{
  "id": "math/0311444",
  "version": "v1",
  "published": "2003-11-25T15:36:50.000Z",
  "updated": "2003-11-25T15:36:50.000Z",
  "title": "Infinite interacting diffusion particles I: Equilibrium process and its scaling limit",
  "authors": [
    "Yuri Kondratiev",
    "Eugene Lytvynov",
    "Michael Röckner"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A stochastic dynamics $({\\bf X}(t))_{t\\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $\\Gamma$ of all locally finite subsets (configurations) in $\\Bbb R$ and which has a Gibbs measure $\\mu$ as an invariant measure. We assume that $\\mu$ corresponds to a symmetric pair potential $\\phi(x-y)$. An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics--the so-called gradient stochastic dynamics, or interacting Brownian particles--has been investigated. By using the theory of Dirichlet forms, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\\cal E}_\\mu^\\Gamma$ on $L^2(\\Gamma;\\mu)$, and under general conditions on the potential $\\phi$, prove its closability. For a potential $\\phi$ having a ``weak'' singularity at zero, we also write down an explicit form of the generator of ${\\cal E}_\\mu^\\Gamma$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\\cal E}_\\mu^\\Gamma$, there exists a diffusion process that is properly associated with it. Finally, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in $C([0,\\infty),{\\cal D}')$, where ${\\cal D}'$ is the dual space of ${\\cal D}{:=}C_0^\\infty({\\Bbb R})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-11-25T15:36:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60B12",
      "60H15",
      "82C22"
    ],
    "keywords": [
      "infinite interacting diffusion particles",
      "scaling limit",
      "stochastic dynamics",
      "dirichlet form",
      "equilibrium process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....11444K"
    }
  }
}