Renormalized differential geometry on path space: Structural equation, curvature
type: Item
instance of: scholarly article
title: Renormalized differential geometry on path space: Structural equation, curvature
zbMATH Open document ID: 0869.60060
author: Paul Malliavin
author: Ana Bela Cruzeiro
published in: Journal of Functional Analysis
publication date: 1997-08-13T00:00:00Z
review text: The structure of manifolds of maps of a compact manifold attracts the interest of mathematicians for more than thirty years starting from the works by Ebin, Marsden, Eliasson and others [see for example \textit{D. G. Ebin} and \textit{J. Marsden}, Ann. Math., II. Ser. 92, 102-163 (1970; Zbl 0211.57401), \textit{H. I. Eliasson}, J. Differ. Geom. 1, 169-194 (1967; Zbl 0163.43901)]. Those manifolds are infinite-dimensional and should be carefully studied to avoid divergence and explosions of some classical differential geometry structures. The authors describe differential geometry of a manifold \(P_{m_0}(M)= \{p(\cdot)\in C([0,1], M):p(0)=m_0\} \) where \(M\) is a compact Riemannian manifold with a model space \(R^d.\) As far as \(P_{m_0}(M)\) comes to be a probability space being endowed with the natural filtration under Wiener measure induced by the Brownian motion \(p\) on \(M\) they use such a powerful tool as Malliavin calculus to describe its differential geometry structures. In particular such notions as connection and especially Levi-Civita connection, parallel translation, curvature tensor, structure equations and others are constructed for the infinite-dimensional manifold \(P_{m_0}(M)\). The main tool the authors develop and work with is the stochastic Stratonovich integral on \(P_{m_0}(M)\) and stochastic differential equation (SDE) theory associated with it. The Stratonovich integral for a vector field \(Z\) on \(M\) is determined as a limit of the Riemannian sum \[ \frac{1}{2}\sum (Z_{p(\sigma)},[\text{exp}^{-1}_{p(\sigma)}(p(\sigma^+)) - \text{exp}^{-1}_{p(\sigma)}(p(\sigma^-))])_{T_{p(\sigma)}(M)} \] where exp is an exponetial map on \(M\), \(p(t)\in M\) is a diffusion process having the Beltrami-Laplace operator \(\Delta_M\) for its generator, \(S\) is a finite part of \([0,1]\) and \[ \sigma^+=\text{inf}\{\sigma'\in S\cup 0:\sigma\leq \sigma'\},\quad \sigma^-=\text{sup}\{\sigma'\in S\cup 0:\sigma'\leq \sigma\}. \] To determine and investigate the nec…
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 944393
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: manifolds
zbMATH Keywords: bundles
zbMATH Keywords: path space
MaRDI profile type: Publication
full work available at URL: jfan.1996.0081
OpenAlex ID: W2028298308
DOI: 10.1006/JFAN.1996.0081
description: scientific article; zbMATH DE number 944393