{
  "id": "1709.00556",
  "version": "v1",
  "published": "2017-09-02T10:10:20.000Z",
  "updated": "2017-09-02T10:10:20.000Z",
  "title": "Nonlinear Fokker-Planck equations for Probability Measures on Path Space and Path-Distribution Dependent SDEs",
  "authors": [
    "Xing Huang",
    "Michael Röckner",
    "Feng-Yu Wang"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker--Planck equations for probability measures $(\\mu_t)_{t \\geq 0}$ on the path space $\\mathcal C:=C([-r_0,0];\\mathbb R^d),$ is analyzed: $$\\partial_t \\mu(t)=L_{t,\\mu_t}^*\\mu_t,\\ \\ t\\ge 0,$$ where $\\mu(t)$ is the image of $\\mu_t$ under the projection $\\mathcal C\\ni\\xi\\mapsto \\xi(0)\\in\\mathbb R^d$, and $$L_{t,\\mu}(\\xi):= \\frac 1 2\\sum_{i,j=1}^d a_{ij}(t,\\xi,\\mu)\\frac{\\partial^2} {\\partial_{\\xi(0)_i} \\partial_{\\xi(0)_j}} +\\sum_{i=1}^d b_i(t,\\xi,\\mu)\\frac{\\partial}{\\partial_{\\xi(0)_i}},\\ \\ t\\ge 0, \\xi\\in \\mathcal C, \\mu\\in \\mathcal P^{\\mathcal C}.$$ Under reasonable conditions on the coefficients $a_{ij}$ and $b_i$, the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-02T10:10:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear fokker-planck equations",
      "path-distribution dependent sdes",
      "probability measures",
      "path space",
      "path-distribution dependent stochastic differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}