Asymptotic log-Harnack inequality and applications for SPDE with degenerate multiplicative noise
type: Item
instance of: scholarly article
title: Asymptotic log-Harnack inequality and applications for SPDE with degenerate multiplicative noise
zbMATH Open document ID: 1454.60094
author: Shihu Li
author: Wei Liu
author: Wei Hong
published in: Statistics \& Probability Letters
publication date: 2020-10-12T00:00:00Z
review text: Let \((H, (.,.), \big\| .\|_{H})\) be a real separable Hilbert space and suppose that \(A\) is a non-negative self-adjoint linear operator and \(V = D\big(A^{\frac12}\big)\) is a reflexive Banach space with the norm \(\big\|u\big\|_{V} = \big\|A^{\frac12} u\big\|_{H}\), which is continuously and densely embedded into \(H\). Assume that there exists an orthonormal basis \(\{ e_k \}_{k \geq 1}\) for \(H\) of eigenfunctions of \(A\), and the associated increasing eigenvalue sequence \(0 <\lambda_1\leq \lambda_2 \leq \cdots \leq \lambda_n \leq \cdots \uparrow \infty\). We thus obtain a Gelfand triple \[V \subset H \equiv H^* \subset V^{*}, \] where \(V^{*}\) (respectively \(H^*\)) is the dual space of \(V\) (resp. \(H\)). The isomorphism \(\equiv\) between \(H\) and \(H^*\) follows from the Riesz representation theorem. Let \(\langle .,. \rangle\) denote the dualization between \(V\) and \(V^{*}\) , it follows that \[\langle u,v \rangle=(u,v),\ u \in H,\ v \in V.\] Let \(\Big(\Omega, \mathcal{F},\big(\mathcal{F}_t\big)_{t \geq 0}, \mathbb{P}\Big)\) be a complete filtered probability space satisfying the usual condition and \((L_2(H , H) , \big\|.\big\|_{L_2})\) be the space consisting of all Hilbert-Schmidt operators from \(H\) to \(H\). For any \(u \in H\), we denote \(u_k := (u , e_k), k \geq 1\). For any \(N \in N\) , we define a projection \(P_N:H \longrightarrow H\) by \[P_Nu :=\sum_{|k| \leq N}u_k e_k ,\quad u \in H .\] The authors consider the following stochastic evolution equation: \[du(t)=[-Au(t)+F(u(t))]dt+B(u(t))dW(t),\quad u(0)=x \in H,\tag{1.1}\] where \(W(t)\) is a cylindrical Wiener process in \(H\) defined on \(\Big(\Omega, \mathcal{F},\big(\mathcal{F}_t\big)_{t \geq 0},\mathbb{P} \Big)\). The function \(F : V\longrightarrow V^*\) and \(B: V\longrightarrow L_2(H,H)\) are measurable and satisfy the following conditions. (1) For all \(u,v,w \in V\), \(s\longmapsto \langle F(u+sv),w \rangle\) is is continuous on \(\mathbb{R}\). (2) For all \(u,v\in V,\) Th…
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 7259056
zbMATH Keywords: coupling method
zbMATH Keywords: Harnack inequality
zbMATH Keywords: strong Feller property
zbMATH Keywords: asymptotically strong Feller property
zbMATH Keywords: ergodicity
zbMATH Keywords: gradient estimate
zbMATH Keywords: degenerate noise
zbMATH Keywords: Krylov-Bogoliubov procedure
reviewed by: El Houcein El Abdalaoui
MaRDI profile type: Publication
full work available at URL: j.spl.2020.108810
OpenAlex ID: W3025184043
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DOI: 10.1016/J.SPL.2020.108810
description: scientific article; zbMATH DE number 7259056