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 important 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.
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