{
  "id": "1802.07533",
  "version": "v1",
  "published": "2018-02-21T12:04:53.000Z",
  "updated": "2018-02-21T12:04:53.000Z",
  "title": "Variational solutions to nonlinear stochastic differential equations in Hilbert spaces",
  "authors": [
    "Viorel Barbu",
    "Michael Röckner"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "One introduces a new variational concept of solution for the stochastic differential equation $dX+A(t)X\\,dt+\\lambda X\\,dt=X\\,dW,$ $t\\in(0,T)$; $X(0)=x$ in a real Hilbert space where $A(t)=\\partial\\varphi(t)$, $t\\in(0,T)$, is a maximal monotone subpotential operator in $H$ while $W$ is a Wiener process in $H$ on a probability space $\\{\\Omega,\\mathcal{F},\\mathbb{P}\\}$. In this new context, the solution $X=X(t,x)$ exists for each $x\\in H$, is unique, and depends continuously on $x$. This functional scheme applies to a general class of stochastic PDE not covered by the classical variational existence theory ([15], [16], [17]) and, in particular, to stochastic variational inequalities and parabolic stochastic equations with general monotone nonlinearities with low or superfast growth to $+\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-21T12:04:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "47H05",
      "47J05"
    ],
    "keywords": [
      "nonlinear stochastic differential equations",
      "variational solutions",
      "maximal monotone subpotential operator",
      "real hilbert space",
      "functional scheme applies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}