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Logarithmic Sobolev inequalities for Gibbs states. (English) Zbl 0801.60056
Dell’Antonio, Gianfausto (ed.) et al., Dirichlet forms. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna, Italy, June 8-19, 1992. Berlin: Springer-Verlag. Lect. Notes Math. 1563, 194-228 (1993).
The article is comprised of several lectures which constitute a condensed exposition of logarithmic Sobolev type inequalities and their connection to Markov semigroups on manifolds. Results presented are mostly author’s own contributions to the subject initiated by L. Gross in the seventies. Especially worth noting are author’s observations regarding a key interplay between Sobolev and Poincaré inequalities of classical analysis and their generalized counterparts in probability (finite- dimensional case, notably uniform ergodicity) and statistical mechanics (infinite-dimensional lattice). In the first two lectures it is shown that if one perturbes the free heat flow on the compact connected Riemannian manifold $$M$$, i.e., the Laplace-Beltrami operator picks an additional force term derived from potential $$U$$, then the Markov semigroup $$P^ U_ t$$ with generator $$\Delta - \nabla U \cdot \nabla$$ is contractive and the following logarithmic Sobolev inequality holds true: $\kappa (U) \int_ Mf \log f d \gamma^ U \leq 4 \int_ M | \nabla f^{1/2} |^ 2d \gamma^ U.$ This in turn implies the uniform ergodicity as follows: $$\| \mu P^ U_ t - \gamma^ U \|_{\text{var}} \leq Ce^{ - 2 \kappa (U)t}$$ for any initial probability measure $$\mu$$, where $$C$$ is independent of $$\mu$$. Here, $$\gamma^ U (dx) = \text{const} e^{-U} \lambda (dx)$$, with $$\lambda$$ being the normalized Riemann measure on $$M$$, is called the Gibbs state and it is the unique probability measure that is invariant for the semigroup $$P_ t^ U$$.
Lectures three and four introduce an infinite-dimensional setting with $$M$$ replaced by $$M^{Z^ d}$$ (to treat lattice gases), an appropriate modification of Gibbs states and Dobrushin-Schlossman mixing condition to ascertain uniqueness of the Gibbs state $$\gamma$$. Lectures five and six are devoted to prove an infinite-dimensional analog of logarithmic Sobolev inequality for potential $${\mathcal J} =$$ family of finite range, lattice shift-invariant potentials. Unlike in finite-dimensional case, a much weaker coercivity (ergodicity) statement holds: $$\| \mu {\mathcal P}_ t - \gamma \|_{\text{var}} \leq C(\mu, \gamma) e^{-2 \kappa ({\mathcal J})t}$$ with $$C(\mu, \gamma)$$ being finite only in the case $$d \mu = fd \gamma$$ and $$\int_ Mf \log fd \gamma < \infty$$.
For the entire collection see [Zbl 0782.00060].

##### MSC:
 60H30 Applications of stochastic analysis (to PDEs, etc.) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 58J65 Diffusion processes and stochastic analysis on manifolds