Uniqueness in law for singular degenerate SDEs with respect to a (sub-)invariant measure

TL;DR

This paper establishes weak existence and uniqueness in law for a broad class of degenerate, discontinuous stochastic differential equations with a prescribed sub-invariant measure, extending classical results to more general settings.

Contribution

It introduces new methods to prove uniqueness and existence for SDEs with degenerate, discontinuous, and unbounded coefficients under a sub-invariant measure.

Findings

Weak uniqueness in law for degenerate SDEs with sub-invariant measure.

Weak existence via martingale problem for broader class.

Applicable to SDEs with discontinuous and unbounded coefficients.

Abstract

We show weak existence and uniqueness in law for a general class of stochastic differential equations in $\mathbb{R}^d$, $d\ge 1$, with prescribed sub-invariant measure $\widehat{\mu}$. The dispersion and drift coefficients of the stochastic differential equation are allowed to be degenerate and discontinuous, and locally unbounded, respectively. Uniqueness in law is obtained via $L^1(\mathbb{R}^d,\widehat{\mu})$-uniqueness in a subclass of continuous Markov processes, namely right processes that have $\widehat{\mu}$ as sub-invariant measure and have continuous paths for $\widehat{\mu}$-almost every starting point. Weak existence is obtained for a broader class via the martingale problem.