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

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 a...

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Bibliographic Details
Published inarXiv.org
Main Authors Lee, Haesung, Trutnau, Gerald
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 23.04.2024
Subjects
Differential equations
Invariants
Markov processes
Martingales
Uniqueness
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Summary: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.
ISSN:2331-8422