On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion
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On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion Songzi Li1 · Xiang-Dong Li2,3
· Yong-Xiao Xie4
Received: 24 April 2019 / Accepted: 14 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 C ([0, T ], P (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 R. Using the optimal transportation theory, we prove that if V ≥ K for some constant K ∈ R, the McKean–Vlasov equation has a unique weak solution in the space of probability measures P (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 . Keywords Generalized Dyson Brownian motion · McKean–Vlasov equation · Gradient flow · Optimal transportation · Voiculescu free entropy · Law of Large Numbers · Propagation of chaos
1 Introduction 1.1 Background In 1962, Dyson [17,18] observed that the eigenvalues of the N × N Hermitian matrix valued Brownian motion is an interacting N -particle system with the logarithmic Coulomb interaction and derived their statistical properties. Since then, the Dyson Brownian motion has been used in various areas in mathematics and physics, including statistical physics and
Communicated by Eric A. Carlen. Songzi Li: Research supported by NSFC No. 11901569. Xiang-Dong Li: Research supported by NSFC No. 11771430, Key Laboratory RCSDS, CAS, No. 2008DP173182. Yong-Xiao Xie: Research supported by NSFC No. 11601287. Extended author information available on the last page of the article
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the quantum chaotic systems. See e.g. [29] and reference therein. In [34], Rogers and Shi proved that the empirical measure of the eigenvalues of the N × N Hermitian matrix valued Ornstein–Uhlenbeck process weakly converges to the nonlinear McKean–Vlasov equation with quadratic external potential as N tends to infinity. This also gave a dynamic proof of Wigner’s famous semi-circle law for Gaussian Unitary Ensemble. See also [2,23]. The purpose of this paper is to study the generalized Dyson Brownian motion and the associated McKean–Vlasov equation with the logarithmic Coulomb interaction and with general external potential. More precisely, let β ≥ 1 be a parameter, V : R → R+ be a continuous function, let (W 1 , . . . , W N )
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