L. Streit - Böcker

'Et moi, ..., si j'avait su comment en revenIT, One service mathematics has rendered the je n'y serais point allt\.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

This monograph presents a framework for infinite dimensional analysis based on white noise. This approach, which has many areas of application is intended to be intuitive and efficient. Among the concepts and structures generalized to an infinite dimensional setting in this book are: spaces of test and generalized functions, differential calculus, Laplacian and Fourier transforms and Dirichlet forms and their Markov processes. A multitude of concepts, such as Brownian motion functionals, falls into this framework. This book presents a simple, yet general theory of stochastic integration and also discusses construction quantum field theory and Feynman's functional integration. This volume should be of interest to mathematicians and scientists who use stochastic methods in their research. The book should be useful for mathematicians in probability theory, functional analysis, measure theory, potential theory, as well as to physicists and scientists in engineering.

The intensive exchange between mathematicians and users has led in recent years to a rapid development of stochastic analysis. Of the users, the physicists form perhaps the most important group, giving direction to the mathematicians' research and providing a source of intuition. White noise analysis has emerged as a viable framework for stochastic and infinite dimensional analysis. Another growth area is the theory of stochastic partial differential equations. Gauge field theories are attracting increasing attention. Dirichlet forms provide a fruitful link between the mathematics of Markov processes and the physics of quantum systems. The deterministic-stochastic interface is addressed, as are Euclidean quantum mechanics, excursions of diffusions and the convergence of Markov chains to thermal states.

This text is dedicated to Professor Takeyuki Hida on the occasion of his 70th birthday. The book is more than a collection of articles, it is a consistent editorial work, devoted to attempting to obtain a unitary picture from the different contributions and to give a comprehensive account of important developments in contemporary white noise analysis and some of its applications. For this reason, not only the results, but also motivations, explanations and connections with previous work have been included. The wealth of applications, from number theory to signal processing, from optimal filtering to information theory, from the statistics of stationary flows to quantum cable equations, show the power of white noise analysis as a tool. Beyond these, the authors emphasize its connections with practically all branches of contemporary probability, including stochastic geometry, the structure theory of stationary Gaussian processes, Neumann boundary value problems, and large deviations.

Volumes 30 and 31 of this series, dealing with "Many Degrees of Freedom," contain the proceedings of the 1976 International Summer Institute of Theoretical Physics, held at the university of Bielefeld from August 23 to September 4, 1976. This institute was the eighth in a series of summer schools devoted to particle physics and organized by universities and research institutes in the Federal Republic of Germany. Many degrees of freedom and collective phenomena play a critical role in the description and understanding of elementary particles. The lectures in this volume were intended to display how these structures occur in various recent developments of mathematical physics. Lectures ranged from classical nonlinear field theory over classical soliton models, constructive quantum field theory with soliton solutions and gauge models to the recent unified description of renormalization group tech­ niques in probabilistic language and to quantum statisti­ cal dynamics in terms of derivations. The Institute took place at the Center for Inter­ disciplinary Research of the University of Bielefeld. On behalf of all participants, it is a pleasure to thank the officials and the administration of the Center for their cooperation and help before and during the Insti­ tute. Special thanks go to V.C. Fulland, M. Kamper, and A. Kottenkamp for their rapid and competent preparation of the manuscripts.

Are we living in a golden age? It is now more than half a century that Einstein and Heisenberg have given us the theories of relativity and of quantum mechanics, but the great challenge of 20th century science remains unre­ solved: to assemble these building blocks into a fundamental theory of matter. And yet, for anyone watching the interplay of mathematics and theoretical physics to-day, developing symbiotically through the stimulus of a lively, even essential interdisciplinary dia­ logue, this is a time of fascination and great satisfaction. It is also a time of gratitude to those who had the courage to in­ sist that "a rudimentary knowledge of the Latin and Greek alpha­ bets" was not enough, and tore down the barriers between the disciplines. On the basis of this groundwork there is now so much progress, and, notably, such strengthening of the dia].ogue with phenomenology that - reaching out for The Great Break­ through - this may indeed turn out to be the golden age.

This monograph presents a framework for infinite dimensional analysis based on white noise. This approach, which has many areas of application is both intuitive and efficient. Among the concepts and structures generalized to an infinite dimensional setting in this book are: spaces of test and generalized functions, differential calculus, Laplacian and Fourier transforms and Dirichlet forms and their Markov processes. A multitude of concepts, such as Brownian motion functionals, falls into this framework. This book presents a simple, yet general theory of stochastic integration and also discusses construction quantum field theory and Feynman's functional integration. This volume will be of interest to mathematicians and scientists who use stochastic methods in their research. The book will be of particular value to mathematicians in probability theory, functional analysis, measure theory, potential theory, as well as to physicists and scientists in engineering.

Recent Developments in Infinite-Dimensional Analysis and Quantum Probability is dedicated to Professor Takeyuki Hida on the occasion of his 70th birthday. The book is more than a collection of articles. In fact, in it the reader will find a consistent editorial work, devoted to attempting to obtain a unitary picture from the different contributions and to give a comprehensive account of important recent developments in contemporary white noise analysis and some of its applications. For this reason, not only the latest results, but also motivations, explanations and connections with previous work have been included. The wealth of applications, from number theory to signal processing, from optimal filtering to information theory, from the statistics of stationary flows to quantum cable equations, show the power of white noise analysis as a tool. Beyond these, the authors emphasize its connections with practically all branches of contemporary probability, including stochastic geometry, the structure theory of stationary Gaussian processes, Neumann boundary value problems, and large deviations.

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.

'Et moi, ..., si j'avait su comment en revenIT, One service mathematics has rendered the je n'y serais point allt\.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.