{
  "id": "2403.01893",
  "version": "v2",
  "published": "2024-03-04T09:54:04.000Z",
  "updated": "2024-10-24T08:02:08.000Z",
  "title": "Particle systems with sources and sinks",
  "authors": [
    "Frank Redig",
    "Ellen Saada"
  ],
  "comment": "Markov Processes And Related Fields, In press",
  "categories": [
    "math.PR"
  ],
  "abstract": "Local perturbations in conservative particle systems can have a non-local influence on the stationary measure. To capture this phenomenon, we analyze in this paper two toy models. We study the symmetric exclusion process on a countable set of sites V with a source at a given point (called the origin), starting from a Bernoulli product measure with density $\\rho$. We prove that when the underlying random walk on V is recurrent, then the system evolves towards full occupation, whereas in the transient case we obtain a limiting distribution which is not product and has long-range correlations. For independent random walkers on V , we analyze the same problem, starting from a Poissonian measure. Via intertwining with a system of ODE's, we prove that the distribution is Poissonian at all later times t \\> 0, and that the system ''explodes'' in the limit t $\\rightarrow$ $\\infty$ if and only if the underlying random walk is recurrent. In the transient case, the limiting density is a simple function of the Green's function of the random walk.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-10-24T08:02:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "particle systems",
      "transient case",
      "symmetric exclusion process",
      "independent random walkers",
      "bernoulli product measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}