Book contents
- Frontmatter
- Contents
- Preface
- List of participants
- An evolution equation for the intersection local times of superprocesses
- The Continuum random tree II: an overview
- Harmonic morphisms and the resurrection of Markov processes
- Statistics of local time and excursions for the Ornstein–Uhlenbeck process
- LP-Chen forms on loop spaces
- Convex geometry and nonconfluent Γ-martingales I: tightness and strict convexity
- Some caricatures of multiple contact diffusion-limited aggregation and the η-model
- Limits on random measures and stochastic difference equations related to mixing array of random variables
- Characterizing the weak convergence of stochastic integrals
- Stochastic differential equations involving positive noise
- Feeling the shape of a manifold with Brownian motion — the last word in 1990
- Decomposition of Dirichlet processes on Hilbert space
- A supersymmetric Feynman-Kac formula
- On long excursions of Brownian motion among Poissonian obstacles
LP-Chen forms on loop spaces
Published online by Cambridge University Press: 31 March 2010
- Frontmatter
- Contents
- Preface
- List of participants
- An evolution equation for the intersection local times of superprocesses
- The Continuum random tree II: an overview
- Harmonic morphisms and the resurrection of Markov processes
- Statistics of local time and excursions for the Ornstein–Uhlenbeck process
- LP-Chen forms on loop spaces
- Convex geometry and nonconfluent Γ-martingales I: tightness and strict convexity
- Some caricatures of multiple contact diffusion-limited aggregation and the η-model
- Limits on random measures and stochastic difference equations related to mixing array of random variables
- Characterizing the weak convergence of stochastic integrals
- Stochastic differential equations involving positive noise
- Feeling the shape of a manifold with Brownian motion — the last word in 1990
- Decomposition of Dirichlet processes on Hilbert space
- A supersymmetric Feynman-Kac formula
- On long excursions of Brownian motion among Poissonian obstacles
Summary
INTRODUCTION
Let M be a Riemannian manifold. Our aim is to study differential forms on the following infinite dimensional manifolds:
(1) the path space PM consisting of paths w : [0, 1] → M,
(2) the loop space LM consisting of paths w such that w(0) = w(1),
(3) the based loop space LxM consisting of loops w such that w(0) = w(1) = x where x is a chosen base point in M.
One consequence of the fact that these manifolds are infinite dimensional is that there are infinite sequences αn of forms with each αn homogeneous of degree n. These infinite sequences are very important in the geometrical applications of loop spaces; for example they are essential in the theory of equivariant cohomology in infinite dimensions as is made quite clear in [25].
In [11] Chen describes the theory of “iterated integrals”; this is a method of constructing differential forms on these infinite dimensional manifolds. We will refer to forms constructed by this means as Chen forms. The purpose of this paper is to study some of the analytical properties of Chen forms; in particular to make estimates for suitable LP-norms and to consider various decay conditions which one might put on the terms in an infinite sequence of the kind mentioned in the previous paragraph.
- Type
- Chapter
- Information
- Stochastic AnalysisProceedings of the Durham Symposium on Stochastic Analysis, 1990, pp. 103 - 162Publisher: Cambridge University PressPrint publication year: 1991
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