Wong–Zakai approximation and support theorem for semilinear stochastic partial differential equations with finite dimensional noise in the whole space

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’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.