On invariant probability measures of regime-switching diffusion processes with singular drifts

TL;DR

This paper investigates the existence and uniqueness of invariant probability measures for regime-switching diffusion processes with singular drifts, introducing integrability conditions and extending the generator in L^1 space.

Contribution

It provides new integrability conditions involving a reference measure and the Q-matrix to establish the existence and uniqueness of invariant measures for such processes.

Findings

Existence of invariant probability measures under new integrability conditions

Extension of the generator to an L^1-space as a generator of a C_0-semigroup

Proof of uniqueness of the invariant measure and its density

Abstract

For regime-switching diffusions processes with singular drifts, we introduce integrability conditions involving a nice reference probability measure and the $Q$-matrix of the jump part to study the existence of the invariant probability measures. Consequently, the generator of the regime-switching diffusions process has an extension in the $L^1$-space w.r.t the invariant probability measure to be a generator of $C_0$-semigroup. Moreover, we prove the uniqueness of the extension. Regularities and the uniqueness of the invariant probability density w.r.t the nice reference probability measure are also considered.