×

Logarithmic Sobolev inequalities for some nonlinear PDE’s. (English) Zbl 1059.60084

Summary: The aim of this paper is to study the behavior of solutions of some nonlinear partial differential equations of MacKean-Vlasov type. The main tools used are, on one hand, the logarithmic Sobolev inequality and its connections with the concentration of measure and the transportation inequality with quadratic cost; on the other hand, the propagation of chaos for particle systems in mean field interaction.

MSC:

60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bakry, D., 1994. L’hypercontractivité et son utilisation en théorie des semigroupes. École d’Été de Probabilités de Saint-Flour XXII—1992, Lectures Notes in Mathematics, Vol. 1581, Springer, Berlin, pp. 1-114.
[2] Bakry, D., 1997. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New Trends in Stochastic Analysis (Charingworth, 1994), Taniguchi symposium, River Edge, NJ, World Scientific, Singapore, pp. 43-75.
[3] Bakry, D., Emery, M., 1985. Diffusions hypercontractives. Séminaire de Probabilités, XIX, 1983/84, Lectures Notes in Mathematics, Vol. 1123, Springer, Berlin, pp. 177-206.
[4] Ben Arous, G.; Zeitouni, O., Increasing propagation of chaos for Mean fields models, Ann. inst. H. Poincaré probab. statist., 35, 1, 85-102, (1999) · Zbl 0928.60092
[5] Benachour, S.; Roynette, B.; Talay, D.; Vallois, D., Nonlinear self-stabilizing processes. I. existence, invariant probability, propagation of chaos, Stochastic process. appl., 75, 2, 173-201, (1998) · Zbl 0932.60063
[6] Benachour, S.; Roynette, B.; Vallois, P., Nonlinear self-stabilizing processes. II. convergence to invariant probability, Stochastic process. appl., 75, 2, 203-224, (1998) · Zbl 0932.60064
[7] Benedetto, D.; Caglioti, E.; Carrillo, J.A.; Pulvirenti, M., A non-Maxwellian steady distribution for one-dimensional granular media, J. statist. phys., 91, 5-6, 979-990, (1998) · Zbl 0921.60057
[8] Bobkov, S., Gentil, I., Ledoux, M., 2001. Hypercontractivity of Hamilton-Jacobi equations. Math. Pures Appl., to appear. · Zbl 1038.35020
[9] Carrillo, J., McCann, R., Villani, C., 2001. Kinetic equilibration rates for granular midia. Work in progress.
[10] Csiszár, I., Sanov property, generalized I-projection and a conditional limit theorem, Ann. probab., 12, 3, 768-793, (1984) · Zbl 0544.60011
[11] Gallot, S.; Hulin, D.; Lafontaine, J., Riemannian geometry, (1990), Springer Berlin · Zbl 0636.53001
[12] Grifone, J., 1990. Algèbre Linéaire. Cepadues Éditions, Toulouse.
[13] Gross, L., Logarithmic Sobolev inequalities, Amer. J. math., 97, 4, 1061-1083, (1975) · Zbl 0318.46049
[14] Ledoux, M., 1999. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités XXXIII, Lectures Notes in Mathematics, Vol. 1709, Springer, Berlin, pp. 120-216. · Zbl 0957.60016
[15] Otto, F.; Villani, C., Generalization of an inequality by talagrand and links with the logarithmic Sobolev inequality, J. funct. anal., 173, 2, 361-400, (2000) · Zbl 0985.58019
[16] Pinsker, M.S., 1964. Information and Information Stability of Random Variables and Processes. Holden-Day Inc., San Francisco, CA (Translated and edited by Amiel Feinstein). · Zbl 0125.09202
[17] Rachev, S.T., Probability metrics and the stability of stochastic models., (1991), Wiley New York · Zbl 0744.60004
[18] Sznitman, A.-S., 1991. Topics in propagation of chaos. École d’Été de Probabilités de Saint-Flour IX—1989, Lectures Notes in Mathematics, Vol. 1464, Springer, Berlin, pp. 165-251.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.