An Initiation to Logarithmic Sobolev Inequalities

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American Mathematical Soc., 2007 - Mathematics - 119 pages
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This book provides an introduction to logarithmic Sobolev inequalities with some important applications to mathematical statistical physics. Royer begins by gathering and reviewing the necessary background material on selfadjoint operators, semigroups, Kolmogorov diffusion processes, solutions of stochastic differential equations, and certain other related topics. There then is a chapter on log Sobolev inequalities with an application to a strong ergodicity theorem for Kolmogorov diffusion processes. The remaining two chapters consider the general setting for Gibbs measures including existence and uniqueness issues, the Ising model with real spins and the application of log Sobolev inequalities to show the stabilization of the Glauber-Langevin dynamic stochastic models for the Ising model with real spins. The exercises and complements extend the material in the main text to related areas such as Markov chains.
 

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Contents

SelfAdjoint Operators
1
12 Spectral decomposition of selfadjoint operators
8
SemiGroups
15
22 Kolmogorov semigroups
19
Logarithmic Sobolev Inequalities
37
32 An application to ergodicity
55
Gibbs Measures
65
42 An Ising model with real spin
73
Stabilization of GlauberLangevin Dynamics
89
52 The case of weak interactions
95
53 Perspectives
101
Appendix A
105
A2 Bounded real measures
109
A3 The topology of weak convergence
111
Bibliography
117
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