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Stochastic and Infinite Dimensional Analysis Christopher C. Bernido Maria Victoria Carpio-Bernido Martin Grothaus Tobias Kuna Maria JoQao Oliveira José Luís da Silva Editors
Editors Christopher C. Bernido Research Center for Theoretical Physics Central Visayan Institute Foundation Jagna, Bohol, Philippines Martin Grothaus Fachbereich Mathematik Technische Universität Kaiserslautern Kaiserslautern, Germany Maria JoQao Oliveira Universidade Aberta Departamento de Ciências e Tecnologia Lisbon, Portugal Maria Victoria Carpio-Bernido Research Center for Theoretical Physics Central Visayan Institute Foundation Jagna, Bohol, Philippines Tobias Kuna Department of Mathematics and Physics University of Reading Reading, United Kingdom José Luís da Silva Universidade da Madeira Faculdade de Ciências Exatas e da Engenharia Funchal, Madeira, Portugal ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISBN 978-3-319-07244-9 ISBN 978-3-319-07245-6 (ebook) DOI 10.1007/978-3-319-07245-6 Library of Congress Control Number: 2016946343 Mathematics Subject Classification (2010): 60-06, 60HXX, 60H40, 60G22 Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser The registered company is Springer International Publishing AG Switzerland (www.birkhauser-science.com)
I am an applied physicist. I apply physics to mathematics. (S. Ulam)
Preface This volume owes its creation to the conference Stochastic and Infinite Dimensional Analysis held at ZiF, Bielefeld, on June 2013 for the 75th birthday of the distinguished scientist Prof. Ludwig Streit. This event was organised to celebrate his long ongoing work in these fields, to which he contributed as a forerunner. This publication collects papers, mostly novel research, spanning the whole range of his interests and activities in the past and future. A sizeable number of the contributions do not stem from participants to the conference, and some of the contributions are only vaguely related to the presentations given during the conference. This book stands out as it combines the many guises in which infinite dimensional analysis can occur like white noise analysis, Malliavin calculus, harmonic analysis, fractional Brownian motion and operator theory. The different guises stem from different viewpoints motivated by particular applications. Fittingly, this collection gathers applications ranging from networks, stochastic partial differential equations, superprocesses, stochastic dynamics on trees, moment problems, and polymer physics to quantum field theory. They play a prominent role as test bed for the mathematics as well as inspiration for the theory. Prof. Ludwig Streit often recalls when he met S. Ulam and heard firsthand the citation given at the beginning of this volume. This phrase may have made a strong impression on Prof. Ludwig Streit since it actually perfectly summarises his working style. The style of this volume can be considered in this spirit. Jagna, Bohol, Philippines Jagna, Bohol, Philippines Kaiserslautern, Germany Reading, UK Lisbon, Portugal Funchal, Madeira, Portugal Christopher C. Bernido M. Victoria Carpio-Bernido Martin Grothaus Tobias Kuna Maria João Oliveira José Luís da Silva vii
Contents Along Paths Inspired by Ludwig Streit: Stochastic Equations for Quantum Fields and Related Systems... 1 Sergio Albeverio Detecting Hierarchical Communities in Networks: A New Approach... 19 Michael J. Barber Transition Probabilities for Processes with Memory on Topological Non-trivial Spaces... 39 Christopher C. Bernido and M. Victoria Carpio-Bernido Generalized Scaling Operators in White Noise Analysis and Applications to Hamiltonian Path Integrals with Quadratic Action... 51 Wolfgang Bock Computer Simulations of Self-repelling Fractional Brownian Motion... 75 Jinky Bornales, Cresente Cabahug, Roel Baybayon, Sim Bantayan, and Beverly Gemao Principal Solutions Revisited... 85 Stephen Clark, Fritz Gesztesy, and Roger Nichols Laplace Operators in Gamma Analysis... 119 Dennis Hagedorn, Yuri Kondratiev, Eugene Lytvynov, and Anatoly Vershik 38 Years with Professor Ludwig Streit... 149 Takeyuki Hida Quasi-analyticity and Determinacy of the Full Moment Problem from Finite to Infinite Dimensions... 161 Maria Infusino ix
x Contents Elements for the Retrieval of the Solar Spectrum on the Surface of Mars from an Array of Photodiodes... 195 Salvador Jiménez and Luis Vázquez Stochastic Processes on Ends of Tree and Dirichlet Forms... 205 Witold Karwowski Completing Canonical Quantization, and Its Role in Nontrivial Scalar Field Quantization... 227 John R. Klauder Stochastic Solutions of Nonlinear PDE s and an Extension of Superprocesses... 243 Rui Vilela Mendes Maximum Likelihood Drift Estimation for the Mixing of Two Fractional Brownian Motions... 263 Yuliya Mishura Existence of Density for Solutions of Mixed Stochastic Equations... 281 Taras Shalaiko and Georgiy Shevchenko