{
  "id": "1904.01564",
  "version": "v1",
  "published": "2019-04-02T17:40:18.000Z",
  "updated": "2019-04-02T17:40:18.000Z",
  "title": "Duality and the well-posedness of a martingale problem",
  "authors": [
    "Andrej Depperschmidt",
    "Andreas Greven",
    "Peter Pfaffelhuber"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "For two Polish state spaces $E_X$ and $E_Y$, and an operator $G_X$, we obtain existence and uniqueness of a $G_X$-martingale problem provided there is a dual process $Y$ on $E_Y$ solving a $G_Y$-martingale problem. Duality here means the existence of a rich function $H$ and transition kernels $(\\mu_t)_{t\\geq 0}$ on $E_X$ such that $\\mathbf E_y[H(x,Y_t)] = \\int \\mu_t(x,dx') H(x',y)$ for all $(x,y) \\in E_X \\times E_Y$ and $t\\geq 0$. While duality is well-known to imply uniqueness of the $G_X$-martingale problem, we give here a set of conditions under which duality also implies existence without using approximation techniques. As examples, we treat branching and resampling models, such as Feller's branching diffusion and the Fleming-Viot superprocess.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-02T17:40:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J25",
      "47D03",
      "60J35"
    ],
    "keywords": [
      "martingale problem",
      "well-posedness",
      "polish state spaces",
      "fellers branching diffusion",
      "approximation techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}