{
  "id": "2411.14936",
  "version": "v1",
  "published": "2024-11-22T13:49:02.000Z",
  "updated": "2024-11-22T13:49:02.000Z",
  "title": "Massive Particle Systems, Wasserstein Brownian Motions, and the Dean--Kawasaki Equation",
  "authors": [
    "Lorenzo Dello Schiavo"
  ],
  "comment": "103 pages, 2 figures. Comments welcome!",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We develop a unifying theory for four different objects: (1) infinite systems of interacting massive particles; (2) solutions to the Dean-Kawasaki equation with singular drift and space-time white noise; (3) Wasserstein diffusions with a.s. purely atomic reversible random measures; (4) metric measure Brownian motions induced by Cheeger energies on $L^2$-Wasserstein spaces. For the objects in (1)-(3) we prove existence and uniqueness of solutions, and several characterizations, on an arbitrary locally compact Polish ambient space $M$ with exponentially recurrent Feller driving noise. In the case of the Dean-Kawasaki equation, this amounts to replacing the Laplace operator with some arbitrary diffusive Markov generator $\\mathsf{L}$ with ultracontractive semigroup. In addition to a complete discussion of the free case, we consider singular interactions, including, e.g., mean-field repulsive isotropic pairwise interactions of Riesz and logarithmic type under the assumption of local integrability. We further show that each Markov diffusion generator $\\mathsf{L}$ on $M$ induces in a natural way a geometry on the space of probability measures over $M$. When $M$ is a manifold and $\\mathsf{L}$ is a drifted Laplace-Beltrami operator, this geometry coincides with the geometry of $L^2$-optimal transportation. The corresponding `geometric Brownian motion' coincides with the 'metric measure Brownian motion' in (4).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-22T13:49:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G57",
      "60H17",
      "60J46",
      "49Q22",
      "70F45"
    ],
    "keywords": [
      "wasserstein brownian motions",
      "dean-kawasaki equation",
      "massive particle systems",
      "metric measure brownian motion",
      "atomic reversible random measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 103,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}