Path continuity of Markov processes and locality of Kolmogorov operators

TL;DR

This paper establishes that the local properties of a Markov process's generator on a rich class of test functions imply the process's path continuity through certain domains, using potential theory and applicable to complex operators.

Contribution

It demonstrates that path continuity can be inferred from local generator properties on test functions, applicable to a broad class of Markov processes and operators.

Findings

Path continuity is linked to local generator properties.

The result applies to time-dependent integro-differential operators.

Potential theoretic tools are used to establish the connection.

Abstract

We prove that if we are given a generator of a cadlag Markov process and an open domain $G$ in the state space, on which the generator has the local property expressed in a suitable way on a class $\mathcal{C}$ of test functions that is sufficiently rich, then the Markov process has continuous paths when it passes through $G$. The result holds for any Markov process which is associated with the generator merely on $\mathcal{C}$. This points out that the path continuity of the process is an a priori property encrypted by the generator acting on enough test functions, and this property can be easily checked in many situations. The approach uses potential theoretic tools and covers Markov processes associated with (possibly time-dependent) second order integro-differential operators (e.g., through the martingale problem) defined on domains in Hilbert spaces or on spaces of measures.