Markov uniqueness and Fokker-Planck-Kolmogorov equations

TL;DR

This paper establishes that Markov uniqueness for symmetric pre-Dirichlet operators can be derived from the uniqueness of the associated Fokker-Planck-Kolmogorov equations, leading to new results in various complex cases.

Contribution

It demonstrates that Markov uniqueness follows from FPKE uniqueness and provides new results for cases with killing and degenerate diffusion coefficients.

Findings

Markov uniqueness follows from FPKE uniqueness.

New results for operators with killing.

Results for degenerate diffusion coefficients.

Abstract

In this paper we show that Markov uniqueness for symmetric pre-Dirichlet operators $L$ follows from the uniqueness of the corresponding Fokker-Planck-Kolmogorov equation (FPKE). Since in recent years a considerable number of uniqueness results for FPKE's have been achieved, we obtain new Markov uniqueness results in concrete cases. A selection of such will be presented in this paper. They include cases with killing and with degenerate diffusion coefficients.