{
  "id": "2003.05195",
  "version": "v1",
  "published": "2020-03-11T10:09:04.000Z",
  "updated": "2020-03-11T10:09:04.000Z",
  "title": "Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces",
  "authors": [
    "D. A. Bignamini",
    "S. Ferrari"
  ],
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "Let $\\mathcal{X}$ be a separable Hilbert space with norm $\\|\\cdot\\|$ and let $T>0$. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $\\mathcal{X}$, let $F:\\mathcal{X}\\rightarrow \\mathcal{X}$ be a (smooth enough) function and let $W(t)$ 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\\in(0,T];\\\\ X(0,x)=x\\in \\mathcal{X}, \\end{array}\\right. \\end{gather*} and in its associated transition semigroup \\begin{align*} P(t)\\varphi(x):=\\mathbb{E}[\\varphi(X(t,x))], \\qquad \\varphi\\in B_b(\\mathcal{X}),\\ t\\in[0,T],\\ x\\in \\mathcal{X}; \\end{align*} where $B_b(\\mathcal{X})$ is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on $Q$ and $F$, $P(t)$ enjoys regularizing properties, along a continuously embedded subspace of $\\mathcal{X}$. More precisely there exists $K:=K(F,T)>0$ such that for every $\\varphi\\in B_b(\\mathcal{X})$, $x\\in \\mathcal{X}$, $t\\in(0,T]$ and $h\\in Q^\\alpha(\\mathcal{X})$ it holds \\[|P(t)\\varphi(x+h)-P(t)\\varphi(x)|\\leq Kt^{-1/2}\\|Q^{-\\alpha}h\\|.\\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-11T10:09:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R60",
      "60G15",
      "60H15"
    ],
    "keywords": [
      "hilbert space",
      "transition semigroup",
      "regularizing properties",
      "semilinear stochastic partial differential equation",
      "non-gaussian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}