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Some fluctuation results for weakly interacting multi-type particle systems

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  • Budhiraja, Amarjit
  • Wu, Ruoyu

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.

Suggested Citation

  • Budhiraja, Amarjit & Wu, Ruoyu, 2016. "Some fluctuation results for weakly interacting multi-type particle systems," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2253-2296.
  • Handle: RePEc:eee:spapps:v:126:y:2016:i:8:p:2253-2296
    DOI: 10.1016/j.spa.2016.01.010
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    References listed on IDEAS

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    1. Kurtz, Thomas G. & Xiong, Jie, 1999. "Particle representations for a class of nonlinear SPDEs," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 103-126, September.
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    Cited by:

    1. Chong, Carsten & Klüppelberg, Claudia, 2019. "Partial mean field limits in heterogeneous networks," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 4998-5036.
    2. Sergey Nadtochiy & Mykhaylo Shkolnikov, 2018. "Mean field systems on networks, with singular interaction through hitting times," Papers 1807.02015, arXiv.org, revised Sep 2019.
    3. Agathe-Nerine, Zoé, 2022. "Multivariate Hawkes processes on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 86-148.
    4. Bhamidi, Shankar & Budhiraja, Amarjit & Wu, Ruoyu, 2019. "Weakly interacting particle systems on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 2174-2206.

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