On the construction of Lyapunov functions for nonlinear Markov processes via relative entropy

We develop an approach to the construction of Lyapunov functions for the forward equation of a nite state nonlinear Markov process. Nonlinear Markov processes can be obtained as a law of large number limit for a system of weakly interacting processes. The approach exploits this connection and the fact that relative entropy de nes a Lyapunov function for the solution of the forward equation for the many particle system. Candidate Lyapunov functions for the nonlinear Markov process are constructed via limits, and veri ed for certain classes of models.

[1]  T. Frank,et al.  Lyapunov and free energy functionals of generalized Fokker–Planck equations , 2001 .

[2]  田村 要造.  Free energy and the convergence of distributions of diffusion processes of McKean type = 自由エネルギーとマッキーン型拡散過程の分布の収束 , 1987 .

[3]  Jean-Yves Le Boudec,et al.  A class of mean field interaction models for computer and communication systems , 2008, Perform. Evaluation.

[4]  K. Oelschlager.  A Martingale Approach to the Law of Large Numbers for Weakly Interacting Stochastic Processes , 1984 .

[5]  F. Martinelli.  Lectures on Glauber dynamics for discrete spin models , 1999 .

[6]  A. Veretennikov,et al.  On Ergodic Measures for McKean-Vlasov Stochastic Equations , 2006 .

[7]  S. Benachour,et al.  Nonlinear self-stabilizing processes – II: Convergence to invariant probability , 1998 .

[8]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[9]  P. Cattiaux,et al.  Probabilistic approach for granular media equations in the non-uniformly convex case , 2006, math/0603541.

[10]  B. Jourdain,et al.  Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf , 2007, math/0701879.

[11]  W. Runggaldier,et al.  Large portfolio losses: A dynamic contagion model , 2007, 0704.1348.

[12]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

[13]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[14]  A. Gottlieb.  Markov Transitions and the Propagation of Chaos , 2000, math/0001076.

[15]  D. Talay,et al.  Nonlinear self-stabilizing processes – I Existence, invariant probability, propagation of chaos , 1998 .

[16]  Philippe Robert,et al.  Stochastic networks with multiple stable points. , 2006, math/0601296.

[17]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[18]  Carl Graham,et al.  McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets , 1992 .

[19]  Carl Graham,et al.  Interacting multi-class transmissions in large stochastic networks , 2008, 0810.0347.

[20]  Yiqiang Q. Zhao,et al.  Balancing Queues by Mean Field Interaction , 2005, Queueing Syst. Theory Appl..

[21]  A. Coolen,et al.  Macroscopic Lyapunov functions for separable stochastic neural networks with detailed balance , 1995 .

[22]  Alexandre Proutière,et al.  A particle system in interaction with a rapidly varying environment: Mean field limits and applications , 2010, Networks Heterog. Media.