{
  "id": "math/0701736",
  "version": "v1",
  "published": "2007-01-25T11:15:44.000Z",
  "updated": "2007-01-25T11:15:44.000Z",
  "title": "Non-equilibrium stochastic dynamics in continuum: The free case",
  "authors": [
    "Y. Kondratiev",
    "E. Lytvynov",
    "M. Röckner"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process $M$ on a Riemannian manifold $X$. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in $X$ such that, with probability one, infinitely many particles will arrive at this set at some time $t>0$. We assume that $X$ has infinite volume and, for each $\\alpha\\ge1$, we consider the set $\\Theta_\\alpha$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $\\alpha$-th power of the volume of the set. We find quite general conditions on the process $M$ which guarantee that the corresponding infinite particle process can start at each configuration from $\\Theta_\\alpha$, will never leave $\\Theta_\\alpha$, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in $X$), and free Kawasaki dynamics on the configuration space. We also show that if $X=\\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-25T11:15:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J65",
      "60J75",
      "60J80",
      "82B21"
    ],
    "keywords": [
      "non-equilibrium stochastic dynamics",
      "free case",
      "configuration space",
      "free kawasaki dynamics",
      "free glauber dynamics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1736K"
    }
  }
}