How large is the mean-field framework ? LLN and CLT for empirical measures of diffusions on (random) graphs

par Eric Lucon (Université de Paris)

mardi 9 janv. 2024, 09:50 → 10:50 Europe/Paris

Amphi Schwartz (IMT)

The aim of the talk it to consider a system of N diffusions interacting on a (possibly random) nontrivial graph. An easy instance corresponds to the case of full-connectivity, that is when the graph of interaction is complete (mean-field case). In this case, the empirical measure of the system converges as $N\to \mathrm{\infty }$ to the solution of a nonlinear Fokker-Planck equation. However, the assumption of full-connectivity is often unrealistic in applications (e.g. neuronal models). The question is then: what happens if one removes edges in the graph of interaction ? Under what condition on the graph of interaction (i.e. the right notion of proximity w.r.t. the complete graph) under which the behavior of the system remains as in the mean-field case? We address this question of universality both at the level of the law of large numbers and fluctuations, for a large class of possibly random graphs, including the Erdös-Rényi class. This is based on joint works with S. Delattre, G. Giacomin and F. Coppini and C. Poquet.