BMO estimates for stochastic singular integral operators and its application to PDEs with Lévy noise

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)=Δ^{\fracα{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 Lévy processes with Lévy 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 $g$.

22 pages

References in corpus (1)

• A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift

Cited by in corpus (2)

• Regularity of stochastic nonlocal diffusion equations
• Schauder estimates for stochastic transport-diffusion equations with Lévy processes