Large deviations for occupation times of Markov processes with $L_{\mathbf{2}}$ semigroups

TL;DR

This paper develops a unified framework for large deviation bounds of occupation times in Markov processes with $L_2$ semigroups, applicable in both continuous and discrete time without requiring the strong Markov property.

Contribution

It extends large deviation bounds for occupation times under minimal conditions, broadening applicability to processes with less restrictive assumptions.

Findings

Unified large deviation bounds for occupation times.

Applicable to processes without strong Markov property.

Methods extend to both continuous and discrete time.

Abstract

Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with $L_2$ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong Markov property is needed. The methods used here apply in both continuous and discrete time. We present the proofs for continuous time only because of the inherent technical difficulties in that situation; the proofs can be adapted for discrete time in a straightforward manner.