http://dx.doi.org/10.1090/psapm/007

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

VOLUME VII

APPLIED PROBABILITY

McGRAW-HILL BOOK COMPANY, INC.

NEW YORK TORONTO LONDON

1957

FOR THE AMERICAN MATHEMATICAL SOCIETY

80 WATERMAN STREET, PROVIDENCE, RHODE ISLAND PROCEEDINGS OF THE SEVENTH SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

Held at the Polytechnic Institute of Brooklyn April 14-15, 1955

COSPONSORED BY THE OFFICE OF ORDNANCE RESEARCH

L. A. MacColl

EDITOR

EDITORIAL COMMITTEE R. V. Churchill A. E. Heins F. J. Murray

Prepared by the American Mathematical Society under Contract No. DA-19-020-ORD-3538 with the Ordnance Corps, U.S. Army.

Copyright © 1957 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permis• sion of the publishers.

Library of Congress Catalog Card Number 50-1183 CONTENTS

EDITOR'S PREFACE v

Brownian Motion Depending on n Parameters: The Particular Case n = 5 1 BY PAUL LEVY

A New Look at the First Boundary-value Problem 21 BY J. L. DOOB

On Boundaries Defined by Stochastic Matrices (Abstract) 35 BY WILLIAM FELLER

On the Application of Functional Calculus to the Statistical Theory of Turbulence . 41 BY EBERHARD HOPF

Stochastic Processes of Astronomical Interest 51 BY GUIDO MUNCH

The Singularity in the Spectrum of Homogeneous Turbulence 67 BY G. K. BATCHELOR

Probability in Classical Physics 73 BY MARK KAC

Infinite Models in Physics , 87 BY S. M. ULAM

Quantum Theory and the Foundations of Probability 97 BY B. O. KOOPMAN

INDEX 103

iii EDITOR'S PREFACE

The Seventh Symposium in Applied Mathematics, sponsored by the Ameri• can Mathematical Society and the Office of Ordnance Research, and devoted to Mathematical Probability and Its Applications, was held at the Polytechnic Institute of Brooklyn on April 14 and 15, 1955. This volume contains the papers (one in abstract form) which were presented at the Symposium. Prolonged consideration by the members of the Program Committee, under the chairmanship of Dr. H. W. Bode, resulted in the decision that the Sym• posium should be concerned with three principal themes, viz., The Theory of Diffusion, The Theory of Turbulence, and Probability in Classical and Modern Physics. However, it was the intention of the Committee that these terms should be interpreted broadly and that the speakers should avail themselves of considerable freedom in determining the actual contents of their papers. In particular, it was understood that the term "theory of diffusion" was to be interpreted so as to cover a wide variety of relations between probability and differential equations. The three themes were dealt with in the order in which they have been men• tioned, and the papers appear here in the order in which they were given. Many individuals have participated, directly and indirectly, in the work of preparing this volume. The editor wishes to express here his sincere thanks to all of these collaborators. The advice and encouragement given by Profes• sor R. V. Churchill, Chairman of the Editorial Committee for the Proceedings of Symposia in Applied Mathematics, has been particularly helpful. All who participated in the Symposium are indebted to the McGraw-Hill Book Com• pany, Inc., which, beginning with the Proceedings of the Symposium on Elas• ticity, has undertaken the task of bringing the Proceedings of these Symposia on Applied Mathematics to the scientific public in book form.

L. A. MACCOLL Editor

v INDEX

Ambarzumian, V. A., 51 Equation, $-, 43ff. Antoine, L., 92 Equilibrium problems, 73ff. Equivalent operators, 38 Bachelier-Wiener function, 2 Escape route, 37 Batchelor, G. K, 47, 50 Events, 98ff. Berkeley Symposium, Third, 8In. incompatible, 100 Berlin, T. H., 76, 85 Everett, C. J., 95 Birkhoff, G., 102 Experimental propositions, 99 Blanc-Lapierre, A., 80 Extremal elements, 36 Bok, B. J., 66 Boltzmann, L., 81 Feynman, R. P., 102 Boltzmann equation, 81, 84 Fortet, R., 80 Boltzmann property, 84 Fourier-Wiener series, 10 Boolean algebras, 101 Free energy per constituent 74 Boundary, adjoint, 39 Function, individual, 3n. entrance, 39 partition, 74ff. exit, 39 random, 3n. natural, 39 regular, 22 regular, 39 resolutive, 24 Boundary points, 37 subregular, 22 Brelot, M., 32, 33 superregular, 22 Brownian function, 2 Functional derivatives, 42 Bruns-Charlier expansion, 47, 48

Chandrasekhar, S., 66 Gibbs, J. W., 81 Characteristic functional, 42 Gordeladse, S. G., 65, 66 Choquet, G., 32, 33 Green, M. S., 85 Clausius, R. J. E., 82 Compatible events, 100 Compatible sets, 101 Harris, T. E., 39 Compatible variables, 102 Heisenberg, W., 48 Heisenberg principle, 101 Darling, D. A., 80 Hierarchy, 90 Densities, asymptotic, 89 Hopf, E., 50 contracted, 84 Derman, C, 39 Imperfect gas, problem of, 77 Diffusion theory, 40 Intrinsic first boundary-value problem, Dirac, P. A. M., 98n., 102 30 Dirichlet problem, 21 ff. Ising models, 74 Doob, J. L., 32, 33

Ehrenfest, P., 81 Kac, M., 75, 85 Ehrenfest, T., 81 Kahn, B., 78, 85 103 104 INDEX

Kaufman, B., 75 Ramakrishnan, A., 66 Khintchine, A., 4 Regular average, 22 Kolmogoroff, A., 39, 40, 47, 48 Reid, W. H., 50 Kolmogoroff differential equations, 38, Robbins, H., 39 Krein, H., 35, 36 Rusakov, G. E., 66

Lee, T. D., 78 Sample space, 99n. Liouville's differential equation, 48, 49 Schrodinger equation, 94 Loeve, M., 13 Semimartingale, 27 Loitsiansky's invariant, 72 Set, regular, 21 Loschmitt paradox, 81 sojourn, 36 Siegert, A. J. F., 80 Markarian, B. E., 66 Specific heat, 75 Martin, M. H., 38 Spectrum tensor, 67ff. Martingale, 27 Spherical models, 76 Master equation, 83-85 Spitzer, L., 66 Mathews, P. M., 66 Steepest descent, method of, 76 Maximal observation, 102 Stochastic boundary function, 30 Maxwell gas, 83 Stone, M. H., 99n. Mayer, J. R. von, 78 Stosszahlansatz, 81-83, 85 Mil'man, D., 35, 36., Stromgren, B., 66 Montroll, E. W., 85 Support of a measure function, 22 Munch, G., 66 Tautz, G., 33 Navier-Stokes equations, 41, 49, 67ff. Titt, E. W., 50 Neumann, J. von, 102 Trajectories to the boundary, 26 Nonequilibrium problems, 8 Iff. Trajectory, co, 27

Onsager, L., 75, 76 w-space, 41 Uhlenbeck, G. E., 78 Ulam, S., 95 Paley-Wiener formula, 10 Ursell, F., 78 Pasta, J., 90 Phase space, 41 ^-equation, 43ff. van Hove, L., 78, 85 Point, irregular, 24 von Neumann, J., 102 regular, 24 Positive cone, 35 Ward, J. C., 75, 85 Probability, laws of, 98 Wave-number space, 43 sojourn, 36 Wiener process, 38 Proudman, I., 50, 67rc., 68, 70 PWB method (Perron-Wiener-Brelot method), 23 Yang, C. N., 78 generalized, 29 Young, T., 97 stochastic, 29 Young's experiment, 97

Quantum mechanics, 97ff. Zermelo paradox, 81