On the Cauchy problem for stochastic integrodifferential parabolic equations in the scale of Lp-spaces of generalized smoothness

TL;DR

This paper establishes existence and uniqueness results for stochastic parabolic integro-differential equations in Lp-spaces characterized by generalized smoothness, using H"ormander conditions and Levy process estimates.

Contribution

It introduces a framework for analyzing stochastic parabolic integro-differential equations in generalized Lp-spaces with Levy measure-based regularity.

Findings

Proved existence and uniqueness of solutions under H"ormander condition.

Derived probability density estimates for associated Levy processes.

Extended analysis to spaces defined by generalized smoothness.

Abstract

Stochastic parabolic integro-differential problem is considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in Lp-spaces of functions whose regularity is defined by a scalable Levy measure. Some rough probability density function estimates of the associated Levy process are used as well.