Annals of Applied Probability

Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime

Martin Hutzenthaler and Daniel Pieper

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Propagation of chaos is a well-studied phenomenon and shows that weakly interacting diffusions may become independent as the system size converges to infinity. Most of the literature focuses on the case of exchangeable systems where all involved diffusions have the same distribution and are “of the same size”. In this paper, we analyze the case where only a few diffusions start outside of an accessible trap. Our main result shows that in this “sparse regime” the system of weakly interacting diffusions converges in distribution to a forest of excursions from the trap. In particular, initial independence propagates in the limit and results in a forest of independent trees.

Article information

Ann. Appl. Probab., Volume 30, Number 5 (2020), 2311-2354.

Received: April 2018
Revised: May 2019
First available in Project Euclid: 15 September 2020

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Mathematical Reviews number (MathSciNet)

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D25: Population dynamics (general)

Propagation of chaos interacting diffusions mean-field approximation many-demes limit measure-valued processes tree of excursions excursion measure altruistic defense


Hutzenthaler, Martin; Pieper, Daniel. Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime. Ann. Appl. Probab. 30 (2020), no. 5, 2311--2354. doi:10.1214/20-AAP1559.

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