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Stochastic Analysis
Edited by: Michael C. Cranston, University of Rochester, NY, and Mark A. Pinsky, Northwestern University, Evanston, IL
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Proceedings of Symposia in Pure Mathematics
1995; 621 pp; hardcover
Volume: 57
ISBN-10: 0-8218-0289-5
ISBN-13: 978-0-8218-0289-2
List Price: US$156
Member Price: US$124.80
Order Code: PSPUM/57
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This book deals with current developments in stochastic analysis and its interfaces with partial differential equations, dynamical systems, mathematical physics, differential geometry, and infinite-dimensional analysis. The origins of stochastic analysis can be found in Norbert Wiener's construction of Brownian motion and Kiyosi Itô's subsequent development of stochastic integration and the closely related theory of stochastic (ordinary) differential equations. The papers in this volume indicate the great strides that have been made in recent years, exhibiting the tremendous power and diversity of stochastic analysis while giving a clear indication of the unsolved problems and possible future directions for development. The collection represents the proceedings of the AMS Summer Research Institute on Stochastic Analysis, held in July 1993 at Cornell University. Many of the papers are largely expository in character while containing new results.

Readership

Researchers in probability theory, partial differential equations, statistical mechanics, dynamical systems, and mathematical physics.

Reviews

"This very rich volume ... is an extremely valuable contribution to the literature on the interplay between stochastics and analysis. The editors have done a marvelous job in collecting a number of papers on subjects which in their entirety constitute the current activities in the field ... This volume has a particularly high standard and appears to be of special significance to current research activity in the ever-increasing field of stochastics and analysis."

-- Metrika, International Journal for Theoretical and Applied Statistics

Table of Contents

Part I. Problems in Analysis
  • R. Bañuelos and T. Carroll -- An improvement of the Osserman constant for the bass note of a drum
  • M. van den Berg -- Heat content asymptotics for some open sets with a fractal boundary
  • E. Bolthausen and U. Schmock -- On self-attracting random walks
  • J. Brossard -- Positivity default for martingales and harmonic functions
  • R. Cairoli and R. C. Dalang -- Optimal switching between two Brownian motions
  • Z. Q. Chen, R. J. Williams, and Z. Zhao -- Nonnegative solutions for semilinear elliptic equations with boundary conditions--a probabilistic approach
  • T.-S. Chiang and Y. Chow -- Simulated annealing and fastest cooling rates for some 1-dim spin glass models
  • T. G. Kurtz and F. Marchetti -- Averaging stochastically perturbed Hamiltonian systems
  • T. J. Lyons -- The interpretation and solution of ordinary differential equations driven by rough signals
  • L. C. G. Rogers -- Time-reversal of the noisy Wiener-Hopf factorisation
  • A.-S. Sznitman -- Some aspects of Brownian motion in a Poissonian potential
  • A. Truman, D. Williams, and K. Y. Yu -- Schrödinger operators and asymptotics for Poisson-Lévy excursion measures for one-dimensional time-homogeneous diffusions
  • S. Watanabe -- Generalized arc-sine laws for one-dimensional diffusion processes and random walks
Part II. Problems in Geometry
  • H. Airault and P. Malliavin -- Semimartingales with values in a Euclidean vector bundle and Ocone's formula on a Riemannian manifold
  • G. B. Arous, M. Cranston, and W. S. Kendall -- Coupling constructions for hypoelliptic diffusions: Two examples
  • I. Benjamini, I. Chavel, and E. A. Feldman -- Heat kernel bounds on Riemannian manifolds
  • M. Brin and Yu. Kifer -- Brownian motion and harmonic functions on polygonal complexes
  • A. Grigor'yan -- Heat kernel of a noncompact Riemannian manifold
  • E. P. Hsu -- Flows and quasi-invariance of the Wiener measure on path spaces
  • N. Ikeda, S. Kusuoka, and S. Manabe -- Lévy's stochastic area formula and related problems
  • Yu. Kifer -- Markov processes and harmonic functions on hyperbolic metric spaces
  • H. Kunita -- Some problems concerning Lévy processes on Lie groups
  • Y. L. Jan -- The central limit theorem for geodesic flows on noncompact manifolds of constant negative curvature
  • F. Ledrappier -- A renewal theorem for the distance in negative curvature
  • P. Malliavin -- A bootstrap proof of the limit theorem for linear SDE
  • W. Zheng -- Diffusion processes on a Lipschitz Riemannian manifold and their applications
Part III. Infinite-Dimensional Problems
  • D. A. Dawson, K. J. Hochberg, and V. Vinogradov -- On path properties of super-2 processes. II
  • B. K. Driver -- Towards calculus and geometry on path spaces
  • E. B. Dynkin -- Branching with a single point catalyst
  • J.-F. Le Gall -- The Brownian path-valued process and its connections with partial differential equations
  • Y. Hu, T. Lindstrøm, B. Øksendal, J. Ubøe, and T. Zhang -- Inverse powers of white noise
  • H. P. McKean and K. L. Vaninsky -- Statistical mechanics of nonlinear wave equations
  • D. Nualart -- Markov properties for solutions of stochastic differential equations
  • I. Shigekawa -- A quasihomeomorphism on the Wiener space
  • A. S. Üstünel and M. Zakai -- Absolute continuity on the Wiener space and some applications
  • K. L. Vaninsky -- Invariant Gibbsian measures of the Klein-Gordon equation
Part IV. Stochastic PDE/ Stochastic Flows
  • S. Albeverio and M. Röckner -- Dirichlet form methods for uniqueness of martingale problems and applications
  • L. Arnold -- Anticipative problems in the theory of random dynamical systems
  • L. Arnold and R. Z. Khasminskii -- Stability index for nonlinear stochastic differential equations
  • D. R. Bell and S.-E. A. Mohammed -- Degenerate stochastic differential equations, flows, and hypoellipticity
  • K. D. Elworthy and X.-M. Li -- Derivative flows of stochastic differential equations: Moment exponents and geometric properties
  • M. Liao -- Invariant diffusion processes in Lie groups and stochastic flows
  • R. Mikulevicius and B. L. Rozovskii -- On stochastic integrals in topological vector spaces
  • C. Mueller and R. Sowers -- Travelling waves for the KPP equation with noise
  • E. Pardoux -- Backward SDEs, quasilinear PDEs, and SPDEs
  • M. A. Pinsky -- Invariance of the Lyapunov exponent under nonlinear perturbations
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