# 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 -particle system with logarithmic Coulomb interaction and general potential . Under reasonable condition on , 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 and all the large 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 . Using the optimal transportation theory, we prove that if for some constant , 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 -convex potentials .

Key words and phrases: Generalized Dyson Brownian motion, McKean-Vlasov equation, gradient flow, optimal transportation, Voiculescu free entropy, Law of Large Numbers, propagation of chaos.

## 1 Introduction

In 1962, F. Dyson [20, 21] observed that the eigenvalues of the Hermitian matrix valued Brownian motion is an interacting -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 the quantum chaotic systems. See e.g. [38] and reference therein. In [47], Rogers and Shi proved that the empirical measure of the eigenvalues of the Hermitian matrix valued Ornstein-Uhlenbeck process weakly converges to the nonlinear McKean-Vlasov equation with quadratic external potential as tends to infinity. This also gave a dynamic proof of Wigner’s famous semi-circle law for Gaussian Unitary Ensemble. See also [2, 27].

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 be a parameter, be a continuous function, let be an -dimensional Brownian motion defined on a filtered probability space satisfying the usual conditions. Let . The generalized Dyson Brownian motion is an interacting -particle system with the logarithmic Coulomb interaction and with external potential , and is defined as the solution to the following SDEs

(1) |

with initial data . It is a SDE for -particles with a singular drift of the form due to the logarithmic Coulomb interaction, and an additional nonlinear drift due to non quadratic external potential. When and , it is the standard Dyson Brownian motion [20, 21]. When and , it has been studied by Chan [16], Rogers and Shi [47], Cépa and Lépingle [15], Fontbona [24, 25], Guionnet [27], Anderson, Guionnet and Zeitouni [2] and references therein. When , see [30].

By Itô’s calculus, is an interacting -particle system with the Hamiltonian

and the infinitesimal generator of is given by

where and .

Under suitable condition on , we prove that the SDEs for admit a unique strong solution with infinite lifetime. See Theorem 1.1 below. Let

Standard argument shows that the family is tight on , and the limit of any weakly convergent subsequence of , denoted by , is a weak solution to the following nonlinear McKean-Vlasov equation: for all ,

(2) |

In the case is absolutely continuous with respect to the Lebesgue measure on , integrating by parts, one can verify that the probability density satisfies the following nonlinear McKean-Vlasov equation (also called nonlinear Fokker-Planck equation in the literature)

(3) |

where

is the Hilbert transform of .

It seems that one can not find well-established result in the literature on the uniqueness of weak solutions to the above McKean-Vlasov equation with general external potential . By lack of this, one can not find established result in the literature on the Law of Large Numbers for the GDBM with non quadratic potentials. One of the main observations of this paper is to find (and prove) the fact that the McKean-Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space equipped with Otto’s infinite dimensional Riemannian structure, and to use the optimal transportation theory to prove the uniqueness of weak solutions to the McKean-Vlasov equation for general potentials with natural condition. This allows us to further derive the Law of Large Numbers for the empirical measures of the generalized Dyson Brownian motion.

Following Voiculescu [54]and Biane [5], for every , we introduce the Voiculescu free entropy as follows

By [29], it is well-known that if satisfies the growth condition

(4) |

then there exists a unique minimizer (called the equilibrium measure) of , denoted by

Moreover, satisfies the Euler-Lagrange equation

The relative free entropy is defined as follows

Following [54, 5], the relative free Fisher information is defined as follows

Note that

We now state the main results of this paper. Our first result establishes the existence and uniqueness of the strong solution to SDEs and the tightness of the associated empirical measure for a class of with reasonable condition.

###### Theorem 1.1

^{1}

^{1}1Under the condition for all , Rogers and Shi [47] proved the non-collision of the strong solution to , but they did not precisely state the condition which is need for the existence of solution. In [26], Graczyk and Malecki proved the existence and uniqueness of strong solution to SDE under the assumption that is global Lipschitz. The conditions in Theorem 1.1 require that satisfies the local monotonicity condition, i.e., , and one-side growth condition at infinity, i.e., . We would like to point out that the local monotonicity condition in Theorem 1.1 is weaker than the condition is local Lipschitz, and the one-side growth condition is also weaker than the condition is global Lipschitz .

Let be a function satisfying the growth condition and the following conditions

(i) For all , there is such that for all
with ,

(ii) There exists a constant such that

(5) |

Then, for all , and for any given , there exists a unique strong solution
taking values in
with infinite lifetime to SDEs with initial value
.

Moreover, suppose that as , and

Then, the family is tight in , and the limit of any weakly convergent subsequence of is a weak solution of the McKean-Vlasov equation .

Inspired by previous works due to Biane [5], Biane-Speicher [6], Carrillo-McCann-Villani [14] (see Theorem 3.1 below) and Sturm [48], we can prove the following result which might be already known by experts even though we cannot find the explicit statement in the literature.

###### Theorem 1.2

For all being a function satisfies the condition , the nonlinear McKean-Vlasov equation , i.e.,

is indeed the gradient flow of on the Wasserstein space .

In the optimal transportation theory, it is well known that if the free energy on the Wasserstein space is -convex, then the -Wasserstein distance between the solutions of the gradient flow with initial datas and satisfies . See [42, 43, 48, 49, 52, 53]. In view of this and Theorem 1.2, and using the Hessian calculation for nonlinear diffusions with interaction on the Wasserstein space as developed in [14, 48], we can prove the following result, which ensures the uniqueness of weak solutions to the McKean-Vlasov equation with general potential satisfying the condition .

###### Theorem 1.3

Suppose that is a function satisfying the same condition as in Theorem 1.1, and there exists a constant such that

Then the Voiculescu free entropy on the Wasserstein space is -convex, i.e., its Hessian on satisfies

Let be two solutions of the McKean-Vlasov equation with initial data , . Then for all , we have

In particular, the Cauchy problem of the McKean-Vlasov equation has a unique weak solution.

We would like to point out that Cśpa and Lépingle [15] proved the uniqueness of weak solution to the McKean-Vlasov equation with quadratic potential function with two constants, and , and Fontbana [25] proved the uniqueness of weak solution to the McKean-Vlasov equation with external potential such that , where is a constant and is a bounded function with bounded derivative. See also [24]. Theorem 1.3 establishes the uniqueness of weak solution to the McKean-Vlasov equation with more general external potentials satisfying the condition for some .

As a consequence of Theorem 1.1 and Theorem 1.3, we can derive the Law of Large Numbers for the empirical measures of the generalized Dyson Brownian motion.

###### Theorem 1.4

^{2}

^{2}2For with , the result in Theorem 1.4 also holds for

Suppose that weakly converges to . Let be a function satisfying the same condition as in Theorem 1.1 and for some constant . Then the empirical measure weakly converges to the unique solution of the McKean-Vlasov equation . Moreover, for all , we have

where the convergence is uniformly with respect to for all fixed .

The notion of propagation of chaos, which was introduced by M. Kac, plays a critical role in the study of the large limit of -particle systems. According to Sznitman-Tanaka’s theorem [50], for exchangeable systems, propagation of chaos is equivalent to the law of large numbers for the empirical measures of the system. In view of this and Theorem 1.4, we have the following result, which is a dynamic version of a result due to Johansson (Theorem 2 in [29]).

###### Theorem 1.5

Assume the conditions in Theorem 1.4 holds. Let be the -th moment measure for the random probability measure , that is, for any Borel sets ,

Then we have

for any continuous, bounded on .

By the ergodic theory of SDE, for a wide class of potentials , and for any fixed , it is known that converge to , as . On the other hand, the large -limit of , i.e., , satisfies the nonlinear Fokker-Planck equation . It is natural to ask the question whether converges to in the weak convergence topology or with respect to the -Wasserstein distance for general potentials . If this is true, then, with respect to the weak convergence on or the -Wasserstein topology on , the following diagram is commutative

In other words, we have

In the literature, Chan [16] and Rogers-Shi [47] proved that this is true for . See also [2, 27]. In particular, this gives a dynamic proof of Wigner’s semi-circle law for the Gaussian Unitary Ensemble. The following result provides some positive answers to this problem for -convex potentials.

###### Theorem 1.6

(i) Suppose that is -convex, i.e., . Then converges to with respect to the Wasserstein distance in , i.e.,

(ii) Suppose that is and there exists a constant such that

Then for all , we have

In particular, if is -uniform convex with , then converges to with the exponential rate in the -Wasserstein topology on .

(iii) Suppose that is a , convex and there exists a
constant such that

Then converges to with an exponential rate in the -Wasserstein topology on . More precisely, there exist two constants and such that

As a corollary of Theorem 1.6, for -convex potentials, we can give a dynamic proof of the well-known result due to Boutet de Monvel-Pastur-Shcherbina [11] and Johansson [29]. Their result says that, for satisfying the growth condition , the empirical measure weakly converges to the equilibrium measure , where , satisfies the following probability distribution

where is a parameter. We would like to mention that, for non-convex potentials , we do not know how to give a dynamic proof of the above result. We would like to mention a recent paper by Bourgade, Erdös, and Yau [10] in which the authors proved the bulk universality of the -ensembles with non-convex regular analytic potentials for any . Whether or not their idea of introducing a “convexified measure” can be used to extend the results in this paper to non-convex case, will be an interesting problem for study in future.

Finally, let us mention that, for and for real analytic function , we can prove that the generalized Dyson Brownian motion can be realized as the eigenvalues process of the real Hermitian matrix valued diffusion process defined by

where is the Hermitian matrix valued Brownian motion. Moreover, we can prove that converges in distribution to the free diffusion process , which was defined by Biane and Speicher [6]. This extends a famous result, due to Voiculescu [55, 56] and Biane [4], which states that the renormalized Hermitian Brownian motion converges in distribution to the free Brownian motion . See [35].

The rest of this paper is organized as follows. In Section , we prove Theorem 1.1. In Section , we prove Theorem 1.2, Theorem 1.3 and Theorem 1.4, Theorem 1.5. In Section , we prove Theorem 1.6. In Section , we discuss the case of double-well potential and raise some conjectures. Finally, let us mention that this paper is an update revised version of our previous paper entitled Generalized Dyson Brownian motion, McKean-Vlasov equation and eigenvalues of random matrices (arxiv.org/abs/1303.1240v1).

## 2 Proof of Theorem 1.1

The proof of Theorem 1.1 is adapted from classical argument coming back to McKean and exposed in [47, 15, 2].

Proof of existence and uniqueness of GDBM. First, for fixed , let if , and if . Since is uniformly Lipschitz and satisfies and , by Theorem 3.1.1 in [45], the following SDE for the truncated Dyson Brownian motion

(6) |

with for , has a unique strong solution. Let

Then is monotone increasing in and for all and .

Second, let on . To prove that is a global solution to SDE , we need only to prove does not explode, and and never collide for all , .

To prove that does not explode, let . By Itô’s formula, and by Levy’s characterization, we can introduce a new Brownian motion , such that

Let be the solution of

with Under the assumption , and using the comparison theorem of one dimensional SDEs, cf. [28], we can derive that

Moreover, by Ikeda and
Watanabe [28] (p. 235-237), the process never explodes. So
the process (and hence ) does not explode in
finite time ^{3}^{3}3 In [47], Rogers and Shi proved the
non-explosion of GDBM for satisfying ,
. .

To prove that and never collide for all , , let us introduce the Lyapunov function

where is the following local martingale

Fix and such that and for all , . Let be such that . Let , and , then for any fixed , and is a supermartingale. Let , we can prove

Letting , and tend to infinity, we can prove , where . This proves that does not collide.

Finally, by the continuity of the trajectory of , we have for all . The same argument as used in the proof of Theorem 12.1 in [27] proves the uniqueness of the weak solution to SDEs . The proof of Theorem 1.1 is completed.

Proof of tightness and identification of McKean-Vlasov limit

We follow the argument used in [47] to prove the tightness of . Let us pick functions which is dense in . Thus

We also pick a function with the properties

Taking test functions in the Schwartz class of smooth functions whose derivatives (up to second order) are rapidly decreasing, we may assume that

By Ethier and Kurtz [22] (p.107), to prove the tightness of , it is sufficient to prove that for each the sequence of continuous real-valued functions

is relatively compact. To this end, note that, by the first part of Theorem 1.1, there is non-collision and non-explosion for the particles for all . By Itô’s formula, we have

(7) | |||||

This yields

(8) | |||||

where

Note that, as is weakly convergent, is convergent. By the assumption that and are uniformly bounded (hence are uniformly bounded) , we can easily show that and converge to zero. Moreover, by the assumption that and are uniformly bounded, the Arzela-Ascoli theorem implies that and are relatively compact in . Thus the sequence is tight in . Tightness also follows for if we have

So let us suppose that the initial distribution have the property for some for all For given we could always find and to satisfy this and the other conditions, and this gives the tightness for also.

Finally, we identify the limit process of any weakly convergent subsequence of . Assuming that is a weakly convergent subsequence in . Then, for all , the Itô’s formula and the above argument show that satisfies the following equation

This proves that is a weak solution to the McKean-Vlasov equation . The proof of Theorem 1.1 is completed.

## 3 McKean-Vlasov equation: gradient flow and uniqueness

To characterize the McKean-Vlasov limit , we need only to use the test function , where , instead of using all test functions in the McKean-Vlasov equation . Let

be the Stieltjes transform of . Then satisfies the following equation

(9) |

In particular, in the case , since

the Stieltjes transform of satisfies the complex Burgers equation

(10) |

In [16, 47], Chan and Rogers-Shi proved that the complex Burgers equation (equivalently, the McKean-Vlasov equation with potential ) has a unique solution, and exists and coincides with the Stieltjes transform of the Wigner semi-circle law . This yields a dynamic proof of the Wigner’s theorem, i.e., weakly converges to .

However, for non quadratic potential , in cannot be expressed in terms of and its derivatives with respect to . Thus, one cannot derive an analogue of the complex Burgers equation for non quadratic potential , and we need to find a new approach to prove the uniqueness of the weak solutions of the Mckean-Vlasov equation for general potential . In this section, we use the theory of gradient flow on the Wasserstein space and the optimal transportation theory to study this problem.

### 3.1 Proof of Theorem 1.2

By Theorem 1.1, we have proved the existence of weak solution to the McKean-Vlasov equation . Assuming that the weak solution of the McKean-Vlasov equation is absolutely continuous with respect to the Lebesgue measure , we derive the existence of the weak solution of the nonlinear Fokker-Planck equation . Thus, to prove the law of large numbers for , we need only to show the uniqueness of the nonlinear Fokker-Planck equation . Note that, letting

then the nonlinear Fokker-Planck equation can be rewritten as follows

(11) |

To study the uniqueness and the longtime behavior of the nonlinear Fokker-Planck equation (i.e., ), we first recall Otto’s infinite dimensional Riemannian structure on the Wasserstein space . Fix , the tangent space of at is given by