Date: 21 Oct 2004

Lectures on Logarithmic Sobolev Inequalities

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Contents.

  • Introduction

  • Chapter 1. Markov semi-groups

  • 1.1 Markov semi-groups and Generators

  • 1.2 Invariant measures of a semi-group

  • 1.3 Markov processes

  • Chapter 2. Spectral gap inequality and L 2 ergodicity

  • Chapter 3. Classical Sobolev inequalities and ultracontractivity

  • Chapter 4. Logarithmic Sobolev inequalities and hypercontractivity

  • 4.1 Properties of logarithmic Sobolev inequality

  • 4.2 Logarithmic Sobolev and Spectral Gap inequalities

  • 4.3 Bakry-Emery Criterion

  • Chapter 5. Logarithmic Sobolev inequalities for spin systems on a lattice

  • 5.1 Notation and definitions, statistical mechanics

  • 5.2 Strategy to demonstrate the logarithmic Sobolev inequality

  • 5.3 Logarithmic Sobolev inequality in dimension 1; an example

  • 5.4 Logarithmic Sobolev inequalities in dimension \(\geq\) 2

  • Chapter 6. Logarithmic Sobolev inequalities and cellular automata

  • Chapter 7. Logarithmic Sobolev inequalities for spin systems with long range interaction. Martingale expansion

  • Chapter 8. Markov semigroup in infinite volume, ergodic properties

  • 8.1 Construction of Markov semi-groups in infinite volume

  • 8.2 Uniform ergodicity of Markov semi-groups in infinite volume

  • 8.3 Equivalence Theorem

  • Chapter 9. Disordered systems; uniform ergodicity in the high temperature regime

  • 9.1 Absence of spectral gap for disordered ferromagnetic Ising model

  • 9.2 Upper bound for the constant of logarithmic Sobolev inequality in finite volume and uniform ergodicity, d=2

  • Chapter 10. Low temperature regime: L 2 ergodicity in a finite volume

  • 10.1 Spectral gap estimate

  • 10.2 L 2 ergodicity in infinite volume

  • Epilogue 2001

  • Bibliography