Mathematical Physics and Stochastic Analysis Essays in Honour of Ludwig Streit This page is intentionally left blank Mathematical Physics and Stochastic Analysis Essays in Honour of Ludwig Streit Editors S. Albeverio, Ph. Blanchard, L Ferreira, T. Hida, Y. Kondratiev and R. Vilela Mendes ttf>e World Scientific Singapore • New Jersey • London • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128. Farrer Road. Singapore 912805 USA office: Suite IB. 1060 Main Street, River Edge. NJ 07661 UK office: 57 Shelton Street. Covent Garden. London WC2H 9HE British Library Cataloguing-in-Publlcation Data A catalogue record for this book is available from the British Library. MATHEMATICAL PHYSICS AND STOCHASTIC ANALYSIS: Essays In Honour of Ludwlg Streit Copyright ® 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means. electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center. Inc.. 222 Rosewood Drive. Danvers, MA 01923. USA. In this case permission to photocopy is not required from the publisher. ISBN 981-02-4440-1 Printed in Singapore by World Scientific Printers V Preface In October 1998 a conference took place in Lisbon to celebrate Ludwig's Streit 60th birthday. This book assembles some of the papers presented at the conference as well as other essays contributed by the many friends and collaborators that wanted to honor Ludwig Streit's scientific career and personality. Ludwig Streit was born June 26, 1938 in Leipzig. He attended the High School "Wilhelmschule" in Kassel, in an experimental course with emphasis on social studies and graduated in 1957 as the best student of the year (with an additional exam with "Grosses Latinum" in 1958). From the Spring semester 1957 he was a student of mathematics and physics at the Faculty of Natural Sciences of Gottingen University. From the Spring 1960 he continued his studies at the University of Graz (Austria), at the Faculty of Philosophy, taking his Diploma Thesis in theoretical physics, with mathematics, philosophy and psychology as minors. In the Fall term 1961-62 he was assistant to Prof. H. Lehmann at the Institute of Theoretical Physics, University of Hamburg. He received his PhD "mit Auszeichung" at the University of Graz in 1962 with the disser­ tation "Analytische Eigenschaften der exakten Streuamplitude" which he wrote under the direction of Prof. W. Zimmermann at Hamburg. After two years, 1963-65, with the position of "Research Assistant" at the University of Hamburg, he was from May 1, 1965 to July 31, 1967 at the Seminar for Theoretical Physics, ETH, Zurich (Prof. M. Fierz and Prof. R. Jost). In 1967-68, resp. 1968-69, he was a Research Associate, resp. an As­ sistant Professor, at the Physics Department of Syracuse University. After a period as Member of the Technical Staff at Bell Telephone Laborato­ ries, Murray Hill, N. J., he was, in 1970, nominated Associate Professor at Syracuse University. He left Syracuse in 1972, to take over a position of Professor at the Physics Department, University of Bielefeld, which he has held ever since. Since 1973 he is also Honorarprofessor at the University of Graz. In 1980, for professional reasons, he declined the offer of a chair in theoretical physics at the University of Minister. From 1989 to 1991 he was titular of the chair for Mathematical Physics at the Universidade do Minho, Braga (Portugal) and since 1991 he holds a chair at the Mathematics Department of the Universidade da Madeira, Funchal (Portugal). In 1980-91 he was given an "Akademie" -Fellowship by the Volkswagen Foundation. He is the author of more than 150 research papers, on topics vi ranging from mathematical and theoretical physics, to mathematics, espe­ cially the theory of stochastic processes, infinite dimensional analysis, the theory of dynamical systems and their applications (in natural sciences and engineering). He is also author of a very influential book on white noise calculus and its applications and editor of 16 proceedings volumes. His outstanding achievements in science are described more closely in two articles (by S. Albeverio and T. Hida) in this volume. Let us stress here that besides his research achievements Ludwig Streit has been very successful in giving stimulus and furthering new research directions, creat­ ing new research centres, organizing research workshops, and international conferences. We mention the creation of the Research Centre BiBoS at the University of Bielefeld (1984 - ), Bochum (1984 - 1997) and Bonn (1997 - ), the organization of a Series of Conferences "Bielefeld Encounters in Mathematics and Physics" at Bielefeld University (started around 1978, of which numerous Proceedings have appeared), the Research Years 1975-76, "Mathematical Problems in Quantum Dynamics", 1983-84, "Project No. 2 in Mathematics and Physics" and "The sciences of complexity: from math­ ematics and technology to a sustainable world", 2000-01 at the Center of Interdisciplinary Research (ZiF) of the University of Bielefeld. He has been Director of ZiF (1981-1985), International Adviser for the International Institute for Advanced Studies at Kansai Science City (Japan), has held executive positions in the International Association for Mathematical Physics (1979-1985), IUPAP Commission for Mathematical Physics (1981-1987), and he is member of several editorial boards of jour­ nals and books. He has given invited lectures at Universities in all continents, with re­ peated long stays in Bulgaria, China, France, Italy, Japan, Philippines, Russia, USA. Ludwig's "engagement" for science in Portugal deserves a special mention. Starting with his nominations in Braga and Madeira, he has developed an impressive activity of research and teaching, with an out­ standing impact on the country's development of scientific research. Let us mention here his creation of a school of Mathematical Physics in Madeira with several PhD students at international level, his influence on other stu­ dents in Portugal who came to Bielefeld and other Central European places to do research work, his creation of the Research Centre "Centro de Cien- cias Matematicas", his participation in the Committee of the Portuguese Secretary of State for Research and Development for a national program in astronomy, his organization of several workshops and international work­ shops, conferences and Summer Schools in Madeira and Lisbon. The contributions of Ludwig Streit for the development of science in vii Portugal were officially acknowledged and praised, at the time of the con­ ference, by the Minister of Science of the Portuguese Government. Ludwig Streit is an instant source of energy and inspiration in a broad area of mathematics and physics and their many apphcations in natural and social sciences. The contributions presented in this volume by stu­ dents, friends, and coworkers reflect, at least partially, the broad interests of Ludwig. The two first contributions by S. Albeverio and T. Hida are particular in the sense that they have as a topic specific aspects of the scientific work of Ludwig and the interaction with the authors over many years. The other contributions, arranged alphabetically by authors, also testify, in one way or another, the great influence of Ludwig in many areas of mathematics, physics and their apphcations. In the name of all friends and contributors, we wish Ludwig many more years of good health and of successful pursuing of his outstanding activities, in science and for the community. This page is intentionally left blank ix Contents Preface v Ludwig Streit's list of publications xiii Some themes of the scientific work of Ludwig Streit S. Albeverio 1 Mathematics, Physics and Streit since 1975 T. Hida 8 Stochastic limit and interacting commutation relations L. Accardi, I. Ya. Aref'eva and I. V. Volovich 18 De Rham complex over product manifolds: Dirichlet forms and stochastic dynamics S. Albeverio, A. Daletskii and Y. Kondratiev 37 Real time random walks on p-adic numbers S. Albeverio and W. Karwowski 54 Characterization of test functions in CKS-space JV. Asai, I. Kubo and H.-H. Kuo 68 Nonlinear Lie algebras in quantum physics and their interest in quantum field theory J. Beckers 79 Continuous percolation: tree approximation and most probable clusters Ph. Blanchard, G. F. Dell'Antonio, D. Gandolfo and M. Sirugue-Collin 84 X Rigged Hilbert space resonances and time asymmetric quantum mechanics A. Bohm and H. Kaldass 99 Q-Ising neural network dynamics: A comparative review of various architectures D. Bolle, G. Jongen and G. M. Shim 114 The relativistic Aharonov-Bohm-Coulomb problem: A path integral solution J. Bornales, C. C. Bernido and M. V. Carpio-Bemido 130 Statistical manifolds: The natural affine-metric structure of probability theory G. Burdet, Ph. Combe and H. Nencka 138 Brydges' operator in renormalization theory P. Cartier and C. DeWitt-Morette 165 Fluctuation of the Bose-Einstein condensate in a trap H. Ezawa, K. Nakamura, K. Watanabe and Y. Yamanaka 169 Time dependent and nonlinear point interactions R. Figari 184 The Cole-Hopf and Miura transformations revisited F. Gesztesy and H. Holden 198 Some models of nonideal Bose gas with Bose-Einstein condensate R. Gielerak 215 Configuration spaces for self-similar random processes and measures quasi-invariant under diffeomorphism groups G. A. Goldin 228 Stochastic processes and the Feynman integral Z. Haba 243