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 $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , $$d\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , with prescribed sub-invariant measure $$\widehat{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> . 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 $$L^1({\mathbb {R}}^d,\widehat{\mu })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -uniqueness in a subclass of continuous Markov processes, namely right processes that have $$\widehat{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> as sub-invariant measure and have continuous paths for $$\widehat{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> -almost every starting point. Weak existence is obtained for a broader class via the martingale problem, by first constructing a sub-Markovian $$C_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup of contractions with respect to $$\widehat{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> and then applying generalized Dirichlet form theory.