The Annals of Applied Probability

Mean field limit for disordered diffusions with singular interactions

Eric Luçon and Wilhelm Stannat

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Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf{R} ^{m}$ in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice $\mathbf{Z} ^{d}$, and the interaction between two diffusions is attenuated by a spatial weight that depends on their positions. For a general class of singular weights (including the case already considered in the physical literature when interactions obey to a power-law of parameter $0<\alpha<d$), we address the convergence as $N\to\infty$ of the empirical measure of the diffusions to the solution of a deterministic McKean–Vlasov equation and prove well-posedness of this equation, even in the degenerate case without noise. We provide also precise estimates of the speed of this convergence, in terms of an appropriate weighted Wasserstein distance, exhibiting in particular nontrivial fluctuations in the power-law case when $\frac{d}{2}\leq\alpha<d$. Our framework covers the case of polynomially bounded monotone dynamics that are especially encountered in the main models of neural oscillators.

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Ann. Appl. Probab. Volume 24, Number 5 (2014), 1946-1993.

First available in Project Euclid: 26 June 2014

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G57: Random measures
Secondary: 60F15: Strong theorems 35Q92: PDEs in connection with biology and other natural sciences 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Disordered models weakly interacting diffusions Wasserstein distance spatially extended particle systems dissipative systems Kuramoto model FitzHugh–Nagumo model


Luçon, Eric; Stannat, Wilhelm. Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Probab. 24 (2014), no. 5, 1946--1993. doi:10.1214/13-AAP968.

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