Lectures in Applied Mathematics 1991; 499 pp; softcover Volume: 27 ISBN10: 0821811339 ISBN13: 9780821811337 List Price: US$245 Member Price: US$196 Order Code: LAM/27
 In recent years, there has been remarkable growth in the mathematics of random media. The field has deep scientific and technological roots, as well as purely mathematical ones in the theory of stochastic processes. This collection of papers by leading researchers provides an overview of this rapidly developing field. The papers were presented at the 1989 AMSSIAM Summer Seminar in Applied Mathematics, held at Virginia Polytechnic Institute and State University in Blacksburg, Virginia. In addition to new results on stochastic differential equations and Markov processes, fields whose elegant mathematical techniques are of continuing value in application areas, the conference was organized around four themes: Systems of interacting particles are normally viewed in connection with the fundamental problems of statistical mechanics, but have also been used to model diverse phenomena such as computer architectures and the spread of biological populations. Powerful mathematical techniques have been developed for their analysis, and a number of important systems are now well understood. Random perturbations of dynamical systems have also been used extensively as models in physics, chemistry, biology, and engineering. Among the recent unifying mathematical developments is the theory of large deviations, which enables the accurate calculation of the probabilities of rare events. For these problems, approaches based on effective but formal perturbation techniques parallel rigorous mathematical approaches from probability theory and partial differential equations. The book includes representative papers from forefront research of both types. Effective medium theory, otherwise known as the mathematical theory of homogenization, consists of techniques for predicting the macroscopic properties of materials from an understanding of their microstructures. For example, this theory is fundamental in the science of composites, where it is used for theoretical determination of electrical and mechanical properties. Furthermore, the inverse problem is potentially of great technological importance in the design of composite materials which have been optimized for some specific use. Mathematical theories of the propagation of waves in random media have been used to understand phenomena as diverse as the twinkling of stars, the corruption of data in geophysical exploration, and the quantum mechanics of disordered solids. Especially effective methods now exist for waves in randomly stratified, onedimensional media. A unifying theme is the mathematical phenomenon of localization, which occurs when a wave propagating into a random medium is attenuated exponentially with propagation distance, with the attenuation caused solely by the mechanism of random multiple scattering. Because of the wide applicability of this field of research, this book would appeal to mathematicians, scientists, and engineers in a wide variety of areas, including probabilistic methods, the theory of disordered materials, systems of interacting particles, the design of materials, and dynamical systems driven by noise. In addition, graduate students and others will find this book useful as an overview of current research in random media. Readership Table of Contents  R. Durrett  The contact process, 19741989
 L. Gray  Is the contact process dead?
 T. Liggett  Limiting behavior of a onedimensional system with long range interactions
 C. Newman  Topics in percolation
 G. Slade  The lace expansion and the upper critical dimension for percolation
 G. Swindle  Mandelbrot percolation in two and three dimensions
 G. Lawler  Intersection probabilities for random walks
 R. Fisch  A summary of results about cyclic cellular automata
 J.P. Fouque  Hydrodynamical behavior of asymmetric attractive particle systems, one example: onedimensional nearest neighbors asymmetric simple exclusion process
 G. Stell  Statistical mechanics applied to randommedia problems
 T. Kurtz  A control formulation for constrained Markov processes
 C. Costantini  Diffusion approximation for a class of transport processes with physical reflection boundary conditions
 Y. Kifer  Random perturbations of dynamical systems: a new approach
 A. Eizenberg and M. Freidlin  Partial differential systems with a small parameter and diffusion process with a discrete component
 D. Ioffe and R. Pinsky  Asymptotics for the solution of the exterior Dirichlet problem for secondorder elliptic operators with small firstorder perturbations
 T. Mikami  Asymptotic behavior of invariant density of diffusion process with small fluctuation
 M. Pinsky and V. Wihstutz  Lyapunov exponents for white and real noise driven twodimensional systems
 B. Matkowsky, Z. Schuss, and C. Tier  Asymptotic methods for Markov jump processes
 T. Naeh, M. Kłosek, B. Matkowsky, and Z. Schuss  Uniform solution of Kramers' problem by a direct approach
 G. Yin and K. Ramachandran  On random differential delay equations with a small parameter
 V. Yurinskiĭ  Homogenization error estimates for random elliptic operators
 O. Bruno  The effective conductivity of composites
 K. A. Lurie  On a general concept in optimal material layout
 S. Torquato  Microstructure and effective properties of random media
 K. Golden  Classical transport in quasiperiodic media
 J. D. Maynard  Tuning up a quasicrystal
 W. Faris  Localization estimates for offdiagonal disorder
 W. Faris  Order statistics and renormalization
 L. Pastur  Wave and particle propagation through onedimensional and quasionedimensional random media
 A. Nachbin and G. Papanicolaou  Water waves in shallow channels of randomly varying depth
 V. Klyatskin  Approximations by deltacorrelated random processes and diffusive approximation in stochastic problems
 V. Goland, V. Klyatskin, and I. Yaroshchuk  Some aspects of wave propagation theory in layer random media
 W. Kohler, G. Papanicolaou, and B. White  Reflection of waves by randomly layered media
