Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces

D. A. Bignamini, S. Ferrari

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 $α\in [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2α-1}:Q^{1-2α}(\mathcal{X})\subseteq \mathcal{X}\rightarrow \mathcal{X}$. 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^αdW(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 $h\in Q^α(\mathcal{X})$ it holds \[|P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-α}h\|.\]

Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces

Abstract

Let be a separable Hilbert space with norm and let . Let be a linear, self-adjoint, positive, trace class operator on , let be a (smooth enough) function and let be a -valued cylindrical Wiener process. For we consider the operator . We are interested in the mild solution 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^αdW(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 is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on and , enjoys regularizing properties, along a continuously embedded subspace of . More precisely there exists such that for every , , and it holds

Paper Structure

This paper contains 18 sections, 18 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Regularization results
  4. Existence and uniqueness
  5. Space regularity
  6. Proof of Theorem \ref{['Hstrong']}
  7. Proof of Theorem \ref{['Hstrongvar']}
  8. Comparisons with some results in the literature
  9. Comparisons with BON-FUR1FUR1
  10. Comparisons with FE-GO2FE-GO1MAS2MAS1
  11. Examples
  12. An example for $\alpha\in [0,1/2)$
  13. An example for $\alpha=1/2$
  14. An example for Remark \ref{['x_dip']}
  15. A gradient type perturbation
  16. ...and 3 more sections

Key Result

Theorem 4

Assume that Hypotheses hyp1 hold. Then, for any $t\in(0,T]$, the semigroup $P(t)$ maps the space $B_b({\mathcal{X}})$ to $\operatorname{{Lip}}_{b,H_\alpha}({\mathcal{X}})$. More precisely for every $\varphi\in B_b({\mathcal{X}})$, $x\in{\mathcal{X}}$, $h\in H_{\alpha}$ and $t\in (0,T]$ it holds

Theorems & Definitions (44)

  • Definition 2
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Definition 7
  • Theorem 8
  • Theorem 9
  • Lemma 10
  • Proposition 11
  • proof
  • ...and 34 more