Schramm-Loewner evolution (SLE for short) is a one-parameter (κ ∈ (0, 8)) family of randomfractal curves which grow in plane domains. For an SLEκ curve γ growing in a domain D, and a single point z0 ∈ D or ∈ ∂D, the Green's function for γ at z0 is the limitG(z) := lim r→0+ r−αP[dist(z0, γ) ≤ r]for some suitable exponent α > 0, provided that the limit exists and is not trivial. TheGreen's function for SLE plays an central role in determining the Hausdorff dimension of SLE, and proving the existence of Minkowski content of SLE.The notion of (one-point) Green's function easily extends to multi-point Green's function.Given n distinct points z1, . . . , zn ∈ D or ∈ ∂D, the n-point Green's function for the SLEκ curve γ at (z1, . . . , zn) is the limitG(z1, . . . , zn) := lim r1,...,rn→0+Yn j=1 r−α j P[dist(zj , γ) ≤ rj , 1 ≤ j ≤ n].In the thesis, we prove that the n-point Green's function exists if γ is a chordal SLE,κ ∈ (0, 8), α = 8/κ −1, z1, . . . , zn ∈ ∂D, and ∂D is smooth near each zj . In addition, we prove that the convergence is uniform over compact sets and the Green's function is continuous. We also give up-to-constant bounds for the Green's function.
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