{
  "id": "1501.03925",
  "version": "v1",
  "published": "2015-01-16T09:37:28.000Z",
  "updated": "2015-01-16T09:37:28.000Z",
  "title": "On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations",
  "authors": [
    "Vassili Kolokoltsov"
  ],
  "comment": "Submitted to Fract. Calc. Appl. Anal",
  "categories": [
    "math.PR"
  ],
  "abstract": "From the point of view of stochastic analysis the Caputo and Riemann-Liouville derivatives of order $\\al \\in (0,2)$ can be viewed as (regularized) generators of stable L\\'evy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. These extensions are introduced and some well-posedness results are obtained that generalize, simplify and unify lots of known facts. This probabilistic analysis leads one to study a class of Markov processes that can be constructed from any given Markov process in $\\R^d$ by blocking (or interrupting) the jumps that attempt to cross certain closed set of 'check-points'.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-16T09:37:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A08",
      "35S15",
      "60J50",
      "60J75"
    ],
    "keywords": [
      "fractional differential equations",
      "related markov processes",
      "riemann-liouville derivatives",
      "multidimensional extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}