## An Initiation to Logarithmic Sobolev InequalitiesThis 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 |

117 | |

### Common terms and phrases

apply arbitrary associated Boltzmann measure bounded function bounded measurable function bounded operator Brownian motion Cl Remark compact configuration consider continuous function convex cr-field deduce defined Definition denote dense density Dirichlet domain equal equation essentially self-adjoint Exercise exists a constant exterior condition filtered Brownian motion finite subset function f Gaussian Gibbs measure Gross inequality Hilbert space hypotheses implies inequality with constant infinitesimal integral interactions invariant measure Ising model isometry Ito's formula Kolmogorov process Kolmogorov semi-group Lemma Let f linear Lipschitz logarithmic Sobolev inequality mapping Markov property martingale measurable function obtain open set Poincare inequality polynomial positive measure preceding probability measure Proof Proposition Prove random variable respect satisfies self-adjoint operator semi-group Nt sense of distributions spectral stochastic sufficient symmetric operator Theorem topology transition kernel Ul,z uniform norm unique upper bound utilizing vector zero