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

TL;DR

This paper derives Morrey-Campanato estimates for moments of stochastic integral operators and applies these results to establish Schauder estimates for solutions of stochastic fractional heat equations driven by Le9vy noise, advancing the understanding of SPDE regularity.

Contribution

It introduces Morrey-Campanato estimates for stochastic integrals and applies embedding theory to obtain Schauder estimates for SPDEs with Le9vy noise, a novel contribution.

Findings

Established Morrey-Campanato estimates for p-moments of stochastic integrals.

Derived Schauder estimates for solutions of stochastic fractional heat equations with Le9vy noise.

Extended regularity results to SPDEs driven by Le9vy processes.

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 H\"older space, we establish the norm of $C^{\theta,\theta/2}(\bar D)$, where $\theta\ge0, \bar D=\bar G\times[0,T]$ 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)=\Delta^\alpha u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ 0\leq t\leq T, x\in G, \eess where $\Delta^\alpha=-(-\Delta)^\alpha$ with $0<\alpha\leq1$ (the fractional Laplacian), $g:[0,T]\times G\times\Omega\to\mathbb{R}$ is a joint measurable coefficient, and $\eta_t, t\in[0,T]$, is either the Brownian motion or a L\'evy process on a given filtered probability space $(\Omega,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. 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 L\'evy noise.