Selected topics in Dirichlet problems for linear parabolic stochastic partial differential equations

This thesis is devoted to the study of Dirichlet problems for some linear parabolic SPDEs. Our aim in it is twofold. First, we consider SPDEs with deterministic coefficients which are smooth up to some order of regularity. We establish some theoretical results in terms of existence, uniqueness and regularity of the classical solution to the considered problem. Then, we provide the probabilistic representations (the averaging-over-characteristic formulas of its solution. We, thereafter, construct numerical methods for it. The methods are based on the averaging-over-characteristic formula and the weak-sense numerical integration of ordinary stochastic differential equations in bounded domains. Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. The Monte Carlo technique is used for practical realization of the methods. Results of some numerical experiments are presented. These results are in agreement with the theoretical findings. Second, we construct the solution of a class of one dimensional stochastic linear heat equations with drift in the first Wiener chaos, deterministic initial condition and which are driven by a space-time white noise and the white noise. This is done by giving explicitly its Wiener chaos decomposition. We also prove its uniqueness in the weak sense. Then we use the chaos expansion in order to show that the unique weak solution is an analytic functional with finite moments of all orders. The chaos decomposition is also utilized as a very useful tool for obtaining a continuity property of the solution.