Morrey-Campanato estimates for the moments of stochastic integral operators and its application to SPDEs

Abstract

In this paper, we are concerned with the estimates for the moments of stochastic convolution integrals. We first deal with the stochastic singular integral operators and we aim to derive the Morrey-Campanato estimates for the $p$-moments (for $p\ge1$). Then, by utilising the embedding theory between the Campanato space and Hölder space, we establish the norm of $C^{θ,θ/2}(\bar D)$, where $θ\ge0, \bar D=\bar G\times[0,T]$ for arbitrarily fixed $T\in(0,\infty)$ and $G\subset\mathbb{R}^d$. As an application, we consider the following stochastic (fractional) heat equations with additive noises \bess du_t(x)=Δ^αu_t(x)dt+g(t,x)dη_t,\ \ \ u_0=0,\ 0\leq t\leq T, x\in G, \eess where $Δ^α=-(-Δ)^α$ with $0<α\leq1$ (the fractional Laplacian), $g:[0,T]\times G\timesΩ\to\mathbb{R}$ is a joint measurable coefficient, and $η_t, t\in[0,T]$, is either the Brownian motion or a Lévy process on a given filtered probability space $(Ω,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. The Schauder estimate for the $p$-moments of the solution of the above equation is obtained. The novelty of the present paper is that we obtain the Schauder estimate for parabolic stochastic partial differential equations with Lévy noise.

19 pages