On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion

Abstract

We study the generalized Dyson Brownian motion (GDBM) of an interacting N-particle system with logarithmic Coulomb interaction and general potential V. Under reasonable condition on V, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on $$\mathcal {C}([0,T],\mathscr {P}(\mathbb {R}))$$ and all the large N limits satisfy a nonlinear McKean–Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, and Blower, we prove that the McKean–Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over $$\mathbb {R}$$ . Using the optimal transportation theory, we prove that if $$V''\ge K$$ for some constant $$K\in \mathbb {R}$$ , the McKean–Vlasov equation has a unique weak solution in the space of probability measures $$\mathscr {P}(\mathbb {R})$$ . This establishes the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM with non-quadratic external potentials which are not necessarily convex. Finally, we prove the longtime convergence of the McKean–Vlasov equation for $$C^2$$ -convex potentials V.