Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise

Raphael Kruse,Stig Larsson

doi:10.1214/ejp.v17-2240

Abstract

This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic parabolic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions and certain linear growth bounds. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is elementary and makes use of existing results on the regularity of the solution, in particular, the Hölder continuity with a non-optimal exponent.

Highlights

Consider the following semilinear stochastic partial differential equation (SPDE) (1.1)dX(t) + [AX(t) + F (X(t))] dt = G(X(t)) dW (t), for 0 ≤ t ≤ T, X(0) = X0, where the mild solution X takes values in a Hilbert space H
Our regularity result for the mild solution of (1.1) coincides with the optimal regularity property of the stochastic convolution but with the restriction r < 1
We study the regularity properties of a stochastic process X : [0, T ] × Ω → H, T > 0, which is the mild solution to the stochastic partial differential equation (1.1)

Summary

Introduction

Consider the following semilinear stochastic partial differential equation (SPDE). dX(t) + [AX(t) + F (X(t))] dt = G(X(t)) dW (t), for 0 ≤ t ≤ T, X(0) = X0, where the mild solution X takes values in a Hilbert space H. SPDE, Holder continuity, temporal and spatial regularity, multiplicative noise, Lipschitz nonlinearities. Our regularity result for the mild solution of (1.1) coincides with the optimal regularity property of the stochastic convolution but with the restriction r < 1. The proof of continuity in the border case requires an additional argument in form of Lebesgue’s dominated convergence theorem. The last section briefly reviews our result in the special case of additive noise and gives an example which demonstrates that the spatial regularity results are optimal

Preliminaries

Spatial regularity

L02 dσ p 2

Regularity in time

Additive noise and optimal regularity