ON THE TREND TO EQUILIBRIUM FOR THE FOKKER PLANCK EQUATION AN INTERPLAY

ON THE TREND TO EQUILIBRIUM FOR THE FOKKER-PLANCK EQUATION : AN INTERPLAY BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS P. A. MARKOWICH? AND C. VILLANI?? Abstract. We present connections between the problem of trend to equilibrium for the Fokker-Planck equation of statistical physics, and several inequalities from functional analysis, like logarithmic Sobolev or Poincare inequalities, together with some inequalities arising in the context of concentration of measures, introduced by Talagrand, or in the study of Gaussian isoperimetry. Contents 1. The Fokker-Planck equation 2 2. Trend to equilibrium 3 3. Entropy dissipation 4 4. Logarithmic Sobolev inequalities 6 5. The Bakry-Emery reversed point of view 7 6. Log Sobolev ? Poincare 8 7. The nonuniformly convex case 9 8. Generalizations to other physical systems 10 9. An example : generalized porous medium equations 12 10. Gaussian isoperimetry 14 11. Talagrand inequalities and concentration of the Gauss measure 15 12. Log Sobolev ? Talagrand ? Poincare 16 13. Related PDE's 18 13.1. The Monge-Ampere equation 18 13.2. The Hamilton-Jacobi equation 19 13.3. The sticky particles system 19 14. HWI inequalities 19 15. Displacement convexity 20 References 22 Acknowledgement: The authors acknowledge support by the EU-funded TMR- network ‘Asymptotic Methods in Kinetic Theory' (Contract _ ERB FMRX CT97 0157) and from the Erwin-Schrodinger-Institute in Vienna.

• elemen- tary inequality


• distance between


• diffusion equations


• relative entropy


• diffusion


• entropy dissipation


• e?v


• known ?


• space ? ?


• lyapunov functional