Parabolic Littlewood-Paley inequality for $\phi(-\Delta)$-type operators and applications to Stochastic integro-differential equations

TL;DR

This paper establishes a parabolic Littlewood-Paley inequality for $\, ext{phi}(- ext{Delta})$-type operators with Bernstein functions, enabling an $L_p$-theory for certain stochastic integro-differential equations, advancing analysis in stochastic PDEs.

Contribution

It introduces a parabolic Littlewood-Paley inequality for $\, ext{phi}(- ext{Delta})$ operators and applies it to develop an $L_p$-theory for related stochastic equations.

Findings

Proved a parabolic Littlewood-Paley inequality for $\, ext{phi}(- ext{Delta})$ operators.

Constructed an $L_p$-theory for stochastic integro-differential equations.

Enabled new analysis techniques for stochastic PDEs with Bernstein functions.

Abstract

In this paper we prove a parabolic version of the Littlewood-Paley inequality for the operators of the type $\phi(-\Delta)$, where $\phi$ is a Bernstein function. As an application, we construct an $L_p$-theory for the stochastic integro-differential equations of the type $du=(-\phi(-\Delta)u+f)dt +gdW_t$.