Some Fluctuation Results for Weakly Interacting Multi-type Particle System

TL;DR

This paper investigates the fluctuation behavior of large systems of weakly interacting particles of multiple types, deriving Gaussian and mixture Gaussian limits, and establishing new limit theorems for multi-type particle statistics.

Contribution

It extends fluctuation analysis to multi-type particle systems with common factors, providing new limit theorems and characterizations of asymptotic distributions.

Findings

Fluctuations are Gaussian without common factor.

Fluctuations are mixture of Gaussians with common factor.

New limit theorems for multi-type particle statistics.

Abstract

A collection of $N$-diffusing interacting particles where each particle belongs to one of $K$ different populations is considered. Evolution equation for a particle from population $k$ depends on the $K$ empirical measures of particle states corresponding to the various populations and the form of this dependence may change from one population to another. In addition, the drift coefficients in the particle evolution equations may depend on a factor that is common to all particles and which is described through the solution of a stochastic differential equation coupled, through the empirical measures, with the $N$-particle dynamics. We are interested in the asymptotic behavior as $N\to \infty$. Although the full system is not exchangeable, particles in the same population have an exchangeable distribution. Using this structure, one can prove using standard techniques a law of large numbers result and a propagation of chaos property. In the current work we study fluctuations about the law of large number limit. For the case where the common factor is absent the limit is given in terms of a Gaussian field whereas in the presence of a common factor it is characterized through a mixture of Gaussian distributions. We also obtain, as a corollary, new fluctuation results for disjoint sub-families of single type particle systems, i.e. when $K=1$. Finally, we establish limit theorems for multi-type statistics of such weakly interacting particles, given in terms of multiple Wiener integrals.