MARK KAC SEMINAR

I will present the last developments of a long-time project involving C. Bahadoran (Clermont-Ferrand), H. Guiol (Grenoble), T. Mountford (Lausanne), K. Ravishankar (SUNY, New Paltz).

We prove a quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on Z in random ergodic environment. The limit is given by the entropy solution to a scalar conservation law with a Lipschitz-continuous macroscopic flux function. Our result is a strong law of large numbers, that we derive by a constructive method. Our approach relies on: (i) a variational formula for entropy solutions, and (ii) an approximation method to prove that the hydrodynamic limit for initial Riemann profiles implies the hydrodynamic limit for general initial profiles.

Our method being quite robust with respect to the model and to the form of the disorder, we illustrate it on various examples such as generalizations of misanthropes processes (including site or bond disorders), and of k-step K-exclusion processes with different types of random environment. I will detail a traffic model and a queueing model.

Logarithmic Sobolev inequalities, introduced by Stam in information theory and Gross in the analysis of infinite dimensional diffusion operators, appear as a main tool in the study of convergence to equilibrium of Markov chains and semigroups in various settings. Their analysis involve a large number of methods, ranging from functional analysis, geometry, optimal transportation, statistical mechanics etc.

The first part of the talk will be devoted to a general introduction to the subject of logarithmic Sobolev inequalities. The second part will present some (old and new) results on logarithmic Sobolev inequalities for continuous unbounded spin systems, both in the perturbative and conservative regimes.