{
  "id": "2409.00609",
  "version": "v1",
  "published": "2024-09-01T04:33:56.000Z",
  "updated": "2024-09-01T04:33:56.000Z",
  "title": "Moduli of continuity for the local times of rebirthed Markov processes",
  "authors": [
    "P. J. Fitzsimmons",
    "Michael B. Marcus",
    "Jay Rosen"
  ],
  "categories": [
    "math.PR"
  ],
  "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.)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-01T04:33:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G17",
      "60J40",
      "60J55",
      "60J60"
    ],
    "keywords": [
      "local times",
      "rebirthed markov processes",
      "continuity",
      "state space",
      "transient symmetric borel right process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}