Yong-Xiao Xie | Searcharxiv

arXiv2015

On the Law of Large Numbers for the empirical measure process of Generalized Dyson Brownian motion

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, 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 $V"\geq K$ for some constant $K\in \mathbb{R}$, the McKean-Vlasov equation has a unique weak solution. This proves the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM. Finally, we prove the longtime convergence of the McKean-Vlasov equation for $C^2$-convex potentials $V$.

math.PR↗

arXiv2013

Generalized Dyson Brownian motion, McKean-Vlasov equation and eigenvalues of random matrices

Using Itô's calculus and the mass optimal transportation theory, we study the generalized Dyson Brownian motion (GDBM) and the associated McKean-Vlasov evolution equation with an external potential $V$. Under suitable condition on $V$, we prove the existence and uniqueness of strong solution to SDE for GDBM. Standard argument shows that the family of the process of empirical measures $L_N(t)$ of GDBM is tight and every accumulative point of $L_N(t)$ in the weak convergence topology is a weak solution of the associated McKean-Vlasov evolution equation, which can be realized as the gradient flow of the Voiculescu free entropy on the Wasserstein space over $\mathbb{R}$. Under the condition $V''\geq -K$ for some constant $K\geq 0$, we prove that the McKean-Vlasov equation has a unique solution $μ(t)$ and $L_N(t)$ converges weakly to $μ(t)$ as $N\rightarrow \infty$. For $C^2$ convex potentials, we prove that $μ(t)$ converges to the equilibrium measure $μ_V$ with respect to the $W_2$-Wasserstein distance on $\mathscr{P}_2(\mathbb{R})$ as $t\rightarrow \infty$. Under the uniform convexity or a modified uniform convexity condition on $V$, we prove that $μ(t)$ converges to $μ_V$ with respect to the $W_2$-Wasserstein distance on $\mathscr{P}_2(\mathbb{R})$ with an exponential rate as $t\rightarrow \infty$. Finally, we discuss the double-well potentials and raise some conjectures.

math.PR↗