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

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

arXiv commentLong version with all details, revised version (only minor revisions such as corrections of typos and minor inconsistencies)

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• Accelerating diffusions
• Analytic theory of Itô-stochastic differential equations with non-smooth coefficients
• Conservativeness criteria for generalized Dirichlet forms
• Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms
• Existence and approximation of Hunt processes associated with generalized Dirichlet forms

Cited by in corpus (1)

• Pointwise well-posedness results for degenerate Itô-SDEs with locally bounded drifts