A Law of Large Numbers and Large Deviations for interacting diffusions on Erdős-Rényi graphs

Fabio Coppini (1) , Helge Dietert (2) , Giambattista Giacomin (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdős-Rényi graph with parameter pn ∈ (0, 1], where n is the size of the graph (i.e., the number of particles). If pn ≡ 1 the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as n → ∞ to the solution of a PDE: a McKean-Vlasov (or Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdős-Rényi graphs with limn pnn = ∞, and properly rescaling the interaction to account for the dilution introduced by pn. However, these results have been proven under strong assumptions on the initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results-Law of Large Numbers and Large Deviation Principle-assuming only the convergence of the empirical measure of the initial condition.

Mots clés: Erdos-Renyi graphs; Interacting diffusions; Vlasov and McKean-Vlasov PDEs.; Mean field

Domaines: Probabilités [math.PR]

Fabio Coppini, Helge Dietert, Giambattista Giacomin. A Law of Large Numbers and Large Deviations for interacting diffusions on Erdős-Rényi graphs. Stochastics and Dynamics, 2020, 20 (2), pp.2050010. ⟨10.1142/S0219493720500100⟩. ⟨hal-02145087⟩