Wong-Zakai approximation and support theorem for semilinear SPDEs with finite dimensional noise in the whole space

TL;DR

This paper establishes convergence of Wong-Zakai approximations and a support theorem for semilinear SPDEs with finite-dimensional noise in the whole space, extending existing theories to more general settings.

Contribution

It introduces a Wong-Zakai approximation scheme and proves a support theorem for semilinear SPDEs with finite-dimensional noise in the whole space.

Findings

Wong-Zakai approximation converges in probability in specified function spaces.

Support theorem analogous to Stroock-Varadhan's for these SPDEs.

Combines ideas from Mackevicius and Krylov to achieve results.

Abstract

In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: $du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum_{k = 1}^m g^k (u(t, x)) dw^k (t).$ We prove the convergence of a Wong-Zakai type approximation scheme of the above equation in the space $ C^{\theta } ([0, T], H^{\gamma}_p (\mathbb{R}^d)) $ in probability, for some $ \theta \in (0,1/2), \gamma \in (1, 2)$, and $p > 2$. We also prove a Stroock-Varadhan's type support theorem. To prove the results we combine V. Mackevicius 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.