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

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.

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• Strong existence and uniqueness for stochastic differential equation with H{ö}lder drift and degenerate noise