On weak uniqueness for some degenerate SDEs by global $L^p$ estimates

TL;DR

This paper establishes the uniqueness in law for certain degenerate stochastic differential equations with linear drift, utilizing global $L^p$ estimates and hypoelliptic operator techniques, extending classical results to more general degenerate cases.

Contribution

It introduces a novel approach combining global $L^p$ estimates with localization techniques to prove weak uniqueness for degenerate SDEs with linear drift.

Findings

Proved weak uniqueness for a class of degenerate SDEs.

Extended classical results to hypoelliptic Ornstein-Uhlenbeck processes.

Developed a general localization principle for martingale problems.

Abstract

We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the H\"ormander hypoellipticity condition. In the proof we use global $L^p$-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti-Cupini-Lanconelli-Priola (Math. Z. 266 (2010)) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems.