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Lectures on logarithmic Sobolev inequalities. (English) Zbl 1125.60111
Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXVI. Berlin: Springer (ISBN 3-540-00072-0/pbk). Lect. Notes Math. 1801, 1-134 (2003).
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 $\ge 2$. 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C20 Dynamic lattice systems and systems on graphs 60J25 Continuous-time Markov processes on general state spaces 82C22 Interacting particle systems