Probability Measures on Groups: Proceedings of the Fifth Conference Oberwolfach, Germany, January 29th – February 4, 1978

361 Pages·1979·4.084 MB·English-French-German

by  P. Baldi (auth.), Herbert Heyer (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
706
Probability Measures
no Groups
Proceedings of the Fifth Conference
Oberwolfach, Germany,
January 29th - February ,4 1978
Edited by .H Heyer
galreV-regnirpS
Berlin Heidelberg New kroY 1979
Editor
Herbert Heyer
Mathematisches Institut
Universit~t TLibingen
Auf der Morgenstelle 10
D-7400 TQbingen
- ,2 ~AI 9791
Library of Congress Cataloging in Publication Data
Main entry under title:
Probability measures on groups.
(Lecture notes in mathematics ; 706)
Bibliography: .p
Includes index.
.i Probabilities--Congresses. .2 Groups,
Theory of--Congresses. .3 Measure theory--
Congresses. .4 Random walks (Mathematics)--
Congresses. .I Heyer, Herbert. II. Title.
III. Series: Lecture notes in mathematics
(Berlin) ; 706.
QAS.L28 no. 706 QA273.AI 510'.8s 519.2
79-10236
AMS Subject Classifications (1970): primary: 43A05, 60K05, 60B15,
60J15 secondary: 60B05, 60J35, 60G50
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INTRODUCTION
Probability theory on groups has become a field of considerable
interest to analysts in recent years. Despite of the progress
which still goes on, the degree of maturity of certain parts of
the theory has encouraged a number of probabilists to write
extended expositions on the subject. These publications prove both
the breadth and the profundity of the problems and their solutions.
We only mention the book I by C. Berg and G. Forst on the
potential theory of continuous convolution semigroups on locally
compact Abelian groups, the Lecture Notes on random walks on Lie
groups by Y. Guivarc'h, M. Keane and B. Roynette 4, on continuous
convolution semigroups and generating distributions by W. Hazod 5
and on convolution products of measures on arbitrary topological
groups and semigroups by A. Mukherjea and N.A. Tserpes i0~
finally the editor's monograph 6 containing a detailed presenta-
tion of the basic topics of the theory for general locally compact
groups. Stimulated by the written work the specialists keep
articulating the legitimate desire of a permanent exchange of
ideas. Meetings and conferences help surveying the published results
and initiate new directions of research. In this spirit the edition
of the contributions collected from the fifth Oberwolfach conference
on Probability Measures on Groups will be more than justified.
When looking into the actual trends of research within the theory
of probability on groups one observes essentially two different
aspects of extending the classical theory:
The one aspect consists in generalizing a given classical theorem
valid for the real line or the torus group to possibly all locally
compact groups. Here the generalization is mostly achieved at the
costs of the strength of the given statement. The second aspect is
based on the idea to preserve the strong formulation of the
classical result as strictly as possible and still extend it to
more general groups. While the first aspect gives rise to an
enlargement of the fundamental concepts of the theory~ the second
aspect enriches the collection of examples which have to be avail-
able in order to test the general theory. Under the first aspect
the theory has been generalized via Lie groups and Lie projective
groups to arbitrary locally compact groups 6 and by various
other methods to completely regular or metric groups and
Vr
semigroups I0. The second aspect led to generalizations for the
affine group~ the motion group~ general nilpotent and solvable
Lie groups as well as their compact extensions 4.
The papers published in this volume represent fairly well the wide
spectrum of the actual theory. We shall arrange the contributions
under five different themes and comment briefly on the subjects
treated.
.1 Infinite convolutione of probability measures on groups and
semigroups from a field of research which is basic for the theory.
Interestingly enough the It~-Kawada theorem still motivates further
work in this direction. By a well-known result of Y. Derriennic and
A. Mukherjea the powers of a probability measure generating a non-
compact second countable locally compact group converge to the zero
measure. Quantitative results on the rate of this convergence are
presented for infinite discrete groups, especially for free groups
by P. GERL using combinatorial methods~ and by P. BOUGEROL using
the method of concentration functions. The last author points out
that it will be promising to generalize his method from groups to
homogeneous spaces. A. MUKHERJEA's contribution based on the work
of G. Brown~ W. Moran and P. Hartman contains purity properties of
the limits of infinite convolutions. The paper of V. LOSERT and
H. RINDLER touches the relationship between random walks on compact
groups and the theory of equidistribution.
.2 Conti~nuous convolution eemigroup8: After the analysis of con-
tinuous convolution semigroups of measures has reached its
culmination with the general L~vy-Khintchine representation, new
detail aspects have moved into the center of discussion. Th. DRISCH
deals in his paper with the analyticity of convolution semigroups.
W. HAZOD's main interest in his contribution is the problem of
mixing convolution and operator semigroups with application to
subordination. Aspects of particular interest are an order relation
defined by the subordination procedure and the Bochner-stable semi-
groups introduced as subordinated semigroups. Convolution semigroups
of matrix-valued measures correspond to contraction semigroups of
invariant operators on the vector-valued continuous functions on
the group. Their representation as contraction semigroups of
positive measures and their generation by operators satisfying a
maximum principle is the object of the paper by J.-P. ROTH.
V
The papers of C. BERG and of C. BERG and G. FORST pertain to the
potential theory of continuous convolution semigroups on an Abelian
group. C. Berg constructs continuous singular Hunt convolution
kernels on the real line and on the infinite-dimensional torus.
Jointly with G. Forst he reviews some results from the theory with
s~ecial emphasis on similarities between potential kernels and
infinitely divisible probabilities on a locally compact Abelian
group.
.3 Special classes of probability measures: As a consequence of the
recent development of the central limit problem infinitely divisible
probability measures received a particular interest. The problem of
embedding infinitely divisible measures in continuous convolution
semigroups has been solved only partially. Very little is known
about the continuous embedding of measures on algebraic groups. In
K.R. PARTHASARATHY's paper it is shown that on SL(2,~) and SL(2,C)
every infinitely divisible probability which generates the group is
diadically embeddable. Another class of probability measures arising
in central limit theory is the class of stable measures, whose
generalization to general groups depends on the automorphism group
of the underlying group. P. BALDI establishes for measures on the
motion group a compactness lemma and obtains a characterization of
stable measures as limits of normed powers. There is no result yet
within this framework on domains of attraction of stable measures.
For Gaussian measures on metric Abelian groups T. BYCZKOWSKI
presents results on the integrability of seminorms and on zero-one
laws. The framework of a metric Abelian group is chosen in order to
apply the theorems obtained also to function spaces. Full measures
as studied by V. LOSERT and K. SCHMIDT have their origin in ergodic
theory. Their relations to cocycles, ergodic flows and eigenfunctions
are described. Moreover the authors include results on the existence
of full measures and of such subgroups which do not carry full
measures. A. JANSSEN's work on the measurability of certain sets of
measures on a topological space has implications to the theory of
admissable translates.
.4 Random walks on groups and homogeneous spaces play an eminent
rSle in the potential theory on these structures. In recent years
special emphasis has been given to noncommutative renewal theory
and to the development of local limit theorems. The paper of G. FORST
concerns renewal measures on the half line. A new proof is given for
IV
Daley's result containing necessary and sufficient conditions for a
given renewal measure to have the property that each of its positive
multiples is again a renewal measure. The aim of L. ELIE's paper
pertaining to renewal theory is a characterization of all almost
compact locally compact groups having property (P). It turns out
that this property (P) is the key tool in establishing renewal
results for spread out transient measures generating the underlying
group. In a second contribution L. Elie studies the cone of ~-invar-
iant positive measures on the affine group and exhibits its extremal
rays. The theory developed yields the study of the Martin boundary
of the corresponding random walk on the group and new information on
its positive harmonic functions. In the paper of H. HENNION a
dichotomy theorem for random walks on homogeneous spaces is proved.
Moreover it is conjectured that for rigid Lie groups a homogeneous
space is recurrent if and only if it is of polynomial growth of
degree ~ 2. The object of the research done by L. HANTSCH and W. V0N
WALDENFELS is a concrete problem from atomic physics. Here the
averaging of atomic decay rates by random impacts leads to a
particular random walk on the general linear group. The limiting
behavior of this random walk can be estimated. A. RAUGI and P. CREPEL
have contributed to central limit theory for random walks on certain
Lie groups. The first author presents a profound result on the con-
vergence of normed powers of a measure to an absolutely continuous
measure lying on a diffusion semigroup. The class of groups involved
consists of semidirect products of rigid solvable Lie groups with
compact groups. In contrast to this rather technical analysis we see
Cr&pel's survey article on the methods and results from central
limit theory for independent and dependent random variables. The well
established methods of Fourier transforms~ moments, those based on
Trotter's and Skorohod's theorems and on the various kinds of
dependence are exemplified on the motion and the Heisenberg groups.
.5 Group representations and probability: In connection with the
theory of ergodic automorphisms of compact groups the problem of
characterizing all locally compact groups whose irreducible, unitary
continuous representations are of uniformly bounded degree appears to
be relevant. C.C. Moore showed that the groups with this property are
exactly those admitting an Abelian normal subgroup of finite index.
In his paper E. SCHLICHTING presents a new and direct proof of the
theorem by reducing the problem to compact groups. Cocycles and
coboundaries on the motion group are central in the contribution of
VII
B.-J. FALKOWSKI. The author exhibits all infinitely divisible
positive definite functions on the group which correspond to the
cocycles of irreducible representations. Weakly stationary processes
on a locally compact Abelian group are exactly the Fourier transforms
of certain vector-valued measures on the group with values in some
Hilbert space. P. RESSEL studies in his paper compact sets of such
measures and derives a L~vy type continuity theorem.
Aside from the open problems stated in some of the papers in this
volume several questions have been brought up at the conference
which might stimulate future research. There is important work still
to be done in the arithmetic of the semigroup of probability measures
on a locally compact group. Fundamental classical results in this
direction have been collected in the monographs 2 of R. Cuppens,
7 and 8 of Yu.V. Linnik and of Yu.V. Linnik and I.V. Ostrovski
resp., and in the article 9 of L.Z. Liv~ic, I.V. 0strovski~ and
G.P. ~istjakov. Another field of interest is the theory of continuous
convolution hemigroups on a locally compact group and its connection
with evolution families of contraction operators and with additive
stochastic processes. Pioneering work has been done in the papers
II and 12 by D.W. Stroock and S.R.S. Varadhan, and 3 by
P. Feinsilver. A wide range of open problems can be looked at in
noncommutative potential theory as it has been exposed at the con-
ference by F. Hirsch. And, after all, the application of the theory
to problems outside mathematics will certainly gain priority in the
course of years. The example presented by R.M. Dudley on the
application of the central limit theorem on GL(2,~) to high energy
scattering gave a realistic insight into the discrepancy between
theory and practice.
Herbert Heyer
References
1 .C Berg, G. Forst: Potential theory on locally compact Abelian
groups.
Berlin-Heidelberg-New York: Springer 1975
2 R. Cuppens: Decomposition of multivariate probability.
New York-San Francisco-London: Academic Press 1975
3 P. Feinsilver: Processes with independent increments on a Lie
group.
Trans. Amer. Math. Soc. 242, 73-121 (1978)
4 Y. Guivarc'h, M. Keane, .B Roynette: Marches al&atoires sur les
groupes de Lie.
Lecture Notes in Math. Vol. 624. Berlin-Heidel-
berg-New York: Springer 1977
5 W. Hazod: Stetige Halbgruppen von WahrscheinlichkeitsmaSen
und erzeugende Distributionen.
Lecture Notes in Math. Vol. 595 Berlin-Heidelberg-
New York: Springer 1977
6 H. Heyer: Probability measures on locally compact groups.
Berlin-Heidelberg-New York: Springer 1977
7 Yu. v. Linnik: D~composition des lois de probabilit~s.
Paris: Gauthier-Villars 1982
8 Yu. V. Linnik, I. V. Ostrovskii: v Decomposition of random
variables and vectors.
English translation: Amer. Math. Soc. 48 (1977)
9 .L .Z Liv~ic, I. V. Ostrovski, G. P. ~istjakov: The arithmetic
of probability laws.
Probability theory. Mathematical Statistics.
Theoretical Cybernetics Vol. 21, pp 5-42
Moskow: Akad. Nauk SSSR Vsesojuz. Inst. Nau~n. .i
Techn. Informacii 1975 Russian
XI
10 A. Mukherjea, N. A. Tserpes: Measures on topological semigroups.
Lecture Notes in Math. Vol. 547. Berlin-
Heidelberg-New York: Springer 1976
11 D. W. Stroock, S.R.S. Varadhan: Diffusion processes with
continuous coefficients I, II.
Comm. Pure Appl. Math. 12, 345-400, 479-530 (1969)
(cid:127)2 D. W. Stroock, S.R.S. Varadhan: Limit theorems for random
walks on Lie groups.
Sankhya Ser. A 35, 27?-294 (1973)
LIST OF PARTICIPANTS
P. Baldi, Paris H. Heyer, TNbingen
C. Berg, Kopenhagen .F Hirseh, Caehan
M.S. Bingham, Hull A. Janssen, Dortmund
P. Bougerol, Paris E. Kaniuth, Paderborn
T. Byczkowski, WroeXaw A. Mukherjea, Tampa
P. Cr~pel, Rennes P. Plaumann, Erlangen
E. Dettweiler, T~bingen A. Raugi, Rennes
Th. Driseh, Dortmund .P Ressel, Hamburg
R.M. Dudley, Cambridge H. Rindler, Wien
L. Elie, Paris J.-P. Roth, Mulhouse
B.-J. Falkowski, MNnchen .B Roynette, Nancy
P.J. Fernandez, Rio de Janeiro E. Schlichting, MNnchen
G. Forst, Kopenhagen K. Schmidt, Coventry
P. Gerl, Salzburg D.W. Strook, Boulder
W. Guth, Konstanz C. Sunyach, Paris
W. Hazod, Dortmund W. von Waldenfels, Heidelberg
H. Hennion, Rennes