{
  "id": "1007.0896",
  "version": "v1",
  "published": "2010-07-06T13:56:44.000Z",
  "updated": "2010-07-06T13:56:44.000Z",
  "title": "Stability of the stochastic heat equation in $L^1([0,1])$",
  "authors": [
    "Nicolas Fournier",
    "Jacques Printems"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the white-noise driven stochastic heat equation on $[0,\\infty)\\times[0,1]$ with Lipschitz-continuous drift and diffusion coefficients $b$ and $\\sigma$. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some {\\it a priori} estimates on solutions. This allows us to prove the strong existence and (partial) uniqueness of weak solutions when the initial condition belongs only to $L^1([0,1])$, and the stability of the solution with respect to this initial condition. We also obtain, under some conditions, some results concerning the large time behavior of solutions: uniqueness of the possible invariant distribution and asymptotic confluence of solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-06T13:56:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15"
    ],
    "keywords": [
      "white-noise driven stochastic heat equation",
      "initial condition belongs",
      "large time behavior",
      "strong existence",
      "uniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.0896F"
    }
  }
}