Stationary Solutions and Forward Equations for Controlled and Singular Martingale Problems

Abstract

Stationary distributions of Markov processes can typically be characterized as probability measures that annihilate the generator in the sense that $int_EAfd\mu =0$ for $fin {cal D}(A)$; that is, for each such $\mu$, there exists a stationary solution of the martingale problem for $A$ with marginal distribution $\mu$. This result is extended to models corresponding to martingale problems that include absolutely continuous and singular (with respect to time) components and controls. Analogous results for the forward equation follow as a corollary.