Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups
type: Item
instance of: scholarly article
title: Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups
zbMATH Open document ID: 1193.47047
author: Guiseppe Da Prato
author: Feng-Yu Wang
author: Michael Röckner
published in: Journal of Functional Analysis
publication date: 2009-07-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/0811.2061
review text: The authors continue earlier investigations [\textit{G.\,Da\,Prato} and \textit{M.\,Röckner}, Probab.\ Theory Relat.\ Fields 124, No.\,2, 261--303 (2002; Zbl 1036.47029)] in order to prove a Harnack inequality for solutions \((X(t))\) of stochastic differential equations (resp., their transition kernels) in the sense of \textit{F.-Y.\thinspace Wang} [Probab.\ Theory Relat.\ Fields 109, No.\,3, 417--424 (1997; Zbl 0887.35012)] of the form \[ d X(t)=(A X(t) + F(X(t)))dt + \sigma d W(t), \quad X(0) = x (\in H), \] where \(H\) is a separable Hilbert space, \((W(t))\) a cylindrical Brownian motion on \(H\), \(\sigma\) a positive definite operator with bounded inverse, \((A, D(A))\) the generator of a \(C_0\)-one-parameter semigroup satisfying the growth condition \(\langle Ax,x\rangle \leq \omega \| x\|^2\) on the domain \(D(A)\), for some real \(\omega\). \(F\) is a set-valued \(m\)-dissipative map \(F:H\supseteq D(F)\to 2^H\). Let \(F_0\) denote a map \(F_0:D(F)\to H\) satisfying \(F_0(x)\in F(x)\) and \(|F_0(x)| = \min_{y\in F(x)}|y|\). The corresponding Kolmogorov operator \(L_0\), defined on a subspace \(\mathcal{E}_A(H)\subseteq B_b(H)\), the space of bounded measurable real functions, is defined by \[ L_0(\varphi)(x)= \tfrac{1}{2} \mathrm{tr}(\sigma^2 D^2 \varphi(x))+ \langle x, A^*D\varphi(x)\rangle + \langle F_0(x), D\varphi(x)\rangle \] for \(x\in D(F)\), \(\varphi\in \mathcal{E}_A(H)\). The investigations rely, as in the aforementioned paper, on several assumptions, \(H_0 - H_5\). In particular, \(H_4\) implies the existence of a infinitesimally invariant probability measure \(\mu\) concentrated on the domain \(D(F)\), and \(L_0\) generates a Markov semigroup of transition kernels, called \(p_t^\mu(\cdot, d x)\) (on \(L^2(H,\mu)\)), such that a Harnack inequality holds for \(p>1\), \(f\in B_b(H)\) (Theorem 1.6): \[ (p_t^\mu f(x))^p \leq p_t^\mu f^p(y)\cdot \exp \left[\|\sigma^{-1}\|^2 p \omega |x-y|^2/\left((p-1)(1-\mathrm{e}^{-2\omega t})\right) \right] \] fo…
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 5586661
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: Harnack inequality
zbMATH Keywords: Kolmogorov operator
zbMATH Keywords: infinitesimally invariant probability
zbMATH Keywords: invariant probability measure
zbMATH Keywords: Markov transition kernels
zbMATH Keywords: monotone coefficients
zbMATH Keywords: Yosida approximation
reviewed by: Wilfried Hazod
MaRDI profile type: Publication
OpenAlex ID: W2962803347
arXiv ID: 0811.2061
cites work: Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below
cites work: Markov processes associated with \(L^{p}\)-resolvents and applications to stochastic differential equations on Hilbert space
cites work: Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators
cites work: Second order PDE's in finite and infinite dimension
cites work: Kolmogorov equations for stochastic PDEs.
cites work: Singular dissipative stochastic equations in Hilbert spaces
cites work: Q4031241
cites work: Stochastic Equations in Infinite Dimensions
cites work: Introduction to the theory of (non-symmetric) Dirichlet forms
cites work: Uniqueness for stochastic evolution equations in Banach spaces
cites work: A concise course on stochastic partial differential equations
cites work: Harnack and functional inequalities for generalized Mehler semigroups.
cites work: Q4943987
cites work: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds
cites work: Harnack inequality and applications for stochastic generalized porous media equations
DOI: 10.1016/J.JFA.2009.01.007
description: scientific article