{
  "id": "1706.05638",
  "version": "v1",
  "published": "2017-06-18T12:12:41.000Z",
  "updated": "2017-06-18T12:12:41.000Z",
  "title": "Invariant Measures for Path-Dependent Random Diffusions",
  "authors": [
    "Jianhai Bao",
    "Jinghai Shao",
    "Chenggui Yuan"
  ],
  "comment": "32 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this work, we are concerned with existence and uniqueness of invariant measures for path-dependent random diffusions and their time discretizations. The random diffusion here means a diffusion process living in a random environment characterized by a continuous time Markov chain. Under certain ergodic conditions, we show that the path-dependent random diffusion enjoys a unique invariant probability measure and converges exponentially to its equilibrium under the Wasserstein distance. Also, we demonstrate that the time discretization of the path-dependent random diffusion involved admits a unique invariant probability measure and shares the corresponding ergodic property when the stepsize is sufficiently small. During this procedure, the difficulty arose from the time-discretization of continuous time Markov chain has to be deal with, for which an estimate on its exponential functional is presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-18T12:12:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariant measures",
      "unique invariant probability measure",
      "continuous time markov chain",
      "path-dependent random diffusion enjoys",
      "time discretization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}