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Book DetailsProceedings of Symposia in Pure MathematicsVolume: 57; 1995; 621 ppMSC: Primary 34; 35; 58; 60; 82; 47; 81;
This book deals with current developments in stochastic analysis and its interfaces with partial differential equations, dynamical systems, mathematical physics, differential geometry, and infinitedimensional 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.
ReadershipResearchers in probability theory, partial differential equations, statistical mechanics, dynamical systems, and mathematical physics.

Table of Contents

Part I. Problems in analysis

Rodrigo Bañuelos and Tom Carroll  An improvement of the Osserman constant for the bass note of a drum [ MR 1335458 ]

M. van den Berg  Heat content asymptotics for some open sets with a fractal boundary [ MR 1335459 ]

Erwin Bolthausen and Uwe Schmock  On selfattracting random walks [ MR 1335460 ]

Jean Brossard  Positivity default for martingales and harmonic functions [ MR 1335461 ]

R. Cairoli and Robert C. Dalang  Optimal switching between two Brownian motions [ MR 1335462 ]

Z. Q. Chen, R. J. Williams and Z. Zhao  Nonnegative solutions for semilinear elliptic equations with boundary conditions—a probabilistic approach [ MR 1335463 ]

TzuuShuh Chiang and Yunshong Chow  Simulated annealing and fastest cooling rates for some $1$dim spin glass models [ MR 1335464 ]

Thomas G. Kurtz and Federico Marchetti  Averaging stochastically perturbed Hamiltonian systems [ MR 1335465 ]

Terry J. Lyons  The interpretation and solution of ordinary differential equations driven by rough signals [ MR 1335466 ]

L. C. G. Rogers  Timereversal of the noisy WienerHopf factorisation [ MR 1335467 ]

AlainSol Sznitman  Some aspects of Brownian motion in a Poissonian potential [ MR 1335468 ]

A. Truman, D. Williams and K. Y. Yu  Schrödinger operators and asymptotics for PoissonLévy excursion measures for onedimensional timehomogeneous diffusions [ MR 1335469 ]

Shinzo Watanabe  Generalized arcsine laws for onedimensional diffusion processes and random walks [ MR 1335470 ]

Part II. Problems in geometry

Hélène Airault and Paul Malliavin  Semimartingales with values in a Euclidean vector bundle and Ocone’s formula on a Riemannian manifold [ MR 1335471 ]

Gérard Ben Arous, Michael Cranston and Wilfrid S. Kendall  Coupling constructions for hypoelliptic diffusions: two examples [ MR 1335472 ]

Itai Benjamini, Isaac Chavel and Edgar A. Feldman  Heat kernel bounds on Riemannian manifolds [ MR 1335473 ]

M. Brin and Yu. Kifer  Brownian motion and harmonic functions on polygonal complexes [ MR 1335474 ]

Alexander Grigor′yan  Heat kernel of a noncompact Riemannian manifold [ MR 1335475 ]

Elton P. Hsu  Flows and quasiinvariance of the Wiener measure on path spaces [ MR 1335476 ]

N. Ikeda, S. Kusuoka and S. Manabe  Lévy’s stochastic area formula and related problems [ MR 1335477 ]

Yuri Kifer  Markov processes and harmonic functions on hyperbolic metric spaces [ MR 1335478 ]

Hiroshi Kunita  Some problems concerning Lévy processes on Lie groups [ MR 1335479 ]

Y. Le Jan  The central limit theorem for geodesic flows on noncompact manifolds of constant negative curvature [ MR 1335480 ]

F. Ledrappier  A renewal theorem for the distance in negative curvature [ MR 1335481 ]

Paul Malliavin  A bootstrap proof of the limit theorem for linear SDE [ MR 1335482 ]

Wei An Zheng  Diffusion processes on a Lipschitz Riemannian manifold and their applications [ MR 1335483 ]

Part III. Infinitedimensional problems

Donald A. Dawson, Kenneth J. Hochberg and Vladimir Vinogradov  On path properties of super2 processes. II [ MR 1335484 ]

Bruce K. Driver  Towards calculus and geometry on path spaces [ MR 1335485 ]

E. B. Dynkin  Branching with a single point catalyst [ MR 1335486 ]

JeanFrançois Le Gall  The Brownian pathvalued process and its connections with partial differential equations [ MR 1335487 ]

Yao Zhong Hu, Tom Lindstrøm, Bernt Øksendal, Jan Ubøe and Tu Sheng Zhang  Inverse powers of white noise [ MR 1335488 ]

H. P. McKean and K. L. Vaninsky  Statistical mechanics of nonlinear wave equations [ MR 1335489 ]

David Nualart  Markov properties for solutions of stochastic differential equations [ MR 1335490 ]

Ichirō Shigekawa  A quasihomeomorphism on the Wiener space [ MR 1335491 ]

Ali Süleyman Üstünel and Moshe Zakai  Absolute continuity on the Wiener space and some applications [ MR 1335492 ]

K. L. Vaninsky  Invariant Gibbsian measures of the KleinGordon equation [ MR 1335493 ]

Part IV. Stochastic PDE/stochastic flows

Sergio Albeverio and Michael Röckner  Dirichlet form methods for uniqueness of martingale problems and applications [ MR 1335494 ]

Ludwig Arnold  Anticipative problems in the theory of random dynamical systems [ MR 1335495 ]

Ludwig Arnold and Rafail Z. Khasminskii  Stability index for nonlinear stochastic differential equations [ MR 1335496 ]

Denis R. Bell and Salah Eldin A. Mohammed  Degenerate stochastic differential equations, flows, and hypoellipticity [ MR 1335497 ]

K. D. Elworthy and X.M. Li  Derivative flows of stochastic differential equations: moment exponents and geometric properties [ MR 1335498 ]

Ming Liao  Invariant diffusion processes in Lie groups and stochastic flows [ MR 1335499 ]

R. Mikulevičius and B. L. Rozovskiĭ  On stochastic integrals in topological vector spaces [ MR 1335500 ]

C. Mueller and R. Sowers  Travelling waves for the KPP equation with noise [ MR 1335501 ]

Étienne Pardoux  Backward SDEs, quasilinear PDEs, and SPDEs [ MR 1335502 ]

Mark A. Pinsky  Invariance of the Lyapunov exponent under nonlinear perturbations [ MR 1335503 ]


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 everincreasing field of stochastics and analysis.
Metrika, International Journal for Theoretical and Applied Statistics


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

Part I. Problems in analysis

Rodrigo Bañuelos and Tom Carroll  An improvement of the Osserman constant for the bass note of a drum [ MR 1335458 ]

M. van den Berg  Heat content asymptotics for some open sets with a fractal boundary [ MR 1335459 ]

Erwin Bolthausen and Uwe Schmock  On selfattracting random walks [ MR 1335460 ]

Jean Brossard  Positivity default for martingales and harmonic functions [ MR 1335461 ]

R. Cairoli and Robert C. Dalang  Optimal switching between two Brownian motions [ MR 1335462 ]

Z. Q. Chen, R. J. Williams and Z. Zhao  Nonnegative solutions for semilinear elliptic equations with boundary conditions—a probabilistic approach [ MR 1335463 ]

TzuuShuh Chiang and Yunshong Chow  Simulated annealing and fastest cooling rates for some $1$dim spin glass models [ MR 1335464 ]

Thomas G. Kurtz and Federico Marchetti  Averaging stochastically perturbed Hamiltonian systems [ MR 1335465 ]

Terry J. Lyons  The interpretation and solution of ordinary differential equations driven by rough signals [ MR 1335466 ]

L. C. G. Rogers  Timereversal of the noisy WienerHopf factorisation [ MR 1335467 ]

AlainSol Sznitman  Some aspects of Brownian motion in a Poissonian potential [ MR 1335468 ]

A. Truman, D. Williams and K. Y. Yu  Schrödinger operators and asymptotics for PoissonLévy excursion measures for onedimensional timehomogeneous diffusions [ MR 1335469 ]

Shinzo Watanabe  Generalized arcsine laws for onedimensional diffusion processes and random walks [ MR 1335470 ]

Part II. Problems in geometry

Hélène Airault and Paul Malliavin  Semimartingales with values in a Euclidean vector bundle and Ocone’s formula on a Riemannian manifold [ MR 1335471 ]

Gérard Ben Arous, Michael Cranston and Wilfrid S. Kendall  Coupling constructions for hypoelliptic diffusions: two examples [ MR 1335472 ]

Itai Benjamini, Isaac Chavel and Edgar A. Feldman  Heat kernel bounds on Riemannian manifolds [ MR 1335473 ]

M. Brin and Yu. Kifer  Brownian motion and harmonic functions on polygonal complexes [ MR 1335474 ]

Alexander Grigor′yan  Heat kernel of a noncompact Riemannian manifold [ MR 1335475 ]

Elton P. Hsu  Flows and quasiinvariance of the Wiener measure on path spaces [ MR 1335476 ]

N. Ikeda, S. Kusuoka and S. Manabe  Lévy’s stochastic area formula and related problems [ MR 1335477 ]

Yuri Kifer  Markov processes and harmonic functions on hyperbolic metric spaces [ MR 1335478 ]

Hiroshi Kunita  Some problems concerning Lévy processes on Lie groups [ MR 1335479 ]

Y. Le Jan  The central limit theorem for geodesic flows on noncompact manifolds of constant negative curvature [ MR 1335480 ]

F. Ledrappier  A renewal theorem for the distance in negative curvature [ MR 1335481 ]

Paul Malliavin  A bootstrap proof of the limit theorem for linear SDE [ MR 1335482 ]

Wei An Zheng  Diffusion processes on a Lipschitz Riemannian manifold and their applications [ MR 1335483 ]

Part III. Infinitedimensional problems

Donald A. Dawson, Kenneth J. Hochberg and Vladimir Vinogradov  On path properties of super2 processes. II [ MR 1335484 ]

Bruce K. Driver  Towards calculus and geometry on path spaces [ MR 1335485 ]

E. B. Dynkin  Branching with a single point catalyst [ MR 1335486 ]

JeanFrançois Le Gall  The Brownian pathvalued process and its connections with partial differential equations [ MR 1335487 ]

Yao Zhong Hu, Tom Lindstrøm, Bernt Øksendal, Jan Ubøe and Tu Sheng Zhang  Inverse powers of white noise [ MR 1335488 ]

H. P. McKean and K. L. Vaninsky  Statistical mechanics of nonlinear wave equations [ MR 1335489 ]

David Nualart  Markov properties for solutions of stochastic differential equations [ MR 1335490 ]

Ichirō Shigekawa  A quasihomeomorphism on the Wiener space [ MR 1335491 ]

Ali Süleyman Üstünel and Moshe Zakai  Absolute continuity on the Wiener space and some applications [ MR 1335492 ]

K. L. Vaninsky  Invariant Gibbsian measures of the KleinGordon equation [ MR 1335493 ]

Part IV. Stochastic PDE/stochastic flows

Sergio Albeverio and Michael Röckner  Dirichlet form methods for uniqueness of martingale problems and applications [ MR 1335494 ]

Ludwig Arnold  Anticipative problems in the theory of random dynamical systems [ MR 1335495 ]

Ludwig Arnold and Rafail Z. Khasminskii  Stability index for nonlinear stochastic differential equations [ MR 1335496 ]

Denis R. Bell and Salah Eldin A. Mohammed  Degenerate stochastic differential equations, flows, and hypoellipticity [ MR 1335497 ]

K. D. Elworthy and X.M. Li  Derivative flows of stochastic differential equations: moment exponents and geometric properties [ MR 1335498 ]

Ming Liao  Invariant diffusion processes in Lie groups and stochastic flows [ MR 1335499 ]

R. Mikulevičius and B. L. Rozovskiĭ  On stochastic integrals in topological vector spaces [ MR 1335500 ]

C. Mueller and R. Sowers  Travelling waves for the KPP equation with noise [ MR 1335501 ]

Étienne Pardoux  Backward SDEs, quasilinear PDEs, and SPDEs [ MR 1335502 ]

Mark A. Pinsky  Invariance of the Lyapunov exponent under nonlinear perturbations [ MR 1335503 ]

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 everincreasing field of stochastics and analysis.
Metrika, International Journal for Theoretical and Applied Statistics