A Law of Large Numbers for interacting diffusions via a mild formulation
Abstract
Consider a system of $n$ weakly interacting particles driven by independent Brownian motions. In many instances, it is well known that the empirical measure converges to the solution of a partial differential equation, usually called McKeanVlasov or FokkerPlanck equation, as $n$ tends to infinity. We propose a relatively new approach to show this convergence by directly studying the stochastic partial differential equation that the empirical measure satisfies for each fixed $n$. Under a suitable control on the noise term, which appears due to the finiteness of the system, we are able to prove that the stochastic perturbation goes to zero, showing that the limiting measure is a solution to the classical McKeanVlasov equation. In contrast with known results, we do not require any independence or finite moment assumption on the initial condition, but the only weak convergence. The evolution of the empirical measure is studied in a suitable class of Hilbert spaces where the noise term is controlled using two distinct but complementary techniques: rough paths theory and maximal inequalities for selfnormalized processes.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.05624
 Bibcode:
 2020arXiv200505624B
 Keywords:

 Mathematics  Probability