{
  "id": "1810.08364",
  "version": "v1",
  "published": "2018-10-19T06:36:29.000Z",
  "updated": "2018-10-19T06:36:29.000Z",
  "title": "Noise reinforcement for L{é}vy processes",
  "authors": [
    "Jean Bertoin"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In a step reinforced random walk, at each integer time and with a fixed probability p $\\in$ (0, 1), the walker repeats one of his previous steps chosen uniformly at random, and with complementary probability 1 -- p, the walker makes an independent new step with a given distribution. Examples in the literature include the so-called elephant random walk and the shark random swim. We consider here a continuous time analog, when the random walk is replaced by a L{\\'e}vy process. For sub-critical (or admissible) memory parameters p < p c , where p c is related to the Blumenthal-Getoor index of the L{\\'e}vy process, we construct a noise reinforced L{\\'e}vy process. Our main result shows that the step-reinforced random walks corresponding to discrete time skeletons of the L{\\'e}vy process, converge weakly to the noise reinforced L{\\'e}vy process as the time-mesh goes to 0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-19T06:36:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vy process",
      "noise reinforcement",
      "elephant random walk",
      "step reinforced random walk",
      "shark random swim"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}