Mehler formula and capacities for infinite dimensional Ornstein-Uhlenbeck processes with general linear drift
type: Item
instance of: scholarly article
title: Mehler formula and capacities for infinite dimensional Ornstein-Uhlenbeck processes with general linear drift
zbMATH Open document ID: 0849.60004
published in: Osaka Journal of Mathematics
publication date: 1996-11-07T00:00:00Z
review text: This article essentially decomposes in the following three main results, that are refinements of previous results by Sugita, Feyel and La Pradelle, Shigekawa, Röckner. Firstly consider a locally convex vector space \(E\) endowed with a centered Gaussian measure and its Mehler semigroup \(p_t\); let \(C_{r,p}\) be the \((r,p)\)-capacity associated with the Gamma transform of \(p_t\); it has been known, for \(E\) Banach, that these \(C_{r,p}\) are tight on compact sets. Here it is shown that they are tight on metrizable compact sets. Secondly consider a positive selfadjoint operator \(A\) on a separable Hilbert space \(H\). A bigger separable Banach space \(E\) carrying a centered Gaussian probability \(\gamma\) (with given covariance commuting with \(A)\) is constructed, with the property that \(e^{- tA}\) extends to a strongly continuous contraction semigroup on \(E\). Then the generalized Mehler semigroup \(p^A_t\) with kernel \((e^{- tA} x + \sqrt {1 - e^{- 2tA}}y)\) and measure \(\gamma\) makes sense on \(E\). Now the third main result is that the capacity \(C^A_{1,2}\) associated (like above) with \(p^A_t\) is tight iff \(p^A_t\) is the semigroup of a diffusion \(X_t\) on \(E\) (that weakly solves \(dX_t = dW_t + A(X_t) dt\), \(W_t\) being Brownian on \(E\) with the covariance of \(\gamma)\).
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 880171
zbMATH Keywords: Mehler semigroup
zbMATH Keywords: capacities
zbMATH Keywords: Gaussian measure
zbMATH Keywords: Ornstein-Uhlenbeck diffusion
zbMATH Keywords: tightness
zbMATH Keywords: self-adjoint operator
author: Vladimir I. Bogachev
author: Michael Röckner
MaRDI profile type: Publication
Recommended article: Approximations of ornstein-uhlenbeck processes with unbounded linear drifts
Recommended article: Capacity estimates, boundary crossings and the Ornstein-Uhlenbeck process in Wiener space
Recommended article: Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes
Recommended article: Ornstein-Uhlenbeck processes with singular drifts: integral estimates and Girsanov densities
Recommended article: On the almost surely asymptotic bounds of a class of Ornstein-Uhlenbeck processes in infinite dimensions
Recommended article: On the continuity of Ornstein-Uhlenbeck processes in infinite dimensions
Recommended article: Q4297333
description: scientific article; zbMATH DE number 880171