In this paper we study coercive inequalities on finite dimensional metric spaces with probability measures which do not have the volume doubling property. Crown
Chapter 1. Introduction and notations Chapter 2. Poincaré-type inequalities Chapter 3. Entropy and Orlicz spaces Chapter 4. LS q and Hardy-type inequalities on the line Chapter 5. Probability measures satisfying LS q-inequalities on the real line Chapter 6. Exponential integrability and perturbation of measures Chapter 7. LS q-inequalities for Gibbs measures with Super Gaussian Tails. Chapter 8. LS q-inequalities and Markov semigroups 8.1. Ergodicity for nonlinear semigroups Chapter 9. Isoperimetry Chapter 10. The localization argument Chapter 11. Infinitesimal version Chapter 12. Proof of Theorem 9.2 Chapter 13. Euclidean distance (proof of Theorem 9.1) Chapter 14. Uniformly convex bodies Chapter 15. From Isoperimetry to LS q-inequalities Chapter 16. Isoperimetric Functional Inequalities Bibliography
For finite range lattice gases with a finite spin space, it is shown that the Dobrushin-Shlosman mixing condition is equivalent to the existence of a logarithmic Sobolev inequality for the associated (unique) Gibbs state. In addition, implications of these considerations for the ergodic properties of the corresponding Glauber dynamics are examined.
PreliminariesWe begin by introducing the setting in which and some of the notation with which we will be working throughout.The Lattice. The lattice Γ underlying our model will be the ^/-dimensional square lattice ΊL d for some fixed deZ + , and, for j e Γ, we will use the norm | k | = max 1 ^ t ^ d |k'|. Given A c Γ, we will use /If = Γ\A to denote the complement of A, \A\ to denote the cardinality oϊΛ, and j + A to denote the translate {j + k:ke/l}of/lby jeΓ. Furthermore, for each i?eR + , we take the R-boundary d R A to be the set {keyl(J: |k -j| ^ R for some ]eA] .We will often use the notation A
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