An Initiation to Logarithmic Sobolev Inequalities (SMF AMS Texts & Monographs)

130 Pages·2007·2.6 MB·English

by  Gilles Royer

An Initiation
to Logarithmic
Sobolev Inequalities
Gilles Royer
An Initiation
to Logarithmic
Sobolev Inequalities

SMF/AMS TEXTS and MONOGRAPHS Volume 14
Cours Specialises Numero 5 1999
An Initiation
to Logarithmic
Sobolev Inequalities
Gilles Royer
Translated by
Donald Babbitt
a 0
N
American Mathematical Society
Societe Mathematique de France
Une Initiation aux Inegalites de Sobolev Logarithmiques
An Initiation to Logarithmic Sobolev Inequalities
Gilles Royer
Originally published in French by Society Mathematique de France.
Copyright © 1999 Societe Matht matique de France
Translated from the French by Donald Babbitt
2000 Mathematics Subject Classification. Primary 60-02; Secondary 35J85, 47B25,
47D07, 60J60, 82C99.
For additional information and updates on this book, visit
www.anis.org/bookpages/smfanLs-14
Library of Congress Cataloging-in-Publication Data
Royer. Gilles.
[Initiation aux in4galites de Sobolev logarithmiques. English]
An initiation to logarithmic Sobolev inequalities / Gilles Royer ; translated by Donald Babbitt.
p. cm. - (SMF/AMS texts and monographs, ISSN 1525-2302 ; v. 14) (Cours specialises,
ISSN 1284-6090 ; 5)
Includes bibliographical references.
ISBN-13: 978-0-8218-4401-4 (alk. paper)
ISBN-10: 0-8218-4401-6 (elk. paper)
1. Ergodic theory. 2. Logarithmic functions. 3. Semigroups of operators. 4. Differential
inequalities. I. Title.
QA313.R6913 2007
515'.48--dc22 2007060798
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Contents
Preface vii
Chapter 1. Self-Adjoint Operators
1
1.1. Symmetric operators
1
1.2. Spectral decomposition of self-adjoint operators 8
Chapter 2. Semi-Groups 15
2.1. Semi-groups of self-adjoint operators 15
2.2. Kolmogorov semi-groups 19
Chapter 3. Logarithmic Sobolev Inequalities 37
3.1. The Poincare and Gross inequalities 37
3.2. An application to ergodicity 55
Chapter 4. Gibbs Measures 65
4.1. Generalities 65
4.2. An Ising model with real spin 73
Chapter 5. Stabilization of Glauber- Langevin Dynamics 89
5.1. The Gross inequality and stabilization 89
5.2. The case of weak interactions 95
5.3. Perspectives 101
Appendix A. 105
A.1. Markovian kernels 105
A.2. Bounded real measures 109
A.3. The topology of weak convergence 111
Bibliography 117
V

Preface
This book contains the material that was essentially covered in a course
"de troisieme cycle"' taught during the second semester of the 1996-1997
academic year at the University d'Orleans. The goal of this course was to
give an exposition of an example of the use of logarithmic Sobolev inequal-
ities coming primarily from two papers by B. Zegarlinski [Zeg9O, Zeg96].
The example is concerned with real spin models with weak interactions on
a lattice where one can apply a classic method due to Dobrushin; see no-
tably [DobTO]. For these models, we give a proof of the uniqueness of the
Gibbs measure by showing the exponential stabilization of the stochastic
evolution of an infinite dimensional diffusion process which generalizes the
case of the Glauber dynamics for the Ising model. Although these models
are technically more complicated than the Ising model, one still uses famil-
iar techniques, e.g., using Ito's stochastic integral calculus to construct and
study diffusion processes, as well as utilizing the well-known properties of
self-adjoint differential operators on iR" and Sobolev and Poincare inequali-
ties in their original setting. These models also utilize in a natural way some
elegant results on logarithmic Sobolev inequalities such as the Bakry-Emery
and Herbst inequalities. Interestingly, these models are simplifications of the
Nelson models of Euclidean fields where Gross first introduced logarithmic
Sobolev inequalities.2
In this book we introduce in a self-contained manner the basic notions of
self-adjoint operators, diffusion processes, and Gibbs measures. The chap-
ter on logarithmic Sobolev inequalities is enriched by adding applications to
Markov chains so as not to remain in too special a setting. The reader will
find indications of some recent applications of logarithmic Sobolev inequal-
ities to statistical mechanics at the end of Chapter 5.
I would like to thank my colleagues S. Roelly and P. Maheux for very
useful discussions as well as the students of the DEA d'Orleans, in particular,
G. Salin.
Note added to the original Preface. The translation presented here
differs from the French original by a small number of corrections. Since the
original course was given, logarithmic Sobolev inequalities have been the
subject of many articles. We recommend that interested readers consult
'Translator's note: "Un cours de troisieme cycle" is equivalent to an advanced grad-
uate course in an American university.
2Translator's note: These are now also called Gross inequalities.
vii
viii PREFACE
[Cor02, OR071, and their bibliographies if they are interested in further
study of the subjects treated here.
CHAPTER 1
Self-Adjoint Operators
We denote by H, a separable complex Hilbert space,' by V a dense
linear subspace of H, and by A an operator from D to H. The space V is
called the domain of the operator A and is denoted D(A). Unlike bounded
operators, 2 in particular, operators on any finite dimensional Hilbert space,
simple consideration of the symmetry of operators does not lead to a theorem
of spectral decomposition. We will introduce directly the notion of self-
adjointness by utilizing spectral conditions based on an expose of P. Cartier
at 1'Ecole Polytechnique.
1.1. Symmetric operators
Definition 1.1.1. We say that the complex number A is in the resolvent
set p(A) of A if (AId-A) is injective, its image (AId-A)V is dense in'H, and
if the inverse operator (A Id - A)-' is a bounded operator from (A Id - A)D
to H. This operator is then uniquely extended to a bounded operator R,,
on H called the resolvent operator.
We often abbreviate A - A Id by A - A.
Proposition 1.1.2 (Resolvent Equation). . For all A, Ec E p(A) we
have:
RA-R1, =(A-u)R,RA.
Note that the Resolvent Equation implies that {RA} is a commutative
family of operators.
Definition 1.1.3. We say that A is closed if V is complete for the norm
IIII.a = (II0II2 + IIA',II2)1,2.
Consider the graph of A:
9A={(1p,ATp)EHxH : 1P ED}.
It is obvious that the projection of the graph of A, with the usual product
norm H x H, onto D, with the norm HA, is an isometry. Thus it is clear
that A is closed if its graph CA is closed in H x H. For a closed operator A
one can express the resolvent set p(A) in a simpler way.
'The scalar product is left linear and right anti-linear.
2Recall that an operator B is bounded if there exists a constant M such that )IBIS <
MJJxJJ for all x in V.
1