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

TL;DR

The paper develops a functional-analytic framework based on generalized Dirichlet forms to study singular degenerate SDEs on ${\mathbb{R}}^d$ with a prescribed sub-invariant measure ${\widehat{\mu}}$. It establishes weak existence via the martingale problem and constructs an $L^1({\widehat{\mu}})$-closed extension of the generator, generating a sub-Markovian semigroup and an associated Hunt process with sub-invariance of ${\widehat{\mu}}$. Under additional regularity, notably locally Hölder coefficients and invariance of ${\widehat{\mu}}$, the paper proves $L^1({\widehat{\mu}})$-uniqueness of the generator and derives conditional uniqueness in law for right processes solving the SDE, including a Lyons–Zheng-type decomposition framework. The results yield well-posed stochastic dynamics for highly singular coefficients and provide a rigorous link between analytic generator uniqueness and probabilistic law uniqueness, with implications for sampling and ergodic control where a target (sub-)invariant measure is prescribed.

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 $\widehatμ$. 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 $L^1(\mathbb{R}^d,\widehatμ)$-uniqueness in a subclass of continuous Markov processes, namely right processes that have $\widehatμ$ as sub-invariant measure and have continuous paths for $\widehatμ$-almost every starting point. Weak existence is obtained for a broader class via the martingale problem.

Theorems & Definitions (37)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Lemma 3.1
• Proposition 3.2
• Remark 3.3
• Lemma 3.4
• Theorem 3.5
• Remark 3.6