Overview

Stochastic analysis and its various applications in physics have to a large extent developed symbiotically. In the past decades mathematics has provided physics witb a vast and rapidly expanding array of tools and methods, while on the other hand physics has counted among the sources of direction and of structural intuition for the mathematical research in stochastics. We hope to have captured some of the focal points of this dialogue in the NATO AS! "Stochastic Analysis and its Applications in Physics" and in the present volume. On the mathematical side White Noise Analysis has emerged as a viable frame- work for stochastic and infinite dimensional analysis (Hida, Streit). Another growth point is the theory of stochastic partial differential equations and their applications (8ertini et aI., 0ksendal, Potthoff, Russo, Sinior). Gauge field theories have in- creasingly attracted the attention not only of physicists but of mathematicians as well (Gross, Liandre, Sengupta). On the other hand the contributions of Lang and of Vilela Mendes show the extent to which stochastic methods have found a place in the physicists' toolbox. Dirichlet forms provide a fruitful link between the mathematics of Markov processes and fields and the physics of quantum systems (Albeverio et al.). The deterministic-stochastic nterface i was addressed by Collet and by Mandrekar, Euclidean quantum mechanics by Cruzeiro and Zambrini, excursions of diffusions by Truman, and Kubo-Martin-Schwinger norms of statistical mechanics by Streater. So much for a rapid synopsis of the material represented in the present volume.