Stochastic Analysis
Proceedings of the Durham Symposium on Stochastic Analysis, 1990
Edited by M. T. Barlow
Edited by N. H. Bingham
Publisher: Cambridge University Press
Print Publication Year: 1991
Online Publication Date:March 2010
Online ISBN:9780511662980
Paperback ISBN:9780521425339
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511662980.012
Subjects: Probability Theory and Stochastic Processes, Abstract Analysis
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Introduction.
In 1984 [P1] we introduced the theme of inverse problems for Brownian motion on Riemannian manifolds, in terms of the mean exit time from small geodesic balls. Since that time a number of works have appeared on related stochastic problems as well as on some classical, non-stochastic quantities which may be treated by the same methods. Most recently H.R. Hughes [Hu] has shown that in six dimensions one cannot recover the Riemannian metric from the exit time distribution, thereby answering in a strong sense the main question posed in [P1].
The general area of “inverse spectral theory” was initiated by Mark Kac in his now famous paper [Ka] on the two-dimensional drumhead. In the intervening years a large literature has developed on inverse spectral problems in higher dimensional Euclidean space and differentiable manifolds; for recent surveys see ([Be], [Br], [Go]). In these approaches one is given the entire spectrum of eigenvalues, from which one asks various geometric questions. Our approach, by contrast, is able to obtain strong geometric information from the sole knowledge of the principal eigenvalue of a parametric family of geodesic balls (see section 6, below).
pp. i-iv
pp. v-vi
pp. vii-vii
List of participants : Read PDF
pp. viii-viii
An evolution equation for the intersection local times of superprocesses : Read PDF
pp. 1-22
The Continuum random tree II: an overview : Read PDF
pp. 23-70
Harmonic morphisms and the resurrection of Markov processes : Read PDF
pp. 71-90
Statistics of local time and excursions for the Ornstein–Uhlenbeck process : Read PDF
pp. 91-102
LP-Chen forms on loop spaces : Read PDF
pp. 103-162
Convex geometry and nonconfluent Γ-martingales I: tightness and strict convexity : Read PDF
pp. 163-178
Some caricatures of multiple contact diffusion-limited aggregation and the η-model : Read PDF
pp. 179-228
Limits on random measures and stochastic difference equations related to mixing array of random variables : Read PDF
pp. 229-254
Characterizing the weak convergence of stochastic integrals : Read PDF
pp. 255-260
Stochastic differential equations involving positive noise : Read PDF
pp. 261-304
Feeling the shape of a manifold with Brownian motion — the last word in 1990 : Read PDF
pp. 305-320
Decomposition of Dirichlet processes on Hilbert space : Read PDF
pp. 321-332
A supersymmetric Feynman-Kac formula : Read PDF
pp. 333-352
On long excursions of Brownian motion among Poissonian obstacles : Read PDF
pp. 353-375