Moduli of continuity for the local times of rebirthed Markov processes

TL;DR

This paper establishes moduli of continuity for local times of rebirthed Markov processes, extending classical results to processes that are recurrent through rebirth mechanisms, with explicit examples involving Lévy processes and diffusions.

Contribution

It introduces new isomorphism theorems linking local times of rebirthed processes to Gaussian processes, enabling precise continuity results for these local times.

Findings

Derived explicit moduli of continuity for local times.

Established isomorphism theorems relating local times to Gaussian processes.

Provided examples with Lévy processes and diffusions.

Abstract

Let $S$ a be locally compact space with a countable base. Let $\cal Y$ be a transient symmetric Borel right process with state space $S$ and continuous strictly positive $p$--potential densities $u^p(x,y)$. Local and uniform moduli of continuity are obtained for the local times of both fully and partially rebirthed versions of $\cal Y$. A fully rebirthed version of $\cal Y$ is an extension of $\cal Y$ so that instead of terminating at the end of its lifetime it is immediately ``reborn'' with a probability measure $\mu $, on $S$. I.e., the process goes to the set $B\subset S$ with probability $\mu(B) $, after which it continues to evolve the way $\YY$ did, being reborn with probability $\mu $ each time it dies. This rebirthed version of $\cal Y$ is a recurrent Borel right process with state space $S$ and $p$-potential densities of form, \[ u^p(x,y)+h(x,y),\qquad x,y\in S,\,\, p>0, \] where $h(x,y)$ is not symmetric. The local times of the rebirthed process are given in terms of the local times of $\cal Y$ and isomorphism theorems in the spirit of Dynkin, Eisenbam and Kaspi are obtained that relate these local times to generalized chi--square processes formed by Gaussian processes with covariances $u^{q}(x,y)$ for different values of $q$. These isomorphisms allow one to obtain exact local and uniform moduli of continuity for the local times of the rebirthed process. Several explicit examples are given in which $\cal Y$ is either a modified L\'evy process or a diffusion. Analogous results are obtained for partially rebirthed versions of $\cal Y$. This is obtained by starting $\cal Y$ in $S$ and when it dies returning it to $S$ with a sub-probability measure $\Xi$. (With probability $1-|\Xi |$ it is sent to a disjoint state space $S'$, where it remains.)