Moduli of continuity for the local times of rebirthed Markov processes

Abstract

Let $S$ a be locally compact space with a countable base. Let $\mathcal{Y}$ 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 $\mathcal{Y}$. A fully rebirthed version of $\mathcal{Y}$ is an extension of $\mathcal{Y}$ 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 $\mathcal{Y}$ 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 $\mathcal{Y}$ 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 $\mathcal{Y}$ is either a modified Lévy process or a diffusion. Analogous results are obtained for partially rebirthed versions of $\mathcal{Y}$. This is obtained by starting $\mathcal{Y}$ 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 $S^{\prime }$, where it remains.)