Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients

TL;DR

This paper establishes the existence, uniqueness, and large deviation principles for degenerate SDEs with irregular Sobolev coefficients, extending classical results to more singular and degenerate cases.

Contribution

It introduces a generalized well-posedness framework for degenerate SDEs with Sobolev coefficients and derives a large deviation principle for these equations.

Findings

Proved existence and uniqueness of solutions for degenerate SDEs with Sobolev coefficients.

Extended classical DiPerna-Lions flow results to degenerate cases.

Established a Freidlin-Wenzell type large deviation principle for these SDEs.

Abstract

In this article we prove the existence and uniqueness for degenerate stochastic differential equations with Sobolev (possibly singular) drift and diffusion coefficients in a generalized sense. In particular, our result covers the classical DiPerna-Lions flows and, we also obtain the well-posedness for degenerate Fokker-Planck equations with irregular coefficients. Moreover, a large deviation principle of Freidlin-Wenzell type for this type of SDEs is established.