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

2018/08/22 by Timur Yastrzhembskiy, Yastrzhembskiy, Timur
Economics, Econometrics and Finance · Engineering · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1808.07584

openalex publication_date 2018/08/22 · openalex created_date 2018/08/31 · openalex updated_date 2026/07/28

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 + ∑_{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^θ ([0, T], H^γ_p (ℝ^d)) in probability, for some θ∈ (0,1/2), γ∈ (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.

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