Integration by parts and quasi-invariance for heat kernel measures on loop groups
type: Item
instance of: scholarly article
title: Integration by parts and quasi-invariance for heat kernel measures on loop groups
zbMATH Open document ID: 0887.58062
author: Bruce K. Driver
published in: Journal of Functional Analysis
publication date: 1998-05-10T00:00:00Z
review text: Let \(G\) be a connected compact Lie group with \(Ad_G\)-invariant inner product in the algebra. Denote by \(L (G)\) the space of continuous loops in \(G\) based at the identity. The author proves two versions of the integration by parts formula: for the Wiener measure on the path space of \(L (G)\) and for the heat kernel measures on \(L (G)\). The former is an infinite-dimensional version of that proved by the author earlier for compact manifolds, it deals with a certain class of vector fields on the path space. The latter is for left-invariant first order differential operators on \(L (G)\), it is an infinite-dimensional analog of the fact that heat kernels have logarithmic derivatives in finite dimension. Both constructions allow the author to show that the heat kernel measures are quasi-invariant under right and left translations by finite energy loops, and to obtain a description of a certain collection of logarithmic Sobolev inequalities for finite-dimensional approximation as a truly infinite-dimensional inequality on the loop group.
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 58C35
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 1078822
zbMATH Keywords: heat kernel
zbMATH Keywords: loop group
zbMATH Keywords: integration by parts
reviewed by: Yuri E. Gliklikh
MaRDI profile type: Publication
OpenAlex ID: W1988949407
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DOI: 10.1006/JFAN.1997.3103
description: scientific article; zbMATH DE number 1078822