Random Partial DiHerential Equations

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2 ISNMI02: International Series of Numerical Mathematics Internationale Schriftenreihe zur Numerischen Mathematik Serie Internationale d1\nalyse Numerique VoI.102 Edited by K.-H. HofTmann, Augsburg; H. D. Mittelmann, Tempe; J. Todd, Pasadena Springer Basel AG

3 Random Partial DiHerential Equations Proceedings of the Conference held at the MathematicaI Research Institute at Oberwolfach, Black Forest, November 19-25,1989 Edited by U. Hornung P. Kotelenez G. Papanicolaou Springer Basel AG 1991

4 Editors Prof. Dr. U. Hornung SCHI P.O. Box 1222 D-W-8014 Neubiberg Germany Prof. G. Papanicolaou Courant Institute 251 Mercer Street New York, NY USA Prof. P. Kotelenez Department of Mathematics and Statistics Case Western Reserve University Cleveland, OH USA Library of Congress Cataloging-in-Publication Data Random partial differential equations : proceedings of the conference held at the Mathematical Research Institute at Oberwolfach, Black Forest, November 19-25,1989/ edited by U. Hornung, P. Kotelenez, G. Papanicolaou. (International series of numerical mathematics = Internationale Schriftenreihe zur numerischen mathematik = Serie internationale d'analyse numerique : vol. 102) Papers from the Conference on "Random Partial Differential Equations." 1. Stochastic differential equations-congresses. 2. Differential equations, Partial-Congresses. 1. Hornung, Ulrich II. Kotelenez, P. (Peter), III. Papanicolaou, George, IV. Conference on "Random Partial Differential Equations" (1989 Mathematic Research Institute at Oberwolfach) V. Series: International series of numerical mathematics ; v QA274.R dc20 Deutsche Bibliothek Cataloging-in-Publication Data Random partial differential equations: proceedings of the conference held at the Mathematical Research Institute at Oberwolfach, Black Forest, November 19-25, 1989/ ed by U. Hornung... - Basel ; Boston ; Berlin : Birkhäuser, 1991 (International series of numerical mathematics ; Vol. 102) NE: Hornung, Ulrich [Hrsg.]; Mathematisches Forschungsinstitut <Oberwolfach>; GT ISBN ISBN (ebook) DOI / This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use a fee is payable to >>Verwertungsgesellschaft Wort«, Munich Springer Basel AG Originally published by Birkhäuser Verlag Basel in Softcover reprint of the hardcover 1 st edition 1991

5 v Preface The present volume contains manuscripts of lectures which were given at the conference on "Random Partial Differential Equations" at the Mathematical Research Institute at Oberwolfach. The conference took place during the week of November 19, 1989, and was organized by the editors. Researchers in probability and differential equations were invited and 37 scientists from 12 count ries participated. Topics of the papers presented at the conference were parabolic Ito equations, Malliavin calculus, Dirichlet forms, Markov fields, Skorohod and Hellinger integrals, random Schrödinger operators, Lyapunov exponents, and optimal control of stochastic systems. Furthermore, limit theorems of particle systems, acoustic waves in random media, and random oscillations were discussed. The thirteen papers in this volume are representative of the topics presented in the lectures. The papers of Chow-Jiang, Jetschke and Kotelenetz are on the analysis of stochastic partial differential equations, primarily with tools from functional analysis. The papers of Grigelionis and Millet-Nualart-Sanz and Manthey rely more on probabilistic ideas and methods. The paper of Golden is on the use of random partial differential equations in composites and on homogenization methods. The paper of Benassi Fouque is on the hydrodynamic behavior of interacting particle systems where both probabilistic and analytical techniques are used. The paper of Wedig is more applied in character and relates directly to problems in mechanics. The papers of Kirsch Pastur, Dell'Antonio, and Alberverio-Blanchard-Ma are motivated from problems in mathematical physics and use a variety of analytical and probabilistic tools; they deal with Schrödinger's equation and Dirichlet forms. The editors wish to thank Prof. M. Barner and his co-workers of the Mathematical Research Institute for their help in creating the conference. The editors also wish to thank Birkhäuser Verlag for their production of this book which will enable many more scientists to enjoy the stimulation of the conference. Ulrich Hornung (Neubiberg) Peter Kotelenetz (Cleveland) George Papanicolaou (New York)

6 VII Contents List of Contributors..... IX Feynman-Kac semigroups in terms of signed smooth measures Sergio Albeverio, Phi/ippe Blanchard and Zhi Ming Ma Fluctuation field for the asymmetrie simple exdusion process Albert Benassi and lean-pierre Fouque Almost sure convergence of some approximate solutions for random parabolic equations Pao-Liu Chow and ling-lin liang Two results on Nelson diffusion Gianfausto Dell'Antonio Bulk conductivity of the square lattice for complex volume fraction Kenneth Golden A stochastic Fubini theorem and equivalence of extended solutions of stochastic evolution equations in Hilbert space Luis G. Gorostiza and lorge A. Leon Hellinger integrals and Hellinger processes for solutions of martingale problems B. Grigelionis Lattice approximation of a nonlinear stochastic partial differential equation with white noise Gottfried letschke Large time behaviour of moments of fundamental solutions of the random parabolic equation Werner Kirsch and L.A. Pastur Acomparison theorem for a dass of stochastic partial differential equations Peter Kotelenez Weak approximation of a nonlinear stochastic partial differential equation Ralf Manthey Small perturbations for quasilinear anticipating stochastic differential equations A. Mi/let, David Nualart and M. Sanz Simulation and analysis of mechanical systems with parameter fluctuations Walter V. Wedig

7 IX List of Contributors Sergio A. ALBERVERIO, Fakultät für Mathematik, Ruhr-Universität, W-4630 Bochum 1, Germany Pao-Liu CHOW, Department of Mathematics, Wayne State University, Detroit, Michigan U.S.A. Gianfausto F. DELL'ANTONIO, Dipartimento di Matematica, Univ. di Roma I, P. Aldo Moro 2, Roma, Italy Jean-Pierre FOUQUE, CNRS, Centre de Mathematique Appliquees, Ecole Polytechnique, F Palaiseau Cedex, France Kenneth GOLDEN, Deptartment of Mathematics, Princeton University, Princeton, NJ U.S.A. Luis G. GOROSTIZA, Centro de Investigaci6n y de Estudios Avanzados, Mexico D.F., Mexico B. GRIGELIONIS, Institute of Mathematics and Cybernetics, Vilnius, Akademijos 4, Lithuania G. JETSCHKE, Institut für Ökologie der Universität, Neugasse 23, Jena, Germany Werner KIRSCH, Fakultät für Mathematik, Ruhr-Universität, Postfach , W Bochum 1, Germany Peter KOTELENEZ, Deptartment of Mathematics, Case Western Reserve University, Cleveland, OH U.S.A. Ralf MANTHEY, Friedrich-Schiller-Universität, Sektion Mathematik, UHH 17. OG, Jena, Germany David NUALART, Universitat de Barcelona, Facultat de Matematiques, Barcelona, Spain W. WEDIG, Institut für Technische Mechanik der Universität, Kaiserstr. 12, W Karlsruhe 1, Germany