Large Deviations and Quasipotential for Finite State Mean Field Interacting Particle Systems.

摘要

We study a general class of mean field interacting particle systems with a finite state space. Particles evolve as exchangeable jump Markov processes, where finite collections of particles are allowed to change their state simultaneously. Such models arise naturally in statistical physics, queueing systems and communication networks literatures. In the first part of the thesis, we establish a large deviation principle for the empirical measure process for the interacting particle systems. The approach is based on a variational representation for functionals of a Poisson random measure. Under an appropriate communication condition, we also prove a locally uniform large deviation principle. The main novelty is that more than one particle is allowed to change its state simultaneously, and so a standard approach to the proof based on change of measure is not possible. Along the way, we establish an LDP for jump Markov processes on the simplex, whose rates decay to zero as they approach the boundary of the domain. This result may be of independent interest. In the second part of the thesis, we focus on the mean field interacting particle systems that only admit single particle jumps. Under the assumption that there exists a unique stationary measure, we construct a Markov chain approximation of the quasipotential function associated with the equilibrium. This is the first example of the Markov chain approximation for problems with non-quadratic running cost (but still convex in the control), which may also have singularities near the boundary.