − t(cid:48)| +

k(cid:88)

j=1

(cid:19)2−β
|xj − x(cid:48)
j|

(5.9)

positive constants C1 and C2 such that

|t − t(cid:48)| +

C1

|xj − x(cid:48)
j|

j=1

for all (t, x), (t(cid:48), x(cid:48)) ∈ [a, a(cid:48)] × [−b, b]k.

The following result establishes the exact uniform modulus of continuity of u(t, x) in the

time and space variables (t, x).

Theorem 5.3.1. Let I = [a, a(cid:48)] × [−b, b]k, where 0 < a < a(cid:48) < ∞ and 0 < b < ∞. Let

γ(cid:2)(t, x), (t(cid:48), x(cid:48))(cid:3) = σ(cid:2)(t, x), (t(cid:48), x(cid:48))(cid:3)(cid:113)

log (1 + σ(cid:2)(t, x), (t(cid:48), x(cid:48))(cid:3)−1).

Then there is a positive finite constant K such that

lim
ε→0+

sup

(t,x),(t(cid:48),x(cid:48))∈I,

0<σ[(t,x),(t(cid:48),x(cid:48))]≤ε

|u(t, x) − u(t(cid:48), x(cid:48))|

γ(cid:2)(t, x), (t(cid:48), x(cid:48))(cid:3) = K,

a.s.

(5.10)

98

Proof. For any ε > 0, let

J(ε) =

|u(t, x) − u(t(cid:48), x(cid:48))|

γ(cid:2)(t, x), (t(cid:48), x(cid:48))(cid:3) .

sup

(t,x),(t(cid:48),x(cid:48))∈I,

0<σ[(t,x),(t(cid:48),x(cid:48))]≤ε

Since ε (cid:55)→ J(ε) is non-decreasing, we see that the limit limε→0+ J(ε) exists a.s. In order to
prove (5.10), we prove the following statements: there exist positive and finite constants K∗

and K∗ such that

and

J(ε) ≤ K∗,

a.s.

lim
ε→0+

J(ε) ≥ K∗,

a.s.

lim
ε→0+

(5.11)

(5.12)

Then the conclusion of Theorem 5.3.1 follows from Lemma 7.1.1 of [40] where τ is chosen
to be the Euclidean metric and d is the canonical metric σ[(t, x), (t(cid:48), x(cid:48))].

[It is a 0-1 law

for the modulus of continuity which is obtained by applying Kolmogorov’s 0-1 law to the

Karhunen–Lo`eve expansion of u(t, x).]

The proof of the upper bound (5.11) is standard. For any ε > 0, denote by N (I, ε, σ)

the smallest number of balls of radius ε in the canonical metric σ(cid:2)(t, x), (t(cid:48), x(cid:48))(cid:3) that are

needed to cover the compact interval I. By the upper bound in (5.9), we have N (I, ε, σ) ≤
Cε−(1+k)/(2−β) and thus

(cid:90) ε

(cid:113)
(cid:112)log N (I, ˜ε, σ) d˜ε ≤ Cε

log(1 + ε−1).

0

99

By Theorem 1.3.5 of [2], there is a positive finite constant K∗ such that

|u(t, x) − u(t(cid:48), x(cid:48))|

ε(cid:112)log(1 + ε−1)

≤ K∗

a.s.

lim sup
ε→0+

sup

(t,x),(t(cid:48),x(cid:48))∈I,

0<σ[(t,x),(t(cid:48),x(cid:48))]≤ε

From this we can deduce (5.11) by considering εn+1 ≤ σ[(t, x), (t(cid:48), x(cid:48))] ≤ εn where εn = 1/n,

and using the fact that the function ε (cid:55)→ ε(cid:112)log(1 + ε−1) is increasing for ε small, and

(cid:113)
(cid:113)
log(1 + ε−1
n )
log(1 + ε−1
n+1)

= 1.

lim
n→∞

εn

εn+1

Next we prove the lower bound (5.12). This is accomplished by applying Proposition

5.2.1, a conditioning argument and the Borel–Cantelli lemma. We first choose δ according
k), a(cid:48) − a, 2b}. Note that δ(cid:48) depends only on
to Proposition 5.2.1 and let δ(cid:48) = min{δ/(1 +
a, a(cid:48) and b. For each n ≥ 1, let

√

εn = [C2((1 + k)δ(cid:48))2−β2−(2−β)n]1/2.

For i = 0, 1, . . . , 2n, let tn,i = a + iδ(cid:48)2−n and xn,i

j = −b + iδ(cid:48)2−n. Then

lim
ε→0+

J(ε) = lim
n→∞

sup

(t,x),(t(cid:48),x(cid:48))∈I,

0<σ[(t,x),(t(cid:48),x(cid:48))]≤εn

|u(t, x) − u(t(cid:48), x(cid:48))|
γ[(t, x), (t(cid:48), x(cid:48))]

|u(tn,i, xn,i) − u(tn,i−1, xn,i−1)|

(cid:113)
log(1 + ε−1
n )

εn

≥ lim inf

n→∞ max
1≤i≤2n

=: lim inf

n→∞ Jn.

To obtain the inequality, we have used the fact that σ[(tn,i, xn,i), (tn,i−1, xn,i−1)] ≤ εn and

100

that the function ε (cid:55)→ ε(cid:112)log(1 + ε−1) is increasing for ε small.

Let K∗ > 0 be a constant whose value will be determined later. Fix n and write tn,i = ti,

xn,i = xi to simplify notations. By conditioning, we can write

P (Jn ≤ K∗)
= P

max
1≤i≤2n

(cid:18)
(cid:34)

= E

1AP

(cid:32)|u(t2n

|u(ti, xi) − u(ti−1, xi−1)|

≤ K∗

(cid:113)
log(1 + ε−1
n )
(cid:113)
) − u(t2n−1, x2n−1)|
log(1 + ε−1
n )

εn
, x2n

εn

(cid:19)

(cid:33)(cid:35)
(cid:12)(cid:12)(cid:12)(cid:12)u(ti, xi) : 0 ≤ i ≤ 2n − 1

,

(5.13)

≤ K∗

where A is the event defined by

(cid:40)

A =

max

1≤i≤2n−1

|u(ti, xi) − u(ti−1, xi−1)|

(cid:113)
log(1 + ε−1
n )

εn

Since |t2n − ti| + |x2n − xi| ≤ δ, by Proposition 5.2.1 we have

(cid:41)

.

≤ K∗

(5.14)

(cid:17)

, x2n

)|u(ti, xi) : 0 ≤ i ≤ 2n − 1

u(t2n

(cid:16)
(cid:90)
(cid:90)
{w∈Sk−1: (1,...,1)·w≥0} min

0≤i≤2n−1

Sk−1

min

Var
≥ C

≥ C

|(t2n − ti) + (x2n − xi) · w|2−β dw
(cid:90)
{w∈Sk−1: (1,...,1)·w≥0} dw

0≤i≤2n−1

≥ C(δ(cid:48))2−β 2−(2−β)n

|δ(cid:48)(2n − i)2−n + δ(cid:48)(2n − i)2−n(1, . . . , 1) · w|2−β dw

= C0 ε2
n

for some constant C0 > 0 depending on a, a(cid:48) and b.

sian with conditional variance Var(cid:0)u(t2n

Since the conditional distribution of u(t2n
, x2n

, x2n

), given u(ti, xi), (0 ≤ i ≤ 2n−1), is Gaus-

)|u(ti, xi) : 0 ≤ i ≤ 2n − 1(cid:1), it follows from An-

101

derson’s inequality [3] and (5.14) that

(cid:32)|u(t2n
(cid:18)

P

, x2n

εn

(cid:113)
) − u(t2n−1, x2n−1)|
(cid:113)
log(1 + ε−1
n )
log (1 + ε−1
n )

C−1

0

(cid:19)

≤ K∗

≤ P

|Z| ≤ K∗

(cid:33)
(cid:12)(cid:12)(cid:12)(cid:12)u(ti, xi) : 0 ≤ i ≤ 2n − 1

√
2π)−1x−1 exp(−x2/2)
where Z is a standard normal random variable. Using P(|Z| > x) ≥ (
for x ≥ 1 and 1+ε−1 < 2/ε for ε small, we deduce that when n is large the above probability

is bounded from above by

K∗(cid:112)log (2/εn)
1 − C(εn/2)K2∗ /(2C0)

≤ exp

(cid:32)
K∗(cid:112)log (2/εn)
−C(εn/2)K2∗ /(2C0)

(cid:33)

(cid:32)

≤ exp

(cid:33)

n

− CK∗2

− (2−β)K2∗
√

4C0

n

where CK∗ > 0 is a constant depending on K∗. Then by (5.13) and induction, we have

P(cid:0)Jn ≤ K∗(cid:1) ≤ exp

(cid:32)

− 2n CK∗2

− (2−β)K2∗
√
n

4C0

n

(cid:33)

.

We can now choose K∗ > 0 to be a sufficiently small constant such that

1 − (2 − β)K2∗

4C0

> 0.

P(cid:0)Jn ≤ K∗(cid:1) < ∞. Hence, by the Borel–Cantelli lemma, lim infn Jn ≥ K∗ a.s.

Then(cid:80)∞

n=1

and the proof is complete.

102

Chapter 6

Propagation of Singularities for the

Stochastic Wave Equation

6.1

Introduction

In this chapter, we consider the stochastic wave equation in one spatial dimension:



∂2

∂t2 u(t, x) − ∂2

∂x2 u(t, x) = ˙W (t, x),
∂
∂t

u(0, x) = 0,

u(0, x) = 0,

t ≥ 0, x ∈ R,

(6.1)

where ˙W is a Gaussian noise that is white in time and colored in space with spatial covariance

E[ ˙W (t, x) ˙W (s, y)] = δ0(t − s)|x − y|−β

(6.2)

with 0 < β < 1. The purpose of this chapter is to study the singularities of the solution
{u(t, x) : t ≥ 0, x ∈ R}. This chapter is based on [36].

In this context, singularity is related to exceptionally large increments of a stochastic

process. By singularity we mean a random point at which the process has local oscillations

that are much larger than those specified by the law of the iterated logarithm (LIL). For

the Brownian motion, this phenomenon was first studied by Orey and Taylor [48].

It is

103

well known that at a fixed time, the increments of a Brownian path satisfies the LIL almost

surely. However, it is not true that the LIL holds simultaneously for all time points with

probability one.

Indeed, according to L´evy’s modulus of continuity, we can find random

points at which the LIL fails and the increments are exceptionally large, and therefore we

can define these exceptional points as singularities. Similarly, we can define singularities for

other general random fields.

The singularities of the Brownian sheet and the one-dimensional stochastic wave equation

driven by the space-time white noise were studied by Walsh [63, 64], and those of semi-

fractional Brownian sheet was studied by Blath and Martin [7]. Based on a simultaneous

law of the iterated logarithm, Walsh [63] showed that the singularities of the Brownian sheet

propagate parallel to the coordinate axis. Moreover, Walsh [64] found an interesting relation

between the Brownian sheet and the solution u(t, x) to (6.1) driven by the space-time white

noise. Specifically, Theorem 3.1 in [64] shows that the solution can be written as the sum of

three components:

u(t, x) =

(cid:20)

B

(cid:16) t − x√

2

1
2

(cid:17)

+ ˆW

(cid:16)t − x√

2

(cid:17)

, 0

(cid:16)

+ ˆW

,

t + x√
2

0,

t + x√
2

(cid:17)(cid:21)

,

(6.3)

where the main component B is a Brownian sheet and ˆW is the modified Brownian sheet
defined in Chapter 1 of Walsh [64], and the processes {B(s, t) : s, t ≥ 0}, { ˆW (s, 0) : s ≥ 0}
and { ˆW (0, t) : t ≥ 0} are independent. This relation implies that the singularities of u(t, x)
propagate along the characteristic curves t − x = c and t + x = c.

Later, Carmona and Nualart [9] extended the study of singularities of the solution to the

linear stochastic wave equation (6.1) driven by space-time white noise in [63, 64] to the case

of one-dimensional nonlinear stochastic wave equations driven by a space-time white noise.

104

Their approach is based on the general theory of semimartingales and two-parameter strong

martingales. In particular, they proved the law of the iterated logarithm for a semimartingale

by the LIL of Brownian motion and a time change. They also proved that, for a class of

two-parameter strong martingales, the law of the iterated logarithm in one variable holds

simultaneously for all values of the other variable.

The main objective of this chapter is to study the existence and propagation of singu-

larities of the solution to (6.1) driven by a Gaussian noise that is white in time and colored

in space with spatial covariance given by (6.2) with 0 < β < 1. In this case, the solution

shares some similarity with the fractional Brownian sheet, but it seems to us that there is

not a natural relation like (6.3) between the solution and the fractional Brownian sheet.

Also, the method in Carmona and Nualart [9] based on semimartingales and two-parameter

strong martingales is not applicable in the case of colored noise. Our approach is based on

a simultaneous LIL for the solution and general methods for Gaussian processes.

This chapter is organized as follows. First, we establish a simultaneous LIL for the

solution of the stochastic wave equation. We prove that after a rotation, the LIL in one

variable holds simultaneously for all values of the other variable. The proof consists of two

parts: proving the upper bound and lower bound. The upper bound is proved in Section

6.2 and the lower bound is proved in Section 6.3. In Section 6.4, we introduce the definition

of singularity for the stochastic wave equation and apply the simultaneous LIL to study

the propagation of singularities. The main result Theorem 6.4.3 shows that singularities

propagate along the characteristic curves.

105

6.2 Simultaneous Law of Iterated Logarithm: Upper

Bound

The noise in (6.1) is defined as the mean zero Gaussian process W (ϕ) indexed by compactly
supported smooth functions ϕ ∈ C∞

c (R+ × R) with covariance function
(cid:90)

(cid:90)

(cid:90)

dy(cid:48) ϕ(s, y)|y − y(cid:48)|−βψ(s, y(cid:48))

R
µ(dξ) F (ϕ(s,·))(ξ)F (ψ(s,·))(ξ)

dy

(cid:90)

R

R

ds

E[W (ϕ)W (ψ)] =

=

(cid:90)

R+
1
2π

ds

R+

(6.4)

c (R+ × R), where µ is the measure whose Fourier transform is | · |−β and
for all ϕ, ψ ∈ C∞
F (ϕ(s,·))(ξ) is the Fourier transform of the function y (cid:55)→ ϕ(s, y) in the following convention:

(cid:90)

R

e−iξyϕ(s, y)dy.

F (ϕ(s,·))(ξ) =

Note that µ(dξ) = Cβ|ξ|−1+βdξ, where

Cβ =

π1/221−βΓ( 1
Γ( β
2 )

2 − β
2 )

see [59, p.117]. We assume that W is defined on a complete probability space (Ω, F , P).

Following [11, 13], for any bounded Borel set A in R+ × R, we can define

W (A) = lim

n→∞ W (ϕn)

in the sense of L2(P)-limit, where (ϕn) is a sequence in C∞
c (R+ × R) with a compact set K
such that supp ϕn ⊂ K for all n and ϕn → 1A. From (6.4), it follows that for any bounded

106

Borel sets A, B in R+ × R, we have

E[W (A)W (B)] =

=

(cid:90)

(cid:90)

R+
1
2π

dy

(cid:90)

R

ds

ds

R+

(cid:90)

(cid:90)

R

dy(cid:48) 1A(s, y)|y − y(cid:48)|−β1B(s, y(cid:48))

Cβdξ
|ξ|1−β

F (1A(s,·))(ξ)F (1B(s,·))(ξ).

R

(6.5)

In dimension one, the fundamental solution of the wave equation is 1

21{|x|≤t}, so the mild

solution of (6.1) is

u(t, x) =

(cid:90) t

(cid:90)

R

0

1
2

1{|x−y|≤t−s}(s, y) W (ds dy) =

1
2

W (∆(t, x)),

(6.6)

where ∆(t, x) = {(s, y) ∈ R+ × R : 0 ≤ s ≤ t, |x − y| ≤ t − s}.

Consider a new coordinate system (τ, λ) obtained by rotating the (t, x)-coordinates by

−45◦. In other words,

(τ, λ) =

(cid:16)t − x√

2

,

t + x√
2

(cid:17)

(cid:16)τ + λ√

2

,

(cid:17)

.

−τ + λ√

2

and (t, x) =

For τ ≥ 0, λ ≥ 0, let us denote

(cid:16)τ + λ√

2

˜u(τ, λ) = u

(cid:17)

.

−τ + λ√

2

,

We are going to prove a simultaneous LIL for the Gaussian random field {˜u(τ, λ) :
τ ≥ 0, λ ≥ 0}. The following result shows an upper bound for the LIL in λ, which holds

simultaneously for all values of τ . By a symmetric argument, we can also prove that the LIL

in τ holds simultaneously for all λ.

107

Proposition 6.2.1. For any λ > 0, we have

lim sup

h→0+

P

where

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
(τ + λ)h2−β log log(1/h)

≤ Kβ for all τ ∈ [0,∞)
(cid:19)1/2

(cid:18) 2(1−β)/2

Kβ =

(2 − β)(1 − β)

.

 = 1,

(6.7)

Lemma 6.2.2. For any 0 < β < 1, a < b and c < d, we have

(cid:90) ∞

−∞

Cβ

|F 1[a,b](ξ)|2 dξ
|ξ|1−β

=

4π

(2 − β)(1 − β)

(b − a)2−β

(6.8)

and

Cβ

=

(cid:90) ∞

−∞

F 1[a,b](ξ)F 1[c,d](ξ)

dξ
|ξ|1−β

(cid:16)|c − b|2−β + |d − a|2−β − |c − a|2−β − |d − b|2−β(cid:17)

.

2π

(2 − β)(1 − β)

Proof. The Fourier transform of the function 1[a,b] is

F 1[a,b](ξ) =

e−iξa − e−iξb

iξ

.

It follows that

Cβ

(cid:90) ∞

−∞

|F 1[a,b](ξ)|2 dξ
|ξ|1−β

(cid:90) ∞

|eiξ(b−a) − 1|2 dξ
|ξ|3−β

−∞

= Cβ
= Cβ(b − a)2−β

(cid:90) ∞

−∞

|eiξ − 1|2 dξ
|ξ|3−β

.

108

The last equality follows by scaling. The proof of Proposition 7.2.8 of [56] shows that

(cid:90) ∞

−∞

|eiξ − 1|2 dξ
|ξ|3−β

=

2π

(2 − β)Γ(2 − β) sin( πβ
2 )

.

Also, using the relations Γ(2z) = 22z−1π−1/2Γ(z)Γ(z + 1), Γ(z)Γ(1 − z) = π/ sin(πz) and

zΓ(z) = Γ(z + 1) (cf. [25, p.895–896]), we can show that

2Γ(2 − β) sin( πβ
2 )

1 − β

.

Cβ =

Hence (6.8) follows.

For the second part,

F 1[a,b](ξ)F 1[c,d](ξ)

dξ
|ξ|1−β

=

(cid:90) ∞

−∞

(cid:0)eiξ(c−a) + eiξ(d−b) − eiξ(c−b) − eiξ(d−a)(cid:1) dξ

|ξ|3−β

.

Note that this integral is real, so we have

F 1[a,b](ξ)F 1[c,d](ξ)

dξ
|ξ|1−β

=

1
2

(cid:90) ∞
(cid:0)eiξ(c−a) + e−iξ(c−a) + eiξ(d−b) + e−iξ(d−b)
− eiξ(c−b) − e−iξ(c−b) − eiξ(d−a) − e−iξ(d−a)(cid:1) dξ

−∞

|ξ|3−β

Since |eiξ(x−y) − 1|2 = 2 − eiξ(x−y) − e−iξ(x−y) for all x, y ∈ R, we have

(cid:90) ∞

−∞

(cid:90) ∞

−∞

(cid:90) ∞

−∞
1
2

=

(cid:90) ∞

−∞

.

.

F 1[a,b](ξ)F 1[c,d](ξ)

(cid:0) − |eiξ(c−a) − 1|2 − |eiξ(d−b) − 1|2 + |eiξ(c−b) − 1|2 + |eiξ(d−a) − 1|2(cid:1) dξ

dξ
|ξ|1−β

|ξ|3−β

Now the result follows from the first part of the proof.

109

Lemma 6.2.3. For any τ, λ, h > 0,

E[(˜u(τ, λ + h) − ˜u(τ, λ))2] =

1
2

K2
β

(cid:104)

(τ + λ)h2−β + (3 − β)−1h3−β(cid:105)

,

where

Proof. Note that

(cid:18) 2(1−β)/2

(2 − β)(1 − β)

(cid:19)1/2

.

Kβ =

E[(˜u(τ, λ + h) − ˜u(τ, λ))2]
√

(cid:20)(cid:16)
(cid:16) τ + λ + h√
(cid:34)(cid:18)
(cid:16)

(cid:16)τ + λ + h√

= E

E

W

∆

u

2

,

,

=

1
4

−τ + λ + h

(cid:17) − u

(cid:16)τ + λ√
−τ + λ√
(cid:17)(cid:15)∆
(cid:16)τ + λ√

(cid:17)(cid:17)2(cid:21)
−τ + λ√

2

2

,

,

2

2

2
−τ + λ + h

√

2

2

(cid:17)(cid:17)(cid:19)2(cid:35)

.

Then by (6.5) and Lemma 6.2.2,

=

1
8π

+

E[(˜u(τ, λ + h) − ˜u(τ, λ))2]

2

0

ds

√

−∞

√
2λ−s,

Cβdξ
|ξ|1−β


(cid:90) τ +λ√
(cid:90) τ +λ+h√

(cid:90) ∞
(cid:90) ∞

(cid:90) τ +λ√

(cid:12)(cid:12)F 1[
(cid:12)(cid:12)F 1[−√
(cid:0)√
2h(cid:1)2−βds +
(τ + λ)h2−β + (3 − β)−1h3−β(cid:105)
(cid:104)

Cβdξ
|ξ|1−β

τ +λ√
2

−∞

ds

1

0

2

2

.

=

2(2 − β)(1 − β)

=

1
2

K2
β

2(λ+h)−s](ξ)(cid:12)(cid:12)2

2(λ+h)−s](ξ)(cid:12)(cid:12)2
(cid:90) τ +λ+h√
(cid:0)√
2(τ + λ + h) − 2s(cid:1)2−βds

2

√
2τ +s,

τ +λ√
2



Recall a standard result for large deviation (cf. [33, 41]): If {Z(t) : t ∈ T} is a continuous

110

centered Gaussian random field which is a.s. bounded, then

lim
γ→∞

1

γ2 log P

Z(t) > γ

sup
t∈T

= −

1

2 supt∈T E(Z(t)2)

.

(6.9)

By symmetry of the distribution of {Z(t) : t ∈ T}, we have

(cid:18)

(cid:18)

(cid:19)

(cid:19)

lim
γ→∞

1

γ2 log P

sup
t∈T

|Z(t)| > γ

= −

1

2 supt∈T E(Z(t)2)

.

(6.10)

Now, we prove Proposition 6.2.1.

lim sup

h→0+

P

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
(τ + λ)h2−β log log(1/h)

Proof of Proposition 6.2.1. It suffices to show that for any 0 ≤ a < b < ∞ and any 0 < ε < 1,

≤ (1 + ε)Kβ for all τ ∈ [a, b]

(6.11)

 = 1.

Let c ∈ [a, b], δ = (a + λ)ε/2 and d = c + δ. Take 0 < θ < 1 such that θ(1 + ε) > 1. Choose
a real number q such that 1 < q < [θ(1 + ε)]1/(2−β). Consider the event

(cid:26)

(cid:12)(cid:12)˜u(τ, λ + h) − ˜u(τ, λ)(cid:12)(cid:12) > γn

(cid:27)

,

An =

sup
τ∈[0,d]

sup

h∈[0,q−n]

where

(cid:113)

(c + λ)(q−n−1)2−β log log qn.

γn = (1 + ε)Kβ

By Lemma 6.2.3,

E(cid:104)
(˜u(τ, λ + h) − ˜u(τ, λ))2(cid:105)

(cid:104)

(τ + λ)h2−β + (3 − β)−1h3−β(cid:105)

.

=

1
2

K2
β

111

By (6.10), for all large n,

log P(An) ≤ −

1
γ2
n

β[(d + λ)(q−n)2−β + (3 − β)−1(q−n)3−β]
K2

θ

.

It follows that

P(An) ≤ exp

(cid:18)

−

θ(1 + ε)2(c + λ)

q2−β[(d + λ) + (3 − β)−1q−n]

(cid:19)

log(n log q)

= (n log q)−pn,

where

pn =

θ(1 + ε)2

q2−β[(1 + δ

c+λ) + (3 − β)−1(c + λ)−1q−n]

.

Recall that δ = (c + λ)ε/2. If n is sufficiently large, then (3 − β)−1(c + λ)−1q−n ≤ ε/2,

which implies that

Hence (cid:80)∞

n=1

pn ≥ θ(1 + ε)
q2−β

> 1.

P(An) < ∞ and by the Borel–Cantelli lemma, we have P(An i.o.) = 0.

It

follows that with probability 1,

sup

h∈[q−n−1,q−n]

sup
τ∈[c,d]

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|

(c + λ)(q−n−1)2−β log log qn

≤ (1 + ε)Kβ

eventually for all large n. Hence

lim sup

h→0+

P

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
(τ + λ)h2−β log log(1/h)

≤ (1 + ε)Kβ for all τ ∈ [c, d]

 = 1.

From this, we can deduce (6.11) by covering the interval [a, b] by finitely many intervals [c, d]

of length δ.

112

6.3 Simultaneous Law of Iterated Logarithm: Lower

Bound

In this section, we prove the lower bound for the simultaneous LIL:

Proposition 6.3.1. For any λ > 0,

lim sup

h→0+

P

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
(τ + λ)h2−β log log(1/h)

 = 1,

≥ Kβ for all τ ∈ [0,∞)

(6.12)

where Kβ is the same constant as in Proposition 6.2.1, i.e.

(cid:18) 2(1−β)/2

(cid:19)1/2

.

Kβ =

(2 − β)(1 − β)

Recall the following version of Borel–Cantelli lemma [55, p.391].

Lemma 6.3.2. Let {An : n ≥ 1} be a sequence of events. If

(i) (cid:80)∞

n=1

(ii) lim inf
n→∞

P(An) = ∞ and

(cid:80)n

(cid:80)n
[(cid:80)n

j=1

j=1

k=1

P(Aj ∩ Ak)
P(Aj)]2

= 1,

then P(An i.o.) = 1.

We will also use the following lemma, which is essentially proved in [57]. For the sake of

completeness, we provide a proof for this result.

Lemma 6.3.3. Let Z1 and Z2 be jointly Gaussian random variables with E(Zi) = 0, E(Z2
i ) =
1 and E(Z1Z2) = r. Then for any γ1, γ2 > 0, there exists a number r∗ between 0 and r such

that

P(Z1 > γ1, Z2 > γ2) − P(Z1 > γ1)P(Z2 > γ2) = rg(γ1, γ2; r∗),

113

where g(x, y; r) is the standard bivariate Gaussian density with correlation r, i.e.

(cid:18)

(cid:19)

.

g(x, y; r) =

1

2π(1 − r2)1/2

exp

−x2 + y2 − 2rxy

2(1 − r2)

(cid:82) ∞

Proof. Let γ1, γ2 > 0 and p(r) =(cid:82) ∞
function f (x, y) as F f (ξ, ζ) =(cid:82)(cid:82)
γ1
R2 e−i(xξ+yζ)f (x, y) dx dy. Note that
(cid:90)(cid:90)

γ2

g(x, y; r) =

1

(2π)2

R2

ei(xξ+yζ)[F g(∗ ; r)](ξ, ζ) dξ dζ

g(x, y; r) dx dy. Define the Fourier transform of a

and

[F g(∗ ; r)](ξ, ζ) = e

− 1
2 (ξ2+2rξζ+ζ2).

By the dominated convergence theorem,

(cid:90)(cid:90)

∂rg(x, y; r) =

−1
(2π)2

ei(xξ+yζ)ξζ[F g(∗ ; r)](ξ, ζ) dξ dζ.

R2

Since (iξ)(iζ) · F f (ξ, ζ) = [F ∂x∂yf ](ξ, ζ), we have

(cid:90)(cid:90)

1

(2π)2

R2

∂rg(x, y; r) =

Therefore,

ei(xξ+yζ)[F ∂x∂yg(∗; r)](ξ, ζ) dξ dζ = ∂x∂yg(x, y; r).

(cid:90) ∞

(cid:90) ∞

γ1

γ2

∂rp =

∂x∂yg(x, y; r) dx dy = g(γ1, γ2; r).

The mean value theorem implies that p(r)− p(0) = rg(γ1, γ2; r∗) for some r∗ between 0 and

r, and hence the result.

114

Let σ and ˜σ be the canonical metric on R+ × R for u and ˜u, respectively, i.e.

σ[(t, x), (t(cid:48), x(cid:48))] = E[(u(t, x) − u(t(cid:48), x(cid:48)))2]1/2,

˜σ[(τ, λ), (τ(cid:48), λ(cid:48))] = E[(˜u(τ, λ) − ˜u(τ(cid:48), λ(cid:48)))2]1/2.

For a rectangle I = [a, a(cid:48)] × [−b, b], where 0 < a < a(cid:48) < ∞ and 0 < b < ∞, recall from [20,

Proposition 4.1] that there exist positive finite constants C1 and C2 such that

(cid:0)|t − t(cid:48)| + |x − x(cid:48)|(cid:1)(2−β)/2 ≤ σ[(t, x), (t(cid:48), x(cid:48))] ≤ C2

(cid:0)|t − t(cid:48)| + |x − x(cid:48)|(cid:1)(2−β)/2

C1

for all (t, x), (t(cid:48), x(cid:48)) ∈ I.

The proof of the following lemma is based on the method in [46, 47].

Lemma 6.3.4. Let τ > 0, λ > 0 and q > 1. Then for all 0 < ε < 1,

(cid:18) ˜u(τ, λ + q−n) − ˜u(τ, λ + q−n−1)

˜σ[(τ, λ + q−n), (τ, λ + q−n−1)]

P

(cid:19)
≥ (1 − ε)(cid:112)2 log log qn infinitely often in n

(6.13)

= 1.

(6.14)

Proof. For n ≥ 1, let An = {Zn > γn}, where

and

˜u(τ, λ + q−n) − ˜u(τ, λ + q−n−1)
˜σ[(τ, λ + q−n), (τ, λ + q−n−1)]

Zn =

γn = (1 − ε)(cid:112)2 log log qn.

We will complete the proof by showing that (i) and (ii) of Lemma 6.3.2 are satisfied. For

115

(i), by using the standard estimate

√
P(Z > x) ≥ (2

2π)−1x−1 exp(−x2/2),

x > 1,

(6.15)

for a standard Gaussian random variable Z, we derive that for large n,

P(Zn > γn) ≥

C

n(1−ε)2√

log n

and hence(cid:80)∞

n=1

P(An) = ∞.

Next, we show that (ii) is satisfied. Since

n(cid:88)

n(cid:88)

(cid:2)P(Aj ∩ Ak) − P(Aj)P(Ak)(cid:3) = E

(cid:34)(cid:18) n(cid:88)

− P(Aj)(cid:1)(cid:19)2(cid:35)

(cid:0)1Aj

≥ 0

j=1

k=1

j=1

and(cid:80)∞

n=1

P(An) = ∞, it is enough to prove that

(cid:80)
1≤j<k≤n[P(Aj ∩ Ak) − P(Aj)P(Ak)]

[(cid:80)n

P(Aj)]2

j=1

lim inf
n→∞

≤ 0.

(6.16)

We are going to use Lemma 6.3.3 to estimate the difference in the numerator. First, we esti-
−τ +λ+q−n

mate the correlation rjk between Zj and Zk for j < k. Let (tn, xn) = ( τ +λ+q−n√

√

).

,

2

2

116

Let ∆j = ∆(tj, xj). By (6.5), we have

E(cid:2)(˜u(τ, λ + q−j) − ˜u(τ, λ + q−j−1))(˜u(τ, λ + q−k) − ˜u(τ, λ + q−k−1))(cid:3)

(cid:1)W(cid:0)∆k\∆k+1

(cid:1)(cid:3)

(cid:90)

E(cid:2)W(cid:0)∆j\∆j+1
(cid:90) ∞
(cid:90)
(cid:90) tk+1
(cid:90)
(cid:90) tk

ds

ds

R

0

0

1
4
1
8π

1
8π

=

=

=

+

1
8π

ds

tk+1

Cβdξ
|ξ|1−β
Cβdξ
|ξ|1−β
Cβdξ
|ξ|1−β

R

R

F (1∆j\∆j+1

(s,·))(ξ)F (1∆k\∆k+1

(s,·))(ξ)

F (1[xk+1+tk+1−s, xk+tk−s])(ξ)F (1[xj+1+tj+1−s, xj +tj−s])(ξ)

F (1[xk−tk+s, xk+tk−s])(ξ)F (1[xj+1+tj+1−s, xj +tj−s])(ξ).

Note that (6.5) also implies that this covariance is nonnegative. Then by Lemma 6.2.2,

E(cid:2)(˜u(τ, λ + q−j) − ˜u(τ, λ + q−j−1))(˜u(τ, λ + q−k) − ˜u(τ, λ + q−k−1))(cid:3)

(q−j−1 − q−k)2−β − (q−j−1 − q−k−1)2−β
(cid:104)

+(q−j − q−k−1)2−β − (q−j − q−k)2−β(cid:105)
(q−j−1 − q−k)2−β − (q−j−1 − s)2−β + (q−j − s)2−β − (q−j − q−k)2−β(cid:105)

ds

= Ctk+1

(cid:104)
(cid:90) q−k

+ C

q−k−1

=: J1 + J2.

(6.17)

Let us consider the first term J1. By the mean value theorem, we can find some a and b

such that

q−j−1 − q−k ≤ a ≤ q−j−1 − q−k−1 < q−j − q−k ≤ b ≤ q−j − q−k−1.

117

and

(q−j−1 − q−k)2−β − (q−j−1 − q−k−1)2−β + (q−j − q−k−1)2−β − (q−j − q−k)2−β
= (2 − β)(b1−β − a1−β)(q−k − q−k−1)
≤ (2 − β)[(q−j − q−k−1)1−β − (q−j−1 − q−k)1−β]q−k.

Suppose j ≤ k−2. By the mean value theorem again, we can find some ξ between q−j−q−k−1
and q−j−1 − q−k such that

(q−j − q−k−1)1−β − (q−j−1 − q−k)1−β = (1 − β)ξ−β[(q−j − q−k−1) − (q−j−1 − q−k)]

≤ (1 − β)(q−j−1 − q−k)−βq−j
≤ (1 − β)(q−j−1 − q−j−2)−βq−j
≤ (1 − β)(q−1 − q−2)(q−j)1−β.

It follows that

J1 ≤ C(q−j)1−βq−k.

Next, we consider the term J2 in (6.17). For every s ∈ [q−k−1, q−k], we can find some ˜a

and ˜b (depending on s) such that

q−j−1 − q−k ≤ ˜a ≤ q−j−1 − s < q−j − q−k ≤ ˜b ≤ q−j − s

118

and

(q−j−1 − q−k)2−β − (q−j−1 − s)2−β + (q−j − s)2−β − (q−j − q−k)2−β
= (2 − β)(˜b1−β − ˜a1−β)(q−k − s)
≤ (2 − β)[(q−j − s)1−β − (q−j−1 − q−k)1−β]q−k.

For j ≤ k − 2, by the mean value theorem again, there exists η between q−j − s and
q−j−1 − q−k such that

(q−j − s)1−β − (q−j−1 − q−k)1−β = (1 − β)η−β[(q−j − s) − (q−j−1 − q−k)]

≤ (1 − β)(q−j−1 − q−j−2)−βqj
≤ (1 − β)(q−1 − q−2)(q−j)1−β.

Then we have

J2 = C

(cid:90) q−k

q−k−1

(cid:104)
(q−j−1 − q−k)2−β − (q−j−1 − s)2−β + (q−j − s)2−β − (q−j − q−k)2−β(cid:105)

ds

≤ C(q−j)1−β(q−k)2 ≤ C(q−j)1−βq−k.

Therefore, by combining (6.17), the upper bounds for J1, J2, and recalling (6.13), we see
that for j ≤ k − 2, the correlation rjk between Zj and Zk satisfies

0 ≤ rjk = E(ZjZk) ≤

C(q−j)1−βq−k

(q−j)1−β/2(q−k)1−β/2

= C0(q−(k−j))β/2 =: ξjk.

(6.18)

By (6.18), we can choose a fixed l ≥ 2 such that r := sup{rjk : j ≤ k − l} < 1. Since

119

(cid:80)∞

n=1

P(An) = ∞, in order to prove (6.16), it suffices to prove that for any δ > 0, there

exists m such that

(cid:80)n

k=m

lim inf
n→∞

(cid:80)k−l
[(cid:80)n
j=1[P(Aj ∩ Ak) − P(Aj)P(Ak)]

P(Aj)]2

j=1

≤ δ.

(6.19)

Let δ > 0 be given and let m be a large integer that will be chosen appropriately depending

on δ. Let ρk =

(β/2) log q log γk, so that for 1 ≤ j ≤ k − ρk,

4

ξjk ≤ C0γ−4
k .

(6.20)

Provided m is large, 1 < k − ρk < k − l for all k ≥ m. By Lemma 6.3.3, we have
n(cid:88)

(cid:98)k−ρk(cid:99)(cid:88)

k−l(cid:88)

n(cid:88)

k−l(cid:88)

[P(Aj ∩ Ak) − P(Aj)P(Ak)] ≤

 rjkg(γj, γk; r∗

 n(cid:88)

+

k=m

j=1

k=m

j=1

k=m

j=(cid:98)k−ρk(cid:99)

jk),

(6.21)

where r∗

jk is a number such that 0 ≤ r∗

jk ≤ rjk for each j, k. Let us consider the two sums

on the right-hand side of (6.21) separately. By (6.18), the first sum is

n(cid:88)
(cid:98)k−ρk(cid:99)(cid:88)
(cid:98)k−ρk(cid:99)(cid:88)
≤ n(cid:88)

j=1

k=m

k=m

j=1

rjk
2π(1 − r∗2

jk)1/2

exp

ξjkγjγk
2π(1 − ξ2

jk)1/2

exp

(cid:33)

j + γ2

(cid:32)
−γ2
(cid:32)−r∗2

k − 2r∗
jkγjγk
2(1 − r∗2
jk)
k) + 2r∗
j + γ2
2(1 − r∗2
jk)

jk(γ2

(cid:33)

−γ2

j /2

γ−1
j e

γ−1
k e

−γ2

k/2.

jkγjγk

Note that γj < γk for j < k. Then by (6.15), (6.18) and (6.20), the sum is

(cid:33)

(cid:32)

C0γ−2
0 γ−8
1 − C2

k

k

P(Aj)P(Ak).

n(cid:88)

(cid:98)k−ρk(cid:99)(cid:88)

k=m

j=1

≤ 4

C0γ−2
k
0 γ−8
k )1/2

(1 − C2

exp

120

Since γk → ∞, we may choose m to be large enough such that this sum is ≤ δ[(cid:80)n

j=1

P(Aj)]2.

By (6.15), the second sum on the right-hand side of (6.21) is

j=(cid:98)k−ρk(cid:99)

k−l(cid:88)
k−l(cid:88)
n(cid:88)

n(cid:88)
≤ n(cid:88)

k=m

k=m

≤ 2√
2π

j=(cid:98)k−ρk(cid:99)

k−l(cid:88)

k=m

j=(cid:98)k−ρk(cid:99)

rjk
2π(1 − r∗2

jk)1/2

exp

rjkγj
2π(1 − r∗2

jk)1/2

exp

γk
(1 − r∗2
jk)1/2

(cid:33)

j + γ2

(cid:32)
k − 2r∗
−γ2
jkγjγk
2(1 − r∗2
(cid:32)
jk)
−(γk − r∗
jkγj)2
2(1 − r∗2
(cid:32)
jk)
−(1 − r∗
jk)2γ2
k
2(1 − r∗2
jk)

exp

(cid:33)
γ−1
(cid:33)
j e

−γ2

j /2

P(Aj).

Recall that r = sup{rjk : j ≤ k − l} < 1. Moreover, if m is large enough, then

(cid:32)

(cid:33)

γk log γk
(1 − r2)1/2

exp

−(1 − r)γ2

k
2(1 + r)

≤ δ

and k − ρk > k/2 for all k ≥ m, so that the last sum above is
(cid:33)

(cid:32)

≤ 2√
2π

≤ C

n(cid:88)
n(cid:88)
n(cid:88)

k=m

k=m

≤ 2Cδ

P(Ak).

ρkγk

(1 − r2)1/2

γk log γk
(1 − r2)1/2

exp

exp

k
2(1 + r)

−(1 − r)γ2
(cid:33)

(cid:32)
−(1 − r)γ2

k
2(1 + r)

P(A(cid:98)k−ρk(cid:99))

P(A(cid:98)k/2(cid:99))

k=1

We get that

n(cid:88)

k−l(cid:88)

[P(Aj ∩ Ak) − P(Aj)P(Ak)] ≤ δ

k=m

j=1

(cid:33)2

P(Aj)

(cid:32) n(cid:88)

j=1

n(cid:88)

j=1

P(Aj).

+ 2Cδ

Hence (6.19) follows and the proof of Lemma 6.3.4 is complete.

121

We now come to the proof of Proposition 6.3.1.

Proof of Proposition 6.3.1. Fix λ > 0. It suffices to show that for any 0 ≤ a < b < ∞ and

0 < ε < 1,

lim sup

h→0+

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
(τ + λ)h2−β log log(1/h)

≥ (1 − ε)Kβ for all τ ∈ [a, b]

 = 1.

(6.22)

P

To this end, let us fix a, b and ε for the rest of the proof.

we can choose and fix a large q > 1 such that

Note that when q is large, q−(2−β)/2(1 + q−n−1
τ +λ )1/2 < ε/4 uniformly for all τ ∈ [a, b]. So
(cid:18)

(cid:19)(2−β)/2 − q−(2−β)/2

(cid:19)1/2 − (1 − ε) > ε/4

(1 − ε/4)

1 +

(6.23)

(6.24)

(cid:18)q − 1

q

q−n−1
τ + λ

for all τ ∈ [a, b]. We also choose δ > 0 small such that

λ(ε/4)2

δ

> 1.

Since we can cover [a, b] by finitely many intervals [c, d] of length δ, we only need to show
(6.22) for τ ∈ [c, d], where [c, d] ⊂ [a, b] and d = c + δ.

Let us define the increment of ˜u over a rectangle (τ, τ(cid:48)] × (λ, λ(cid:48)] by

∆˜u((τ, τ(cid:48)] × (λ, λ(cid:48)]) = ˜u(τ(cid:48), λ(cid:48)) − ˜u(τ, λ(cid:48)) − ˜u(τ(cid:48), λ) + ˜u(τ, λ).

122

Then for all τ ∈ [c, d] we can write

˜u(τ, λ + q−n) − ˜u(τ, λ) = ˜u(d, λ + q−n) − ˜u(d, λ + q−n−1)

+ ˜u(τ, λ + q−n−1) − ˜u(τ, λ)
− ∆˜u((τ, d] × (λ + q−n−1, λ + q−n]).

(6.25)

By Lemma 6.3.4, we have

|˜u(d, λ + q−n) − ˜u(d, λ + q−n−1)|
˜σ[(d, λ + q−n), (d, λ + q−n−1)]

≥ (1 − ε/4)(cid:112)2 log log qn

infinitely often in n with probability 1. By Lemma 6.2.3,

˜σ[(d, λ + q−n), (d, λ + q−n−1)]

(cid:113)

(d + λ + q−n−1)(q−n − q−n−1)2−β + (3 − β)−1(q−n − q−n−1)3−β,

=

Kβ√
2

so we have

|˜u(d, λ+q−n)− ˜u(d, λ+q−n−1)| ≥ (1−ε/4)Kβ

(cid:113)

(d + λ)(q−n − q−n−1)2−β log log qn (6.26)

infinitely often in n with probability 1. Also, by Proposition 6.2.1, with probability 1, for all
τ ∈ [c, d] simultaneously,

|˜u(τ, λ + q−n−1) − ˜u(τ, λ)| ≤ Kβ

(τ + λ + q−n−1)(q−n−1)2−β log log qn

(6.27)

(cid:113)

eventually for all large n.

Next, we derive a bound for the term ∆˜u((τ, d] × (λ + q−n−1, λ + q−n]). For τ ∈ [c, d],

123

let

φ(τ ) = (1− ε/4)

(cid:18) q − 1

(cid:19)(2−β)/2

q

(d + λ)1/2− q−(2−β)/2(τ + λ + q−n−1)1/2− (1− ε)(τ + λ)1/2.

Consider the events

(cid:40)

An =

sup
τ∈[c,d]

where

|∆˜u((τ, d] × (λ + q−n−1, λ + q−n])| > γn

(cid:41)

,

(cid:113)

γn = Kβ φ(d)

(q−n)2−β log log qn.

Note that ∆˜u((τ, d]× (λ + q−n−1, λ + q−n]) = 1
(τ, d] × (λ + q−n−1, λ + q−n] under the rotation (τ, λ) (cid:55)→ ( τ +λ√
we have Q = Q1 ∪ Q2 ∪ Q3, where

2

2 W (Q), where Q is the image of the rectangle

, −τ +λ√

2

). Provided n is large,

(cid:110)
(cid:110)

Q1 =

Q2 =

(t, x) :

(t, x) :

τ + λ + q−n−1

√

2

< t ≤ τ + λ + q−n

√

2

√

√
2(λ + q−n−1) − s < x < −

,

2τ + s

τ + λ + q−n

√

2

< t ≤ d + λ + q−n−1

,

√
√

2
2(λ + q−n−1) − s < x ≤

√

2(λ + q−n) − s

(cid:111)

,

,

(cid:111)
(cid:111)

2(λ + q−n) − s

.

(cid:110)

(t, x) :

Q3 =

d + λ + q−n−1

√

2

< t ≤ d + λ + q−n

√

2

√
,−

2d + s ≤ x ≤

√

124

By (6.5), it follows that

=

2
√

1
8π

τ +λ+q−n−1

E(cid:2)(∆˜u((τ, d] × (λ + q−n−1, λ + q−n]))2(cid:3) =
(cid:12)(cid:12)(cid:98)1[
(cid:12)(cid:12)(cid:98)1[
(cid:12)(cid:12)(cid:98)1[−√

(cid:40)(cid:90) τ +λ+q−n√
(cid:90) d+λ+q−n−1
2
√
τ +λ+q−n√
(cid:90) d+λ+q−n√

(cid:90) ∞
(cid:90) ∞
(cid:90) ∞

Cβdξ
|ξ|1−β

Cβdξ
|ξ|1−β

d+λ+q−n−1

Cβdξ
|ξ|1−β

−∞

−∞

√

√

ds

ds

ds

+

+

2

−∞

2

2
√

2

1
4

E(cid:2)W (Q)2(cid:3)
2(λ+q−n−1)−s,−√

2τ +s](ξ)(cid:12)(cid:12)2
2(λ+q−n)−s](ξ)(cid:12)(cid:12)2

2(λ+q−n−1)−s,

√

2(λ+q−n)−s](ξ)(cid:12)(cid:12)2

√

(cid:41)

.

2d+s,

Then by Lemma 6.2.2,

1

=

2(2 − β)(1 − β)

(cid:40)(cid:90) τ +λ+q−n√

E(cid:2)(cid:0)∆˜u((τ, d] × (λ + q−n−1, λ + q−n])(cid:1)2(cid:3)
2(τ + λ + q−n−1)(cid:1)2−βds
(cid:0)2s −
(cid:90) d+λ+q−n−1
2(q−n − q−n−1)(cid:1)2−βds
(cid:0)√
√
τ +λ+q−n√
(cid:41)
(cid:90) d+λ+q−n√
2(d + λ + q−n) − 2s(cid:1)2−βds
(cid:0)√

τ +λ+q−n−1

2
√

√

+

+

2

2

2

=

1

2(2 − β)(1 − β)

2 · 2

(cid:40)

2

2
√

d+λ+q−n−1
1−β
2
3 − β
1−β

+ 2

=

1
2

β(q−n − q−n−1)2−β
K2

(q−n − q−n−1)3−β

2 (q−n − q−n−1)2−β(cid:0)d − τ − (q−n − q−n−1)(cid:1)(cid:41)
(cid:26)
(d − τ ) − 1 − β
3 − β

(q−n − q−n−1)(cid:1)(cid:27)

.

125

Since d − τ ≤ d − c = δ, we have

E(cid:2)(∆˜u((τ, d] × (λ + q−n−1, λ + q−n]))2(cid:3) ≤ 1

2

sup
τ∈[c,d]

β(q−n − q−n−1)2−βδ.
K2

By (6.10), for all large n,

log P(An) ≤ −

1
γ2
n

1

β(q−n − q−n−1)2−βδ
K2

.

It follows that

(cid:18)

P(An) ≤ exp

where

(cid:19)

= (n log q)p,

− φ(d)2(q−n)2−β log log qn
(q−n − q−n−1)2−βδ

(cid:19)2−β

(cid:18) q

q − 1

p =

1
δ

φ(d)2.

By (6.23) and (6.24),

p ≥ d + λ

δ

(1 − ε/4)

(cid:20)

(cid:18) q − 1

q

(cid:19)(2−β)/2 − q−(2−β)/2

(cid:18)

1 +

(cid:19)1/2 − (1 − ε)
(cid:21)2

q−n−1
d + λ

>

λ(ε/4)2

δ

> 1.

Hence P(An i.o.) = 0 by the Borel–Cantelli lemma. Then the symmetry of u and the

monotonic decreasing property of φ imply that with probability 1, simultaneously for all
τ ∈ [c, d],

(cid:113)
(cid:12)(cid:12)∆˜u((τ, d] × (λ + q−n−1, λ + q−n])(cid:12)(cid:12) ≤ Kβφ(τ )

(q−n)2−β log log qn

(6.28)

126

eventually for all large n. By (6.25) and the triangle inequality,

(cid:12)(cid:12)˜u(τ, λ + q−n) − ˜u(τ, λ)(cid:12)(cid:12) ≥(cid:12)(cid:12)˜u(d, λ + q−n) − ˜u(d, λ + q−n−1)(cid:12)(cid:12)
−(cid:12)(cid:12)˜u(τ, λ + q−n−1) − ˜u(τ, λ)(cid:12)(cid:12)
−(cid:12)(cid:12)∆˜u((τ, d] × (λ + q−n−1, λ + q−n])(cid:12)(cid:12).

Then (6.26), (6.27) and (6.28) together imply that with probability 1, for all τ ∈ [c, d]

simultaneously,

(cid:12)(cid:12)˜u(τ, λ + q−n) − ˜u(τ, λ)(cid:12)(cid:12)
(cid:18) q − 1
(cid:20)
(cid:19)(2−β)/2
(cid:113)
q
(cid:113)
(q−n)2−β log log qn
× Kβ
(τ + λ)(q−n)2−β log log qn

(1 − ε/4)

(cid:21)
(d + λ)1/2 − q−(2−β)/2(τ + λ + q−n−1)1/2 − φ(τ )

≥

≥ (1 − ε)Kβ

infinitely often in n. This yields (6.22) for τ ∈ [c, d] and concludes the proof of Proposition

6.3.1.

6.4 Singularities and Their Propagation

In this section, we study the existence and propagation of singularities of the stochastic wave

equation (6.1). The main result is Theorem 6.4.3.

Let us first discuss the interpretation of singularities and how they may arise. Proposition

127

6.2.1 and 6.3.1 imply that LIL holds at any fixed point (t, x):

|u(t + h√

(cid:113)

lim sup
h→0+

, x + h√
2

2

) − u(t, x)|

h2−β log log(1/h)

√
= Kβ(

2t)1/2

a.s.

It indicates the size of oscillation of u when (t, x) is fixed. However, the behavior will be

different when (t, x) is not fixed. Indeed, from the modulus of continuity in Theorem 5.3.1,
we know that for I = [a, a(cid:48)] × [−b, b], where 0 < a < a(cid:48) and b > 0, there exists a positive

finite constant K such that

σ[(t, x), (t(cid:48), x(cid:48))](cid:112)log(1 + σ[(t, x), (t(cid:48), x(cid:48))]−1)

|u(t(cid:48), x(cid:48)) − u(t, x)|

= K a.s.

lim
h→0+

sup

(t,x),(t(cid:48),x(cid:48))∈I:

0<σ[(t,x),(t(cid:48),x(cid:48))]≤h

(cid:113)

(cid:113)

Recalling (6.13), this result shows that the largest oscillation in I is of order

h2−β log(1/h),
h2−β log log(1/h) specified by the LIL. It suggests that the LIL does
which is larger than
not hold simultaneously for all (t, x) ∈ I and there may exist random exceptional points

with much larger oscillation. Therefore, we can define singularities as such points where the

LIL fails. More precisely, we say that (τ, λ) is a singular point of ˜u in the λ-direction if

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
h2−β log log(1/h)

= ∞

lim sup
h→0+

and a singular point in the τ -direction if

(cid:113)

|˜u(τ + h, λ) − ˜u(τ, λ)|
h2−β log log(1/h)

= ∞.

lim sup
h→0+

Our goal is to justify the existence of random singular points and study their propagation.

128

Fix τ0 > 0. Let us decompose ˜u into ˜u1 + ˜u2, where

(cid:16) τ + λ√

2

−τ + λ√

2

,

(cid:17)

,

i = 1, 2,

˜ui(τ, λ) = ui

and

(cid:16)
(cid:16)

u1(t, x) =

∆(t, x) ∩(cid:8)0 ≤ t < τ0/
2(cid:9)(cid:17)
∆(t, x) ∩(cid:8)t ≥ τ0/
Let Fτ0 be the σ-field generated by {W(cid:0)B ∩ {0 ≤ t < τ0/

u2(t, x) =

W

W

1
2
1
2

√

.

√

2(cid:9)(cid:17)

,

√

2}(cid:1) : B ∈ Bb(R2)} and the

P-null sets. Note that Fτ0 is independent of the process ˜u2.

Following the approach of Walsh [63] and Blath and Martin [7], we will use Meyer’s

section theorem to prove the existence of a random singularity. Let us recall Meyer’s section

theorem ([21], Theorem 37, p.18):

Let (Ω, G , P) be a complete probability space and S be a B(R+) × G -measurable subset
of R+ × Ω. Then there exists a G -measurable random variable T with values in (0,∞] such

that

(a) the graph of T, denoted by [T ] := {(t, ω) ∈ R+ × Ω : T (ω) = t}, is contained in S;

(b) {T < ∞} is equal to the projection π(S) of S onto Ω.

Lemma 6.4.1. Let τ0 > 0. Then there exists a positive, finite, Fτ0-measurable random

variable Λ such that

(cid:113)
|˜u1(τ0, Λ + h) − ˜u1(τ0, Λ)|

h2−β log log(1/h)

lim sup
h→0+

= ∞ a.s.

129

Proof. Note that

(cid:113)
|˜u1(τ0, Λ + h) − ˜u1(τ0, Λ)|

h2−β log log(1/h)

lim sup
h→0+

(cid:113)
|˜v1(τ0, Λ + h) − ˜v1(τ0, Λ)|

h2−β log log(1/h)

,

= lim sup
h→0+

where ˜v1(τ0, λ) = ˜u1(τ0, λ) − ˜u1(τ0, 0). The covariance for the process {˜v1(τ0, λ) : λ ≥ 0} is

E[˜v1(τ0, λ)˜v1(τ0, λ(cid:48))] =

1
4

E[W (Aλ)W (Aλ(cid:48))]

√
2,−t < x ≤ √
for λ, λ(cid:48) ≥ 0, where Aλ = {(t, x) : 0 ≤ t < τ0/

2λ − t}. By (6.5) and Lemma

6.2.2,

=

ds

−∞

√

2

1
8π

(cid:90) τ0/

E[˜v1(τ0, λ)˜v1(τ0, λ(cid:48))]
(cid:90) ∞
(cid:90) τ0/
(cid:16)|λ|2−β + |λ(cid:48)|2−β − |λ − λ(cid:48)|2−β(cid:17)

4(2 − β)(1 − β)
2−(3+β)/2 τ0
(2 − β)(1 − β)

F 1[−s,
√

2λ−s](ξ)F 1[−s,
2λ(cid:48)|2−β − |

√
2λ|2−β + |

Cβdξ
|ξ|1−β
√

2

(cid:16)|

=

=

√

0

1

0

√
2λ(cid:48)−s](ξ)
√
√
2λ −

2λ(cid:48)|2−β(cid:17)

ds

.

It follows that {C0˜v1(τ0, λ) : λ ≥ 0} is a fractional Brownian motion of Hurst parameter
(2 − β)/2 for some constant C0 depending on τ0 and β.

Let

S =

(cid:40)

(λ, ω) ∈ R+ × Ω : lim sup
h→0+

(cid:113)

|˜v1(τ0, λ + h)(ω) − ˜v1(τ0, λ)(ω)|

h2−β log log(1/h)

(cid:41)

= ∞

.

Then S is B(R+) × Fτ0-measurable. Using Meyer’s section theorem, we can find a positive
Fτ0-measurable random variable Λ such that (a) [Λ] ⊂ S, and (b) π(S) = {Λ < ∞}.

130

We claim that Λ < ∞ a.s. Indeed, by the modulus of continuity for fractional Brownian

motion (cf. [30], Theorem 1.1), for any 0 ≤ a < b,

(cid:113)

|˜v1(τ0, λ + h) − ˜v1(τ0, λ)|

h2−β log(1/h)

lim sup
h→0+

sup
λ∈[a,b]

√
2 a.s.

= C−1

0

(6.29)

We now use an argument with nested intervals (cf. [48], Theorem 1) to show the existence
of a random λ∗ such that

(cid:113)
|˜v1(τ0, λ∗ + h) − ˜v1(τ0, λ∗)|

h2−β log log(1/h)

= ∞

lim sup
h→0+

(6.30)

with probability 1. First take an event Ω∗ of probability 1 such that (6.29) holds for all

intervals [a, b], where a and b are rational numbers. Let

(cid:113)

ϕ(h) =

C−1

0

1
2

2h2−β log(1/h).

Let h0 > 0 be small such that ϕ is increasing on [0, h0]. For an ω ∈ Ω∗, we define two
1, say in [1, 2], with λ1 < λ(cid:48)
sequences (λn), (λ(cid:48)
such that λ(cid:48)

n) as follows. By (6.29), we can choose λ1, λ(cid:48)

1 − λ1 < h0 and

1

|˜v1(τ0, λ(cid:48)

1) − ˜v1(τ0, λ1)| > ϕ(λ(cid:48)

1 − λ1).

Suppose n ≥ 1 and λn and λ(cid:48)

n are chosen with λn < λ(cid:48)

n and

|˜v1(τ0, λ(cid:48)

n) − ˜v1(τ0, λn)| > ϕ(λ(cid:48)

n − λn).

131

Since ˜v1 is continuous and ϕ(h) is increasing for h small, we can find some ˜λn such that
λn < ˜λn < min{λ(cid:48)

n, λn + 2−n} and

|˜v1(τ0, λ(cid:48)

n) − ˜v1(τ0, λ)| > ϕ(λ(cid:48)

n − λ)

for all λ ∈ [λn, ˜λn].

(6.31)

Then we can apply (6.29) for a rational interval [a, b] ⊆ [λn, ˜λn] to find λn+1 and λ(cid:48)
that λn ≤ λn+1 < λ(cid:48)

n+1 ≤ ˜λn and

n+1 such

|˜v1(τ0, λ(cid:48)

n+1) − ˜v1(τ0, λn+1)| > ϕ(λ(cid:48)

n+1 − λn+1).

We obtain a sequence of nested intervals [λ1, λ(cid:48)
2] ⊃ ··· with lengths λ(cid:48)
2−n+1. Therefore, the intervals contain a common point λ∗ ∈ [1, 2] such that λ(cid:48)
Since λ∗ ∈ [λn, ˜λn] for all n, by (6.31) we have

1] ⊃ [λ2, λ(cid:48)

n − λn ≤
n+1 ↓ λ∗.

|˜v1(τ0, λ(cid:48)

n) − ˜v1(τ0, λ∗)| > ϕ(λ(cid:48)

n − λ∗).

Hence, for each ω ∈ Ω∗, there is at least one λ∗ > 0 (depending on ω) such that (6.30) holds.
It implies that Ω∗ ⊂ π(S). Then from (b) we deduce that Λ < ∞ a.s., and from (a) we

conclude that

(cid:113)
|˜v1(τ0, Λ + h) − ˜v1(τ0, Λ)|

h2−β log log(1/h)

lim sup
h→0+

= ∞ a.s.

The proof of Lemma 6.4.1 is complete.

132

Lemma 6.4.2. For any τ0 > 0 and λ > 0,

lim sup

h→0+

P

(cid:113)
|˜u2(τ, λ + h) − ˜u2(τ, λ)|
h2−β log log(1/h)

= Kβ(τ − τ0 + λ)1/2 for all τ ≥ τ0

 = 1.

Proof. By Proposition 6.2.1 and 6.3.1,

lim sup

h→0+

P

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
h2−β log log(1/h)

= Kβ(τ + λ)1/2 for all τ ≥ 0

 = 1.

Then the result can be obtained by the observation that {˜u2(τ0 + τ, λ) : τ, λ ≥ 0} has the
same distribution as {˜u(τ, λ) : τ, λ ≥ 0}. Indeed, for any bounded Borel sets A, B in R+ × R
and c = (c1, c2) ∈ R+ × R, by (6.5) and change of variables we have

c1

0

=

=

R

R

ds

dy

(cid:90)
(cid:90)

E(cid:2)W (A + c)W (B + c)(cid:3)
(cid:90)
(cid:90) ∞
(cid:90)
(cid:90) ∞
= E(cid:2)W (A)W (B)(cid:3).
(cid:16)τ0 + τ + λ√

√

dy

R

R

ds

,

2

2

−τ0 − τ + λ

Since

∆

dy(cid:48) 1A(s − c1, y − c2)|y − y(cid:48)|−β1B(s − c1, y − c2)
dy(cid:48) 1A(s, y)|y − y(cid:48)|−β1B(s, y)

(cid:17) ∩(cid:8)t ≥ τ0/

2(cid:9) = ∆

(cid:16) τ + λ√

√

2

(cid:17)

+ c,

−τ + λ√

2

,

133

where c = (

τ0√
2

2

,− τ0√

), it follows that for any τ, λ, τ(cid:48), λ(cid:48) ≥ 0,

=

E

(cid:20)

E(cid:2)˜u2(τ0 + τ, λ)˜u2(τ0 + τ(cid:48), λ(cid:48))(cid:3)
(cid:16)τ0 + τ + λ√
(cid:16)τ0 + τ(cid:48) + λ(cid:48)
(cid:16)τ + λ√
−τ + λ√

(cid:16)
(cid:16)
(cid:16)

2
√

× W

(cid:20)

1
4

W

E

∆

W

∆

∆

2

,

=

,

1
4

√

(cid:17) ∩(cid:8)t ≥ τ0/
2(cid:9)(cid:17)
2(cid:9)(cid:17)(cid:21)
(cid:17) ∩(cid:8)t ≥ τ0/
(cid:17)(cid:17)(cid:21)
(cid:16)τ(cid:48) + λ(cid:48)
−τ(cid:48) + λ√

√

√

,

√

−τ0 − τ + λ
−τ0 − τ(cid:48) + λ(cid:48)
(cid:16)
(cid:17)(cid:17)

2
√

2

,

W

∆

2

2

2

2
= E[˜u(τ, λ)˜u(τ(cid:48), λ(cid:48))].

The result follows immediately.

We are now in a position to state and prove our main theorem. The first part of the

theorem justifies the existence of a random singularity. It shows that if we fix τ0 > 0, then

based on the information from the σ-field Fτ0, we can actually find a random variable Λ

such that (τ0, Λ) is a singularity in the λ-direction. The second part says that if (τ0, Λ) is
a singularity in the λ-direction, then (τ, Λ) is also a singularity for all τ ≥ τ0.

In other

words, singularities in the λ-direction propagate orthogonally, towards the τ -direction. By

a symmetric argument, one can show that singularities in the τ -direction propagate towards
the λ-direction. These are the directions of the characteristic curves t + x = c and t− x = c.

Theorem 6.4.3. Let τ0 > 0. The following statements hold.

(i) There exists a positive, finite, Fτ0-measurable random variable Λ such that

(cid:113)
|˜u(τ0, Λ + h) − ˜u(τ0, Λ)|
h2−β log log(1/h)

lim sup
h→0+

= ∞ a.s.

134

(ii) If Λ is any positive, finite, Fτ0-measurable random variable, then on an event of prob-

ability 1, we have

(cid:113)
|˜u(τ0, Λ + h) − ˜u(τ0, Λ)|
h2−β log log(1/h)

lim sup
h→0+

= ∞ ⇔ lim sup
h→0+

(cid:113)

|˜u(τ, Λ + h) − ˜u(τ, Λ)|
h2−β log log(1/h)

= ∞

for all τ > τ0 simultaneously.

Proof. To simplify notations, let

and

L(τ, λ) = lim sup
h→0+

(cid:113)

|˜u(τ, λ + h) − ˜u(τ, λ)|
h2−β log log(1/h)

Li(τ, λ) = lim sup
h→0+

(cid:113)
|˜ui(τ, λ + h) − ˜ui(τ, λ)|
h2−β log log(1/h)

for i = 1, 2. As in [63, 7], we will use the property that for any two functions f and g,

lim sup
h→0

|f (h)|− lim sup
h→0

|g(h)| ≤ lim sup
h→0

|f (h) + g(h)| ≤ lim sup
h→0

|f (h)| + lim sup
h→0

|g(h)| (6.32)

provided that lim suph→0 |g(h)| < ∞.

(i). By Lemma 6.4.1, we can find a positive, finite, Fτ0-measurable random variable Λ

such that

L1(τ0, Λ) = ∞ a.s.

Since Λ is independent of the process ˜u2, Lemma 6.4.2 implies that

L2(τ0, Λ) = KβΛ1/2

135

which is finite a.s. Since ˜u = ˜u1 + ˜u2, it follows from the lower bound of (6.32) that

L(τ0, Λ) ≥ L1(τ0, Λ) − L2(τ0, Λ) = ∞ a.s.

This proves (i).

(ii). Suppose Λ is a positive, finite, Fτ0-measurable random variable. By (6.32), we have

L1(τ, Λ) − L2(τ, Λ) ≤ L(τ, Λ) ≤ L1(τ, Λ) + L2(τ, Λ)

(6.33)

for all τ ≥ τ0, provided that L2(τ, Λ) < ∞. Note that for τ ≥ τ0,

˜u1(τ, Λ + h) − ˜u1(τ, Λ) = ˜u1(τ0, Λ + h) − ˜u1(τ0, Λ),

hence L1(τ, Λ) = L1(τ0, Λ). Also, by Lemma 6.4.2 and independence between Λ and ˜u2, we

L2(τ, Λ) = Kβ(τ − τ0 + Λ)1/2 for all τ ≥ τ0

= 1.

(cid:17)

Since Λ is finite a.s., it follows from (6.33) that

P(cid:16)

have

P(cid:16)

L1(τ0, Λ) − Kβ(τ − τ0 + Λ)1/2 ≤ L(τ, Λ) ≤ L1(τ0, Λ) + Kβ(τ − τ0 + Λ)1/2 for all τ ≥ τ0

(cid:17)

= 1,

and it implies

P(cid:16)

L(τ0, Λ) = ∞ ⇔ L(τ, Λ) = ∞ for all τ ≥ τ0

(cid:17)

= 1.

136

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