We study a system of $N$ particles interacting through their empirical distribution on a finite state space in continuous time. Formally, in the limit as $N\to\infty$, the system takes the form of a nonlinear (McKean-Vlasov) Markov chain, and we rigorously establish this limit. Specifically, under the assumption that the mean field system has a unique, exponentially stable stationary distribution, we show that the weak error between the empirical measures of the $N$-particle system and the law of the mean field system is of order $1/N$ uniformly in time. Our analysis makes use of a master equation for test functions evaluated along the measure flow of the mean field system, and we demonstrate that the solutions of this master equation are sufficiently regular. We then show that exponential stability of the mean field system is implied by exponential stability of solutions to the linearized Kolmogorov equation with a source term. We apply our results to study the convergence of large population games to mean field games and give a new condition for the existence of a unique, exponentially stable stationary distribution for a nonlinear Markov chain.
Building: | East Hall |
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Event Type: | Workshop / Seminar |
Tags: | Mathematics |
Source: | Happening @ Michigan from Financial/Actuarial Mathematics Seminar - Department of Mathematics, Department of Mathematics |