Ray-Knight theorems for the local times of rebirthed Markov processes

TL;DR

This paper generalizes Ray-Knight theorems to a broad class of non-symmetric Markov processes created through a rebirthing mechanism, linking local times with Gaussian processes and analyzing their continuity properties.

Contribution

It introduces new Ray-Knight theorems for non-symmetric Markov processes obtained via rebirthing, connecting local times with Gaussian processes instead of permanental ones.

Findings

Established generalized Ray-Knight theorems for rebirthed processes

Linked local times to Gaussian processes for these non-symmetric processes

Analyzed the modulus of continuity of local times in spatial variables

Abstract

We prove generalizations of the first and second Ray-Knight theorems, for a large class of non-symmetric strong Markov processes. These results link the local times of the Markov process with the squares of associated Gaussian processes. This connection allows us to establish results about the exact modulus of continuity (in the spatial variable) of the local times. Our approach is different from earlier treatments which were based on associated permanental processes rather than Gaussian processes. The type of process with which we work can be described as follows. Start with a symmetric Markov process with finite lifetime; upon its death resurrect it at a place in the state space chosen at random, independent of the past. Continue in this way, resurrecting at each death, to obtain a recurrent process. The rebirthing procedure destroys the symmetry of the original process, leading to a large class of non-symmetric processes. The main results are illustrated by many examples.