{
  "id": "2011.07535",
  "version": "v1",
  "published": "2020-11-15T14:03:59.000Z",
  "updated": "2020-11-15T14:03:59.000Z",
  "title": "The heat equation with order-respecting absorption and particle systems with topological interaction",
  "authors": [
    "Rami Atar"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "A PDE formulation is proposed, referred to as a heat equation with order-respecting absorption, aimed at characterizing hydrodynamic limits of a class of particle systems on the line with topological interaction that have so far been described by free boundary problems. It consists of the heat equation with measure-valued injection and absorption terms, where the absorption measure respects the usual order on $\\R$ in the sense that, for all $r\\in\\R$, it charges $(-\\iy,r)$ only at times when the solution vanishes on $(r,\\iy)$. The formulation is used to obtain new hydrodynamic limit results for two models. One is a variant of the main model studied by Carinci, De Masi, Giardin\\`a and Presutti \\cite{CDGPbook} where Brownian particles undergo injection according to a general injection measure, and removal that is restricted to the rightmost particle of the configuration. This partially addresses a conjecture of \\cite{CDGPbook}. Next a Brownian particle system is considered where the $Q$-quantile member of the population is removed until extinction, where $Q$ is a given $[0,1]$-valued continuous function of time. Here, unlike in earlier work on the subject, the removal mechanism acts on particles that are `at the boundary' but are not rightmost or leftmost. Finally, further potential uses of order-respecting absorption are mentioned.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-15T14:03:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heat equation",
      "order-respecting absorption",
      "topological interaction",
      "brownian particles undergo injection",
      "free boundary problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}