Strong regularization by noise for kinetic SDEs

TL;DR

This paper proves strong well-posedness for a class of degenerate stochastic differential equations with irregular drift, demonstrating a regularization by noise effect in kinetic SDEs under weak Hörmander conditions.

Contribution

It establishes strong well-posedness for kinetic SDEs with Hölder continuous drifts, extending the understanding of regularization by noise in degenerate systems.

Findings

Strong well-posedness proven for degenerate SDEs with irregular drift.

Allows Hölder continuous velocity drifts with any positive exponent.

Demonstrates regularization by noise in systems lacking deterministic well-posedness.

Abstract

In this paper we prove strong well-posedness for a system of stochastic differential equations driven by a degenerate diffusion satisfying a weak-type H\"ormander condition, assuming H\"older regularity assumptions on the drift coefficient. This framework encompasses, as particular cases, stochastic Langevin systems of kinetic SDEs. The drift coefficient of the velocity component is allowed to be $\alpha$-H\"older continuous without any restriction on the index $\alpha$, which can be any positive number in $]0,1[$. As the deterministic counterparts of these differential systems are not well-posed, this result can be viewed as a phenomenon known as regularization by noise.