{
  "id": "2010.03580",
  "version": "v1",
  "published": "2020-10-07T18:00:07.000Z",
  "updated": "2020-10-07T18:00:07.000Z",
  "title": "Propagation of minimality in the supercooled Stefan problem",
  "authors": [
    "Christa Cuchiero",
    "Stefan Rigger",
    "Sara Svaluto-Ferro"
  ],
  "categories": [
    "math.PR",
    "q-fin.MF"
  ],
  "abstract": "Supercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean-Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. Our main contributions are: (i) we prove a general tightness theorem for the Skorokhod M1-topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) We prove propagation of chaos for a perturbed version of the particle system for general initial conditions. (iii) We prove a conjecture of Delarue, Nadtochiy and Shkolnikov, relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean-Vlasov equation are physical whenever the initial condition is integrable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-07T18:00:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "propagation",
      "minimality",
      "one-phase one-dimensional supercooled stefan problem",
      "particle system",
      "general initial conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}