On the parabolic Martin boundary of the Ornstein-Uhlenbeck operator on Wiener space
type: Item
instance of: scholarly article
title: On the parabolic Martin boundary of the Ornstein-Uhlenbeck operator on Wiener space
zbMATH Open document ID: 0761.60067
published in: The Annals of Probability
publication date: 1993-01-16T00:00:00Z
review text: Let \(E\) be a separable real Banach space, \(\mu\) be a Gaussian mean zero measure on \((E,{\mathcal B}(E))\) and \((E,H,\mu)\) be an abstract Wiener space. The function \(p(t,\cdot)\) is called \(L\)-parabolic if \(p(t,\cdot)\in L^ 2(E,\mu)\cap D(L)\) and \(Lp(t,\cdot)={d\over dt}p(t,\cdot)\), where \(L\) is a generator of the Ornstein-Uhlenbeck process on \(E\). A positive \(L\)- parabolic function \(p(\cdot,\cdot)\) is called normalized if \(\int_ E p(t,z)\mu(dz)=1\), and let \(\mathcal P\) denote the convex set of all normalized, positive parabolic functions on \(E\). The square-integrable parabolic functions on the space \((E,H,\mu)\) are studied. The author studies the extreme points of the set \(\mathcal P\) and the possibilities to have the integral representation for any \(p\in{\mathcal P}\) in terms of them. The investigation method is similar to the approach developed by Dynkin.
reviewed by: N. S. Bratijchuk
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 89802
zbMATH Keywords: abstract Wiener space
zbMATH Keywords: Ornstein-Uhlenbeck process
zbMATH Keywords: parabolic functions
author: Michael Röckner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989818
OpenAlex ID: W1970762621
DOI: 10.1214/AOP/1176989818
Recommended article: Q3319533
Recommended article: Q3350425
Recommended article: The Ornstein-Uhlenbeck process over the space of Riemannian paths and the martingale problem
Recommended article: On the Ornstein-Uhlenbeck operators on Wiener-Riemannian manifolds
Recommended article: On a Dirichlet problem involving an Ornstein--Uhlenbeck operator
Recommended article: Martingale representation and logarithmic-Sobolev inequality for the fractional Ornstein-Uhlenbeck measure
Recommended article: L p -Parabolic Regularity and Non-degenerate Ornstein-Uhlenbeck Type Operators
Recommended article: Sobolev spaces over the Wiener space based on an Ornstein-Uhlenbeck operator
description: scientific article; zbMATH DE number 89802