{
  "id": "2010.03908",
  "version": "v1",
  "published": "2020-10-08T11:36:39.000Z",
  "updated": "2020-10-08T11:36:39.000Z",
  "title": "On semilinear SPDEs with nonlinearities with polynomial growth",
  "authors": [
    "D. A. Bignamini",
    "S. Ferrari"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "Let $\\mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $\\mathcal{X}$, let $F:\\X\\rightaarrow\\X$ be a (smooth enough) function and let $\\{W(t)\\}_{t\\geq 0}$ be a $\\mathcal{X}$-valued cylindrical Wiener process. For $\\alpha\\in [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2\\alpha-1}:Q^{1-2\\alpha}(\\mathcal{X})\\subseteq\\mathcal{X}\\rightarrow\\mathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation \\begin{gather} \\left\\{\\begin{array}{ll} dX(t,x)=\\big(AX(t,x)+F(X(t,x))\\big)dt+ Q^{\\alpha}dW(t), & t>0;\\\\ X(0,x)=x\\in \\mathcal{X}, \\end{array} \\right. \\end{gather} and in its associated transition semigroup \\begin{align} P(t)\\varphi(x):=E[\\varphi(X(t,x))], \\qquad \\varphi\\in B_b(\\mathcal{X}),\\ t\\geq 0,\\ x\\in \\mathcal{X}; \\end{align} where $B_b(\\mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $\\mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(\\mathcal{X},\\nu)$, where $\\nu$ is the unique invariant probability measure of \\eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincar\\'e inequalities and we study the maximal Sobolev regularity for the stationary equation \\[\\lambda u-N_2 u=f,\\qquad \\lambda>0,\\ f\\in L^2(\\mathcal{X},\\nu);\\] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(\\mathcal{X},\\nu)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-08T11:36:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28C10",
      "28C20",
      "35J15",
      "46G12",
      "60G15",
      "60H15"
    ],
    "keywords": [
      "polynomial growth",
      "semilinear spdes",
      "semilinear stochastic partial differential equation",
      "nonlinearities",
      "unique invariant probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}