{
  "id": "2306.03585",
  "version": "v1",
  "published": "2023-06-06T11:03:08.000Z",
  "updated": "2023-06-06T11:03:08.000Z",
  "title": "Selection principle for the Fleming-Viot process with drift $-1$",
  "authors": [
    "Oliver Tough"
  ],
  "comment": "25 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We consider the Fleming-Viot particle system consisting of $N$ identical particles evolving in $\\mathbb{R}_{>0}$ as Brownian motions with constant drift $-1$. Whenever a particle hits $0$, it jumps onto another particle in the interior. It is known that this particle system has a hydrodynamic limit as $N\\rightarrow\\infty$ given by Brownian motion with drift $-1$ conditioned not to hit $0$. This killed Brownian motion has an infinite family of quasi-stationary distributions (QSDs), with a Yaglom limit given by the unique QSD minimising the survival probability. On the other hand, for fixed $N<\\infty$, this particle system converges to a unique stationary distribution as time $t\\rightarrow\\infty$. We prove the following selection principle: the empirical measure of the $N$-particle stationary distribution converges to the aforedescribed Yaglom limit as $N\\rightarrow\\infty$. The selection problem for this particular Fleming-Viot process is closely connected to the microscopic selection problem in front propagation, in particular for the $N$-branching Brownian motion. The proof requires neither fine estimates on the particle system nor the use of Lyapunov functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-06T11:03:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35C07",
      "35K57",
      "35Q92",
      "60J80",
      "60J85"
    ],
    "keywords": [
      "fleming-viot process",
      "selection principle",
      "brownian motion",
      "particle stationary distribution converges",
      "yaglom limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}