Stochastic Analysis: (Proceedings of Symposia in Pure Mathematics)

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 Ito'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 Comell University. Many of the papers are largely expository in character while containing new results.

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