Table Of ContentStochastic Processes in Physics and Engineering
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, UniversitiI degli Studi di Roma, Italy
Yu. I. MANIN, Sleklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-Co ROTA, M.I.T., Cambridge, Mass., U.S.A.
Stochastic Processes
in Physics
and Engineering
edited by
Sergio Albeverio
Fachbereich Mathematik. Ruhr"Universitiit Bochum. FR.C.
Philip Blanchard
Fakultiitfur Physik. Universitiit Bielefeld. FR.C.
Michiel Hazewinkel
Centrum voor Wiskunde en Informatica.
Amsterdam. The Netherlands
artd
Ludwig Streit
Fakultiitfur Physik. UniversitiitBielefeld. FR.C.
D. Reidel Publishing Company
A ME.MBE.R or THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht! Boston I Lancaster I Tokyo
Library of Congress Cataloging in Publication Data
Stochastic processes in physics and engineering I edited by S. Albeverio ... [et al.].
p. cm. - (Mathematics and its applications)
Results of a Meeting, BIB OS IV, held April 1986, in Bielefeld.
Includes index.
TSBN-13: 978-94-010-7803-0 e-TSBN-13: 978-94-009-2893-0
DOl: 10.1007/978-94-009-2893-0
1. Stochastic processes---Congresses. 2. Mathematical physics---Congresses.
3. Engineering mathematics---Congresses. I. Albeverio, Sergio. II. BIBOS IV
(1986: Bielefeld, Germany) III. Series: Mathematics and its applications (D. Reidel
Publishing Company)
QC20.7.S8S77 1987
519.2---dc 19 87-31865
CIP
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322,3300 AH Dordrecht, Holland.
All Rights Reserved
© 1988 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover I st edition 1988
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
SERIES EDITOR'S PREFACE
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perhaps you will find the final question.
O.K. Chesterton. The Scandal of Father
'The Hermit Qad in Crane Feathers' in R. Brown 'The point of a Pin'.
van Gu!ik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and textbooks on
increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields
does not grow only by putting forth new branches. It also happens, quite often in fact, that
branches which were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various sciences has
changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical
economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the
structure of water meet one another in packing and covering theory; quantum fields, crystal defects
and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering;
and prediction and electrical engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable
systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the
existing classification schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such
(new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical and/or
scientific specialized areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have and have had on
the development of another.
The Mathematics and Its Applications programme tries to make available a careful selection of
books which fit the philosophy outlined above. With such books, which are stimulating rather than
definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better
communication among the practitioners in diversified fields.
Stochastics is, as a well-developed specialism in mathematics, a relatively young discipline and
probably the fastest growing one. It is extraordinarily important in applications. There are, of
course, many phenomena in nature and engineering which have a random component. But the
applications of stochastic thinking are not limited to such cases. Stochastic models also can be, and
are, used to model other phenomena which do not have an obvious random component. For exam
ple, very complicated deterministic phenomena (turbulence, chaos) can sometimes fruitfully be han
dled in this way, or unknown aspects of a given class of models can be incorporated by means of
stochastic ideas. Further, there are the interrelations between potential theory and probability and,
of course, the evolution equations for the density of suitable continuous-time Markovian processes
are partial differential equations, and such equations may well arise in different contexts. And in
v
vi SERIES EDITOR'S PREFACE
such cases a stochastic interpretation can be very fruitful. Quite generally there seems to be a ten
dency in mathematics as a whole for ideas from stochastics to penetrate into other areas. For exam
ple, stopping times are a powerful tool in certain parts of analysis.
As already remarked, stochastics has very many 'applications': in physics, in chemistry, in geol
ogy, in astronomy, in all kinds of engineering, .... These application areas have all evolved their
own corresponding intuitions. In this book, the ways physicists think about these things and the
way engineers do are brought together. That was the idea of the meeting on which it is based. There
is more on this theme in the preface to the book. And it is clear from the meeting itself that such an
interaction can be remarkably useful and stimulating. This book will help to further enhance this
beneficial interaction.
The unreasonable effectiveness of matbemat As long as algebra and geometry proceeded
ics in science ... along separate patbs, their advance was slow
and tbeir applications limited.
Eugene Wigner But when these sciences joined company
tbey drew from each otber fresh vitality and
Well, if you know of a better 'ole, go to it. tbenceforward marched on at a rapid pace
towards perfection.
Bruce Bairnsfatber
Joseph Louis Lagrange.
What is now proved was once only ima-
gined.
William Blake
Bussum, September 1987 Michiel Hazewinkel
TABLE Of CONTENTS
Preface LX.
J. Bernasconi, Statistica1 physics and optimization: binary sequences with small
autocorrelations
GL Bla"kenship, A. LaVigna, D,C. MacEnony, J.P. Quadrat, l. Yon, A MACSYMA/expert 17
system for nonlinear filtering
P.L Chaw, Generalized solution of some stochastic P.D.E.'s in turbulent diffusion 63
N.J. Cut/and, Nonstandard techniques in stochastic optimal control theory 75
T Fresewinkel, Recursive estimalion of order and parameters of a pilot plant turbo 91
generator
GL GOmez, Attainability and reversibility of a golden age for the labour-surplus economy: 107
a stochastic variational approach
A. Grossmann, Wavelet transforms and edge detection 149
M. Hazewinke/, Lie algebraic method in filtering and identification 159
T. HitkJ, K. Saito, White noise analysis and the Levy Laplacian 177
R. Hut, G.J. Ouder, An application of adaptive optimal control 185
A.G.Z. Kemna, A.C.£. Vorst, The value of an option based on an average security value 193
T. Koski, On asymptotic localiz.ation by perturbing operators for partial observation of a 205
stochastic evolution equation
G.P. Papavassi/opou/os, Adaptive games 223
J. Picard. Nonlinear filtering and smoothing with high signal-Ie-noise ratio 237
L Pie/ronero, C. EverlSZ, A.P. Siebesma, Fractal and multifractal structures in kinetic 253
critical phenomena
w.R.
Sch"eider, Generalized Levy distributions as limit laws 279
W. Schempp. The oscillator representation of the metaplectic group applied to quantum 305
electronics and computerized tomography
T. Shiga, Stepping stone models in population genetics and population dynamics 345
Y. Takahashi, Power spectrum and Fredholm determinant related to intermittent chaos 357
P. Tautu, On the qualitative behaviour of interacting biological cell systems 381
List of panicipants 403
Index 409
PREFACE
BIBOS, of course, stands for a rather large research project stretching over a number of years involv
ing the universities of Bochum and Bielefeld. The'S' stands for stochastics. The project is funded by
the Stiftung Volkswagenwerk. The present book is the result of the meeting BIBOS IV which took
place in April 1986 at ZiF in Bielefeld. The basic philosophy at the back of this meeting was as fol
lows:
'Stochastics in physics has a strong intuitive background. Also in order to deal with the
various stochastic ideas which arise in physics an enormous amount of technique and
expertise has been developed. It is quite likely that much of all this could also profitably
be used in the kind of problems met in engineering and industrial mathematics, in chemis
try and geology, as well as in economics, planning and management. The inverse is also
likely true. For instance the engineering community has much more of a habit in thinking
about problems involving exterior forces (often called inputs or controls in certain set
tings) and observations (outputs), i.e. about situations in which only indirect data are
available about the true physical state of the process involved. Thus the basic philosophy
of BIBOS IV is the serve as a meeting ground and forum where engineers, planners,
analysts, physicists and mathematicians meet and try to solve each other problems and
describe and discuss each others solutions and techniques'.
Let us elaborate a bit on this theme at the hand of the various contributions in this volume.
Simulated annealing (Bernasconi) is a global optimization technique and is a perfect example of
how introducing ideas from a completely different, and at first sight totally unrelated, field may yield
large and unexpected benefits. Annealing is the process of heating a solid and cooling it slowly to
remove strain and crystal imperfections. During this process the free energy of the solid is minimized.
The initial heating is necessary to avoid becoming trapped in a local minimum. Virtually every func
tion can be viewed as the free energy of some system and thus imitating on a computer whatever goes
on physically during annealing yields optimization algorithms. The result is a powerful new technique
called simulated annealing. Still very young; the first book on the topic (Aarts, van Laarhoven, Simu
lated annealing: theory and practice, Reidel, 1987) just appeared. It is clearly a very powerful tool in
some cases. It is almost totally unknown for what kind of optimization problem it will give good
results. "Potholes" e.g. are bad (cf. the Bernasconi article). Perhaps here another look at the original
physics will prove illuminating.
As already noted the engineering world (signal processing, control) has paid a great deal of atten
tion to situations where the state of a certain process is not directly observable but must be infered
from - usually noise corrupted - indirect (derived) observations. Here we take this problem in the
large sense of including parameter identification. This problem area, not surprisingly, involves all
kinds of deep mathematics and sophisticated computer science and is well represented by five articles
(Blankenship, Fresewinkel, Hazewinkel, Koski, Picard) all more or less from the engineering side.
Now of course also in physics the true state is rarely directly accessible and must be infered. It would
seem likely that the large engineering literature on filtering and identification could be very useful in
the natural sciences. On the other hand some of the series expansions in the Blankenship and
Hazewinkel articles are very similar to Dyson-Feynrnan expansions and it is far from unlikely that
some of the ideas of theoretical physics to deal with divergent series will be most useful here.
ix
x PREFACE
Wavelets of course come out of physics. It is already clear that a wavelet-based Fourier-type
analysis (Grossmann) will become most important in such computer science and engineering subjects
as pattern recognition and signal and image processing.
Optimal control, stochastic (optimal) control and adaptive control mainly came out of engineering
but the applications definitely are not limited to engineering and physics. There are for instance all
kinds of applications to economics (Gomez, Kemna-Vorst, Papavassilopoulos to some extent). A more
traditional engineering adaptive control problem is discussed by Hut-Olsder. Quite generally adaptive
type control should be important in very many fields. So far it seems not to be employed systemati
cally outside of the control electrical engineering and signal processing worlds. The idea is simple:
built in extra feedback loops which serve to adapt the control loops to unknown or changing parame
ters of the system. There are a number of adaptive schemes available and implemented. Many work
quite well. The underlying theory is still almost totally missing. Thus there are but a few conver
gence and stability proofs, and we have no idea to what extent the various available schemes are in
any sense near optimal.
By and large the engineering community seems to be more conservative that the physics one, and
slower to adopt such bits of weird mathematics as nonstandard analysis, superspace and fractals and
also slower to see that the simplest approximation to 'large' may well be 'infinite'. However nonstan
dard techniques definitely have found their way into optimal stochastic control (Cutland), and in this
field yield standard, i.e. implementable, results.
One traditional difficulty with infinite dimensional things is that usually there are many potential
infinite dimensional analogues for well studied finite dimensional concepts. For example the Laplacian
and harmonic analysis. Here the stochastic approach via Hida's powerful white noise analysis ideas
suggests the right analogues for infinite dimensional harmonic analysis, an infinite dimensional Lapla
cian and an infinite dimensional rotation group (Hida-Saito).
Many applications of stochastic ideas rest in one form or another on a some law of large numbers.
This makes any and all studies on limit distributions important also from the applied point of view
(Schneider).
Socalled IIf noise occurs frequently in engineering situations (cooled electric circuits e.g., or in
road modeling (11£2 for large f seems more frequent there )) and has been the subject of many confer
ences. Beyond phenomenological results, descriptions etc. not much could be done until recently. The
start was a suggestion, independently by several people, that deterministic chaos could well have some
of the characteristics of lIf-noise. Now with Takahashi's impressive paper on semi-Bemouilli systems
the road seems to be open for a great deal of progress in this area. It is rather pleasant to observe
that the engineering community thus will have to master yet another area of sophisticated mathemat
ics (viz. Fredholm determinants and such; and no doubt, as always, they will find yet other uses for
these concepts).
Fractals, like deterministic chaos, also are but hesitantly finding a niche in the engineering worlds.
Possibly because they are primarily a descriptive device with no physical dynamics behind them and
(hence) no control parameters. The word 'are' in the previous sentence is not quite correct anymore. It
is one of the most fascinating merits of the Pietronero--Evertsz-Siekesma paper in this volume that
they do manage to give an underlying 'genetic' dynamics to fractal structures. This really opens up
all kinds of highly promising control applications. For of course approximately fractal structures tum
up just about anywhere.
Th~lapplications of stochastics are by no means limited to phenomena with a clear stochastic com
ponent in them. Many phenomena which are purely deterministic can up to a point be modeled well
or even better by means of stochastic ideas. Turbulence for instance can be approached this way
(Chow). Here, in the general area of modeling, the engineering communities seem to put rather less
stress on the need to interprete the various internal variables of the model in terms of physically exist
ing entities. This leads sometimes to remarkably useful, very flexible and mathematically attractive
PREFACE xi
(exotic) models such as a completely 1-0 model for the heating of a gas flowing through a pipe.
Sometimes building a model based on whatever is known about the physics is preferable; sometimes a
more 'black-box' type approach yields more useful results.
We already mentioned simulated annealing above. One gift, so to speak, from statistical mechan
ics to other parts of mathematics. Interacting particle systems constitute another idea from this gen
eral field. And these in turn have lots of applications, e.g. in several ways to biology (stepping-stone
models for distributed biological populations (Shiga) and models for interacting biological cell systems
including cancer growth models (Tautu).
Finally, in this volume, there is the metaplectic group and its 'most singular representation' the
oscillator representation or Segal-Shale-Weil representation. It comes up in the (reduction) theory of
quadratic forms, i.e. in number theory (Weil), it arises naturally when studying symmetries of boson
quantum fields (Segal, Shale), it is definitely one of the most important representations from the
purely representation theoretic and harmonic analysis points of view (R. Howe's theory of dual reduc
tive pairs) and it is narrowly related to linear filtering and the Kalman-Bucy filters and hence the
matrix Riccati equation (cf. the Hazewinkel article in this volume). That would appear to various
enough for anyone mathematical object. But no, it is also of fundamental and for reaching impor
tance in laser optics and computerized tomography (Schempp).
Thus we feel that the contributions in this volume form a nice interesting cross-section of stochas
tics (taken to include some 'probabilistic-like' aspects of quantum mechanics) as it manifests itself
across a rather wide area of science and engineering. May these contributions be as stimulating to
many readers as they were to the participants.
Organizing a meeting like this is work. It is a real pleasure to thank the most efficient staff of ZiF
in Bielefeld for their courteous and plentiful help. It is also a pleasure to thank the Stiftung
Volkswagenwerk for the BIBOS funds which went into this meeting.
S. Albeverio
Ph. Blanchard
M. Hazewinkel
L. Streit
