Moderate Deviation Principles for Weakly Interacting Particle Systems

TL;DR

This paper establishes moderate deviation principles for empirical measures of weakly interacting Markov processes, covering diffusions and jump processes, using large deviation techniques in functional spaces.

Contribution

It introduces moderate deviation principles for two classes of weakly interacting Markov processes, extending large deviation theory to empirical measure paths in new functional spaces.

Findings

Moderate deviation principles are proved for diffusion-based models.

Moderate deviation principles are proved for jump process models.

The results are formulated in Schwartz distribution space and Hilbert space.

Abstract

Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in the path space of an appropriate Schwartz distribution space whereas for the second family the LDP is proved in the space of $l_2$ (the Hilbert space of square summable sequences)-valued paths. Proofs rely on certain variational representations for exponential functionals of Brownian motions and Poisson random measures.