On The Wong-Zakai And Support Theorems For Stochastic Partial Differential Equations

Abstract

The purpose of this thesis is to demonstrate how to obtain convergence results of Wong-Zakai type by using the $L_p$-theory of SPDE. Let $m$ be a positive integer, $\{w^k, k = 1, \ldots, m\}$ be a sequence of independent Wiener processes, and $W$ be a cylindrical Wiener process on $L_2 (\mathbb{R})$. We consider two parabolic stochastic partial differential equations (SPDEs) on the whole space: \begin{align*} du (t, x)& = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum_{k = 1}^m g^k (u(t, x)) dw^k (t), x \in \mathbb{R}^d,\\ % % dv (t, x)& = [a (t, x) D^2_{x} v (t, x) + b (t, x) D_x v (t, x) + f (v, t, x)] dt, \\ % & + v (t, x) dW(t), x \in \mathbb{R}. \end{align*} For the first equation we prove the convergence of a Wong-Zakai type approximation scheme in some H\"older-Bessel space in probability. We also prove a Stroock-Varadhan's type support theorem in H\"older-Bessel space for both equations. To prove the results we combine V. Mackevi\v cius's ideas from his papers on Wong-Zakai theorem and the support theorem for diffusion processes with N.V. Krylov's $L_p$-theory of SPDEs.