2014/01/14 by Amarjit Budhiraja, Budhiraja, Amarjit, Abhishek Pal Majumder +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1401.3423
openalex publication_date 2014/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study long time behavior of a discrete time weakly interacting particle\nsystem, and the corresponding nonlinear Markov process in \ℝd,\ndescribed in terms of a general stochastic evolution equation. In a setting\nwhere the state space of the particles is compact such questions have been\nstudied in previous works, however for the case of an unbounded state space\nvery few results are available. Under suitable assumptions on the problem data\nwe study several time asymptotic properties of the N-particle system and the\nassociated nonlinear Markov chain. In particular we show that the evolution\nequation for the law of the nonlinear Markov chain has a unique fixed point and\nstarting from an arbitrary initial condition convergence to the fixed point\noccurs at an exponential rate. The empirical measure \μnN of the\nN-particles at time n is shown to converge to the law \μn of the\nnonlinear Markov process at time n, in the Wasserstein-1 distance, in\nL1, as N\→ \∞, uniformly in n. Several consequences of this\nuniform convergence are studied, including the interchangeability of the limits\nn\→ \∞ and N\→\∞ and the propagation of chaos property at n =\n\∞. Rate of convergence of \μnN to \μn is studied by\nestablishing uniform in time polynomial and exponential probability\nconcentration estimates.\n