Lectures on Logarithmic Sobolev Inequalities

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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 L2 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: L2 ergodicity in a finite volume 10.1 Spectral gap estimate 10.2 L2 ergodicity in infinite volume Epilogue 2001 Bibliography