Some fluctuation results for weakly interacting multi-type particle systems

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 → ∞. 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.

Weakly interacting particle systems on inhomogeneous random graphs

“…For example, in [10,11,23,24] a setting where a common noise process influences the dynamics of every particle is considered and limit theorems of the above form are established. For a setting with K -different subpopulations within each of which particle evolution is exchangeable, LLN and POC have been studied in [2], and a corresponding CLT has been established in [11]. Mean field results for heterogeneous populations have also been studied in [12,13].…”

Moderate deviation principles for weakly interacting particle systems

“…Although originally motivated by problems in statistical physics, over the past few decades, such models have arisen in many different application areas, including communication networks, mathematical finance, chemical and biological systems, and social sciences. For an extensive list of references to such applications see [8,13].…”

“…For example, in [27,28,12,13] limit theorems of the above form are established for a setting where there is a common noise process that influences the dynamics of each particle. LLN and POC in a setting with K-different subpopulations where particle evolution within each subpopulation is exchangeable, has been studied in [1] and a corresponding CLT has been established in [13]. Other works that have studied mean field results for such heterogeneous populations include [15,14].…”

Partial mean field limits in heterogeneous networks

“…Under (1)- (9) it is easy to prove that the rates r N 1 (T ), r N 2 (T ), r N 5 (T ) and r N 6 (T ) converge to 0 when N → ∞. For the latter two, this can be deduced in the same way as in Section 3.1 because the driving noises of different particles are uncorrelated.…”

“…Another restriction we will relax in our analysis concerns the driving noises of the interacting SDEs: instead of independence we explicitly allow for different degrees of dependence in the noise terms, even asymptotically. Until now there exists only a very small amount of literature that generalizes (1.1) in these directions: in [9,23,30,31] the particles are divided into finitely many groups within which they are homogeneous (and the number of members in both groups must tend to infinity for the law of large numbers), and [8,10], where one major agent exists and propagation of chaos for the minor agents is considered conditioned on the major one. Other papers that consider general heterogeneous systems include [13], where the propagation of chaos result is assumed, and [16,17,21], where a law of large numbers for the empirical measure is proved under various conditions.…”