Cauchy Problem of Stochastic Kinetic Equations

TL;DR

This paper develops optimal regularity estimates for stochastic kinetic equations with random coefficients, applying these results to nonlinear filtering, conditional densities, and super-linear growth equations with measure initial data.

Contribution

It introduces new regularity estimates for stochastic kinetic equations and demonstrates their applications to filtering, well-posedness, and kinetic Parabolic Anderson models.

Findings

Established optimal regularity estimates in anisotropic Besov spaces.

Proved existence and regularity of conditional densities for degenerate diffusions.

Showed well-posedness for super-linear stochastic kinetic equations with white noise.

Abstract

In this paper we establish the optimal regularity estimates for the Cauchy problem of stochastic kinetic equations with random coefficients in anisotropic Besov spaces. As applications, we study the nonlinear filtering problem for a degenerate diffusion process, and obtain the existence and regularity of conditional probability densities under few assumptions. Moreover, we also show the well-posedness for a class of super-linear growth stochastic kinetic equations driven by velocity-time white noises, as well as a kinetic version of Parabolic Anderson Model with measure as initial values.