BMO estimates for stochastic singular integral operators and its application to PDEs with L\'{e}vy noise

TL;DR

This paper establishes BMO estimates for stochastic singular integral operators and applies these results to analyze PDEs driven by Lévy noise, specifically fractional Laplacian equations with jump processes.

Contribution

It introduces new BMO estimates for stochastic singular integral operators and applies them to PDEs with Lévy noise, extending existing methods to jump processes.

Findings

BMO estimates for stochastic singular integral operators are derived.

The q-th order BMO quasi-norm of derivatives of solutions is controlled by the noise term.

Application to fractional Laplacian equations with Lévy noise demonstrates the estimates' effectiveness.

Abstract

In this paper, we consider the stochastic singular integral operators and obtain the BMO estimates. As an application, we consider the fractional Laplacian equation with additive noises \bess du_t(x)=\Delta^{\frac{\alpha}{2}}u_t(x)dt+\sum_{k=1}^\infty\int_{\mathbb{R}^m}g^k(t,x)z\tilde N_k(dz,dt),\ \ \ u_0=0,\ 0\leq t\leq T, \eess where $\Delta^{\frac{\alpha}{2}}=-(-\Delta)^{\frac{\alpha}{2}}$, and $\int_{\mathbb{R}^m}z\tilde N_k(t,dz)=:Y_t^k$ are independent $m$-dimensional pure jump L\'{e}vy processes with L\'{e}vy measure of $\nu^k$. Following the idea of \cite{Kim}, we obtain the $q$-th order BMO quasi-norm of the $\frac{\alpha}{q_0}$-order derivative of $u$ is controlled by the norm of $g$.