Stochastic Analysis in Mathematical Physics

Transcription

1 Stochastic Analysis in Mathematical Physics

2 This page intentionally left blank

3 Stochastic Analysis in Mathematical Physics Proceedings of a Satellite Conference of ICM 2006 Lisbon, Portugal 4 8 September 2006 Editors Gerard Ben Arous Courant Institute, New York University, USA Ana Bela Cruzeiro GFMUL & IST (TUL), Portugal Yves Le Jan Universitè Paris 11, France Jean-Claude Zambrini GFMUL, University of Lisbon, Portugal World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Stochastic Analysis in Mathematical Physics Downloaded from British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. STOCHASTIC ANALYSIS IN MATHEMATICAL PHYSICS Proceedings of the Satellite Conference of the ICM 2006 Copyright 2008 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 ISBN X Printed in Singapore.

5 v PREFACE Stochastic Analysis in Mathematical Physics Downloaded from In the Summer of 2006, an International Conference, satellite of the International Congress of Mathematicians (ICM2006, Madrid), took place in Lisbon and was organized by the Group of Mathematical Physics of the University of Lisbon (GFMUL). We decided to entitle Stochastic Analysis in Mathematical Physics the elaboration of the ideas presented there by some of the participants. The last ten years were witness to a remarkable penetration of the methods of Stochastic Analysis in all fields of Mathematical Physics. What was regarded by many, not so long ago, as a set of esoteric tools, turned into a fundamental component of our understanding of natural phenomena. The works collected here illustrate the versatility of those stochastic methods and we warmly thank the authors. The Group of Mathematical Physics is sponsored by the Portuguese Foundation for Science and Technology, specifically, for this conference, by Project POCTI/0208/2003, co-financed by FCT/OE and FEDER, as well as Project POCI/MAT/55977/2004. We thank also the Foundation Calouste Gulbenkian and the FLAD (Luso-American Foundation) for their support. The Editors, G. Ben Arous (New York) A. B. Cruzeiro (Lisbon) Y. Le Jan (Paris) J. C. Zambrini (Lisbon) October 11, 2007

6 This page intentionally left blank

7 vii LIST OF PARTICIPANTS Stochastic Analysis in Mathematical Physics Downloaded from Gérard Ben Arous Jean Bertoin Erwin Bolthausen Fernanda Cipriano Ana Bela Cruzeiro Krzysztof Gawedzki Patrícia Gonçalves Maria Gordina Vadim Iourinski Antii Kupiainen Yves Le Jan Thierry Lévy Wu Liming Piotr Lugiewicz Paul Malliavin Jonathan Mattingly Stanislav Molchanov Ana Filipa Pontes Rolando Rebolledo Jorge Rezende Christophe Sabot Ambar Sengupta Alain-Sol Sznitman Eric Vanden-Eijnden Jean-Claude Zambrini École Polytechnique Fédérale de Lausanne, Switzerland Université Paris VI, France Universität Zürich, Switzerland GFMUL/Univ. Nova de Lisboa, Portugal GFMUL/Instituto Superior Técnico, Portugal École Normale Supérieure de Lyon, France IMPA, Brazil University of Connecticut, USA Universidade da Beira Interior, Portugal University of Helsinki, Finland Université Paris-Sud 11, France École Normale Supérieure, France Université Blaise Pascal-Clermont II, France Imperial College London, UK Université Paris VI, France Duke University, USA University of Northern Carolina Charlotte, USA ISEP, Portugal Pontificia Universidad Católica, Chile GFMUL/Universidade de Lisboa, Portugal École Normale Supérieure de Lyon, France Louisiana State University, USA ETH-Zürich, Switzerland Courant Institute of Math. Sciences, USA GFMUL/Universidade de Lisboa, Portugal

8 This page intentionally left blank

9 ix CONTENTS Stochastic Analysis in Mathematical Physics Downloaded from Preface Participants Stochastic parallel transport on the d-dimensional torus 1 Ana Bela Cruzeiro and Paul Malliavin Riemannian geometry of Diff(S 1 )/S 1 revisited 19 Maria Gordina Ergodic theory of SDE s with degenerate noise 30 Antti Kupiainen Dynkin s isomorphism without symmetry 43 Yves Le Jan Large deviations for the two-dimensional Yang-Mills measure 54 Thierry Lévy Laplace operator in networks of thin fibers: Spectrum near the threshold 69 S. Molchanov and B. Vainberg v vii Adiabatic limits and quantum decoherence 94 Rolando Rebolledo and Dominique Spehner Gauge theory in two dimensions: Topological, geometric and probabilistic aspects 109 Ambar N. Sengupta

10 x Near extinction of solution caused by strong absorption on a fine-grained set 130 V. V. Yurinsky and A. L. Piatnitski