Regularity of Stochastic Kinetic Equations

TL;DR

This paper investigates the regularity of solutions to stochastic kinetic equations with multiplicative noise, demonstrating that such SPDEs maintain Sobolev regularity and are uniquely weakly differentiable, unlike their deterministic counterparts.

Contribution

It establishes the existence of unique weakly differentiable solutions for stochastic kinetic equations with mixed regularity, extending understanding beyond deterministic cases.

Findings

SPDE solutions preserve initial Sobolev regularity

Solutions do not develop discontinuities over time

Constructs a stochastic flow for the equations

Abstract

We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity ($L^p$-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow.