Strong solutions to degenerate SDEs and uniqueness for degenerate Fokker-Planck equations

TL;DR

This paper establishes the existence of strong solutions for a broad class of possibly degenerate SDEs with Sobolev coefficients, and proves new uniqueness results for related Fokker-Planck equations, advancing understanding of degenerate stochastic systems.

Contribution

It introduces a novel approach combining Fokker-Planck existence with restricted pathwise uniqueness to prove strong solutions for degenerate SDEs with Sobolev coefficients.

Findings

Existence of strong solutions for degenerate SDEs with Sobolev coefficients.

New restricted pathwise uniqueness results for SDEs.

Uniqueness results for degenerate Fokker-Planck equations.

Abstract

We prove the existence of probabilistically strong solutions for large classes of possibly degenerate stochastic differential equations with locally Sobolev-regular coefficients, using the restricted Yamada-Watanabe theorem. Our approach relies on existence results for the corresponding Fokker-Planck equation, combined with both novel and existing restricted pathwise uniqueness results for SDEs. Here, restricted pathwise uniqueness means pathwise uniqueness among a subclass of weak solutions to the SDE. Furthermore, we derive new uniqueness results for the Fokker-Planck equation.