# Recent advances on eigenvalues of matrix-valued stochastic processes

Jian Song  Jianfeng Yao  Wangjun Yuan  School of Mathematics, Shandong
University  Department of Statistics and Actuarial Science,The University of
Hong Kong  Department of Mathematics and Statistics, University of Ottawa

###### Abstract

Since the introduction of Dyson’s Brownian motion in early 1960’s, there have
been a lot of developments in the investigation of stochastic processes on the
space of Hermitian matrices. Their properties, especially, the properties of
their eigenvalues have been studied in great details. In particular, the
limiting behaviors of the eigenvalues are found when the dimension of the
matrix space tends to infinity, which connects with random matrix theory. This
survey reviews a selection of results on the eigenvalues of stochastic
processes from the literature of the past three decades. For most recent
variations of such processes, such as matrix-valued processes driven by
fractional Brownian motion or Brownian sheet, the eigenvalues of them are also
discussed in this survey. In the end, some open problems in the area are also
proposed.

###### keywords:

Brownian sheets , Dyson Brownian motion , Eigenvalue distribution , Fractional
Brownian motion , Matrix-valued process , Squared Bessel particle system ,
Wishart process.

###### MSC:

[2020] Primary 62H10, 60F05 , Secondary 60H15

††journal: Journal of Multivariate Analysis

##  1 Introduction

Stochastic processes with values in the space of symmetric matrices have been
attracting the attention for some years. Their introduction is commonly
attributed to the celebrated work [16] by F. J. Dyson.By that time, Gaussian
matrix ensembles were well known; the distribution of their eigenvalues has a
density function of the form

$G(x_{1},\ldots,x_{n})=C\exp\\{-\beta W\\},\quad
W=W(x_{1},\ldots,x_{n})=-\sum_{i<j}\ln|x_{i}-x_{j}|+\sum_{i}(x_{i}^{2}/2a^{2}),$
(1)

where $\beta,a>0$ are parameters, and $C=C(\beta,a)$ is a normalization
constant. The distribution (1) also appears in the Coulomb gas model: it is
the probability distribution of the positions of $n$ point charges which are
free to move on the real line $\mathbb{R}$ under the forces derived from the
potential energy $W$ and in a state of thermodynamic equilibrium at a
temperature $T=(k\beta)^{-1}$ ($k$ is the Boltzmann constant). Note that the
equation (1) is static and does not describe the evolution of the position of
the point charges before reaching the equilibrium. Dyson brought in the
Brownian motion to get a time-dependent model that describes the evolution of
the positions from an initial distribution $F_{0}$. The Brownian motion, also
called the time-dependent Coulomb gas has a simple structure, and the joint
density function $F(x_{1},\ldots,x_{n};t)$ of the positions of the $n$ point
charges at time $t>0$ is fully characterized as a solution to the Smoluchowski
equation

$\displaystyle c\frac{\partial F}{\partial
t}=\sum_{i}\left[\frac{1}{\beta}\frac{\partial^{2}F}{\partial
x_{i}^{2}}-\frac{\partial}{\partial x_{i}}\\{E(x_{i})F\\}\right],$ (2)

where $c$ is a constant and

$\displaystyle E(x_{i})=-\frac{\partial W}{\partial x_{i}}=\sum_{j\neq
i}\frac{1}{x_{j}-x_{i}}-\frac{x_{i}}{a^{2}}$

is an external electric force. In particular, $F(x_{1},\ldots,x_{n};t)$ tends
to the Coulomb gas distribution $G$ in (1) as $t\to\infty$.

This extension of Coulomb gas from the static equilibrium state to a dynamical
version also applies to the associated Gaussian matrix ensembles. More
precisely, Dyson introduced a stochastic process with values in the space of
symmetric matrices, the eigenvalues of which coincide with the dynamical
Coulomb gas model. Amazingly, this process is extremely simple: its elements
are independent Ornstein-Uhlenbeck processes on the underlying field! (The
underlying field is $\mathbb{R}$ for $\beta=1$, $\mathbb{C}$ for $\beta=2$,
and the quaternion field for $\beta=4$).

This deep connection between stochastic processes with values in the space of
symmetric (Hermitian) matrices and the induced dynamical system of its
eigenvalues, had been however quite ignored during a while afterwards. It was
revived in the papers [50, 6] where the idea of Dyson was extended to the
space of positive-definite matrices (ellipsoids). In the subsequent three
decades, the study of these stochastic processes and their associated
eigenvalue processes has been developed in much depth. Particularly, the
symmetric (Hermitian) matrix-valued processes have covered Brownian motion,
Ornstein-Uhlenbeck process and fractional Brownian motion.

Instead of considering the $N$ particles (eigenvalues) with $N$ fixed, the
limits of the empirical measures of particles when $N$ tends to infinity
(high-dimensional limits) were studied in several models. In particular, the
high-dimensional limit of the empirical measures of the Dyson’s Brownian
motion is the famous Wigner’s semicircle law, which provides a dynamical
version of Wigner’s Theorem for GOE and GUE (see, e.g., [1]). In this aspect,
the study of large particle systems is closely related to the random matrix
theory. Moreover, the equation satisfied by the limits of the empirical
measures of the Dyson’s Brownian motion is the so-called McKean-Vlasov
equation, which appears in the study of propagation of chaos for large systems
of interacting particles (see [3, 30, 62]).

Another motivation for studying high-dimensional limits of the empirical
measures of eigenvalues arises from free probability theory. By [4], the free
additive Brownian motion can be viewed as the high-dimensional limit of a
matrix Brownian motion with appropriate scaling. Moreover, [4, 5] developed
the stochastic calculus for free Brownian motion. Besides, the non-commutative
fractional Brownian motion was introduced in [51].

There is also a deep connection between matrix-valued stochastic processes and
multivariate statistical analysis. Here are a few applications of these
processes in recent statistical literature:

  1. 1.

Financial data analysis: multivariate volatility/co-volatility
(variance/covariance) between stock returns or interest rates from different
markets have been studied recently through Wishart processes, see [22, 23, 18,
13, 19, 20, 70].

  2. 2.

Machine learning: an important task in machine learning using kernel functions
is the determination of a suitable kernel matrix for a given data analysis
problem ([61]). Such determination is referred as the kernel matrix learning
problem. A kernel matrix is in fact a positive definite Gram-matrix of size
$N\times N$ where $N$, the sample size of the data, is usually large. An
innovative method for kernel learning is proposed by [72] where unknown kernel
matrix is modelled by a Wishart process prior. This approach has been followed
in [40, 44].

  3. 3.

Computer vision: real-time computer vision often involves tracking of objects
of interest. At each time $t$, a target is encoded into a $N$-dimensional
vector $a_{t}\in\mathbb{R}^{N}$ (feature vector). It is therefore clear that
measuring “distance” between these vectors, say $a_{t}$ and $a_{t+dt}$ at two
consecutive time spots $t$ and $t+dt$, is of crucial importance for object
tracking. Because the standard Euclidean distance $\|a_{t+dt}-a_{t}\|^{2}$ is
rarely optimal, it is more satisfactory to identify a better metric of the
form $(a_{t+dt}-a_{t})^{\intercal}M_{t}(a_{t+dt}-a_{t})$ using a suitable
positive definite matrix $M_{t}$. An innovative model where the process
$M_{t}$ follows a Wishart process is proposed in [42].

This survey reviews a selection of results from the last three decades. In
Section 2, we provide a study of Dyson’s Brownian motion with full details.
This includes a modern derivation of the process using Itô calculus. A limit
for the processes of empirical eigenvalue measures is derived when the number
of eigenvalues, or electric charges, tends to infinity. Besides, a limiting
Gaussian process is derived in order to characterize the fluctuation of the
empirical eigenvalue measures around their limit. In Section 3, we discuss two
specific classes of stochastic processes with values in the space of positive-
definite matrices, that is, Brownian motions of ellipsoids and Wishart
processes. In Section 4, a more general form of stochastic processes on the
space of Hermitian matrices is studied, and a link is also made with some
familiar systems of interacting particles. The following Sections 5 and 6
concern extensions of Dyson’s Brownian motion in two different directions. The
first extension replaces the Brownian motions in the matrix by fractional
Brownian motions, and the second one by Brownian sheets. Finally in Section 7,
we conclude with a discussion on open problems related to the results
introduced in the preceding sections.

##  2 Dyson’s Brownian motion

In this section, we mainly focus on the Dyson’s Brownian motion. We discuss
the system of SDEs satisfied by Dyson’s Brownian motion in Section 2.1 and the
limiting behaviors of the eigenvalue empirical measure process in Section 2.2.

###  2.1 Finite-dimensional results

Throughout the survey, we denote the complex imaginary by $\iota=\sqrt{-1}$.

######  Definition 1.

Let $\\{B_{i,j}(t),\tilde{B}_{i,j}(t),1\leq i\leq j\leq N\\}$ be a family of
i.i.d. real valued standard Brownian motions. Let
$H^{N,\beta}(t)=\left(H^{N,\beta}_{k,l}(t)\right)_{1\leq k\leq l\leq N}$ be a
real symmetric ($\beta=1$) or complex Hermitian ($\beta=2$) $N\times N$
matrix-valued process with entries

$\displaystyle H^{N,\beta}_{k,l}(t)=\dfrac{1}{\sqrt{\beta
N}}\left(B_{k,l}(t)+\iota(\beta-1)\tilde{B}_{k,l}(t)\right){\large
1}_{\\{k<l\\}}+\dfrac{\sqrt{2}}{\sqrt{\beta N}}B_{l,l}(t){\large
1}_{\\{k=l\\}}.$

Then $H^{N,1}(t)$ is a real symmetric matrix Brownian motion and $H^{N,2}(t)$
is a complex Hermitian matrix Brownian motion.

The following results state that the eigenvalue processes of real symmetric or
complex Hermitian matrix Brownian motion never collide almost surely and are
characterized by a system of stochastic differential equations (SDEs).

######  Theorem 1 ([1], Theorem 4.3.2).

Let $X^{N,\beta}(0)$ be a real symmetric ($\beta=1$) or complex Hermitian
($\beta=2$) $N\times N$ deterministic matrix and let
$X^{N,\beta}(t)=X^{N,\beta}(0)+H^{N,\beta}(t)$. Let
$\lambda_{1}^{N,\beta}(t)\geq\lambda_{2}^{N,\beta}(t)\geq\cdots\geq\lambda_{N}^{N,\beta}(t)$
be the ordered eigenvalue processes of $X^{N,\beta}(t)$. Denote the first
collision time of the eigenvalue processes by

$\displaystyle\tau_{N,\beta}=\inf\left\\{t>0:\exists\ i\neq
j,~{}\lambda_{i}^{N,\beta}(t)=\lambda_{j}^{N,\beta}(t)\right\\}.$ (3)

Then $\mathbb{P}\left(\tau_{N,\beta}=+\infty\right)=1.$ Furthermore, the
ordered eigenvalue processes
$\lambda_{1}^{N,\beta}(t)>\cdots>\lambda_{N}^{N,\beta}(t)$ are the unique
solution to the following system of SDEs:

$\displaystyle d\lambda_{i}^{N,\beta}(t)=\dfrac{\sqrt{2}}{\sqrt{\beta
N}}dW_{i}(t)+\dfrac{1}{N}\sum_{j:j\neq
i}\dfrac{dt}{\lambda_{i}^{N,\beta}(t)-\lambda_{j}^{N,\beta}(t)},~{}i\in\\{1,\ldots,N\\}.$
(4)

Here, $\\{W_{1}(t),\ldots,W_{N}(t)\\}$ is a family of independent standard
Brownian motions.

###### Proof:.

The proof is motivated by [24, Theorem 3,5] and [1, Lemma 4.3.3]. We only
consider the real symmetric case $\beta=1$. The complex Hermitian case
$\beta=2$ is similar and thus is omitted. Since the dimension $N$ is fixed, we
may omit both $N$ and $\beta$ on subscript and superscript without ambiguity.
For simplicity, we only give a proof under the condition
$\lambda_{1}(0)>\cdots>\lambda_{N}(0)$. For the case that $X(0)$ has collision
eigenvalues, we refer the interested readers to [1, Page 257]. We divide the
proof into three steps.

Step 1: Derivation of the system of SDEs for eigenvalue processes before the
first collision time by Itô calculus and martingale theory.

We may use the Stratonovich differential notation, which can be founded in,
for example, [29, Chapter III]. For two $N\times N$ matrices $X$ and $Y$, we
have $X\circ dY=XdY+\frac{1}{2}dXdY$, where $XdY$ is the Itô differential,
$X\circ dY$ is the Stratonovich differential and $dXdY=d\langle X,Y\rangle$.
By matrix multiplication, for three $N\times N$ matrices $X,Y$ and $Z$, one
can verify that

$\displaystyle dX\circ(YZ)=(dX\circ Y)\circ
Z=dXYZ+\dfrac{1}{2}\left(dXdYZ+dXYdZ\right),$ $\displaystyle(X\circ dY)\circ
Z=X\circ(dY\circ Z)=XdYZ+\dfrac{1}{2}\left(dXdYZ+XdYdZ\right),$
$\displaystyle(X\circ dY)^{\intercal}=dY^{\intercal}\circ X^{\intercal}.$

Moreover, by Itô formula and matrix multiplication, one can verify that

$\displaystyle d(XYZ)=dX\circ YZ+X\circ dY\circ Z+XY\circ dZ.$ (5)

For a real symmetric matrix process $X(t)$, consider its spectral
decomposition $X(t)=P(t)D(t)P(t)^{\intercal}$, where $D(t)$ is a diagonal
matrix of eigenvalues of $X(t)$ ordered decreasingly, and $P(t)$ is an
orthogonal matrix of eigenvectors of $X(t)$. According to [50], the matrices
$D(t),P(t)$ can be chosen as smooth functions of $X(t)$ for
$t<\tau_{N,\beta}$. Let $Q(t)$ be the matrix-valued processes satisfying

$\displaystyle dQ(t)=P(t)^{-1}\circ dP(t)=P(t)^{\intercal}\circ dP(t).$

The process $Q(t)$ is known as the stochastic logarithm of $P(t)$. By the Itô
formula (5), we have the following identity

$\displaystyle 0=dI_{N}=d(P(t)^{\intercal}P(t))=dQ(t)+dQ(t)^{\intercal}.$ (6)

Applying the Itô formula (5) to the spectral decomposition of $X(t)$ and using
(6), we have

$\displaystyle dD(t)$ $\displaystyle=dP(t)^{\intercal}\circ
X(t)P(t)+P(t)^{\intercal}\circ dX(t)\circ P(t)+P(t)^{\intercal}X(t)\circ
dP(t)$ $\displaystyle=dP(t)^{\intercal}\circ P(t)D(t)+P(t)^{\intercal}\circ
dX(t)\circ P(t)+D(t)P(t)^{\intercal}\circ dP(t)$ $\displaystyle=-dQ(t)\circ
D(t)+P(t)^{\intercal}\circ dX(t)\circ P(t)+D(t)\circ dQ(t).$ (7)

By considering the non-diagonal entries of (2.1), we have

$\displaystyle dQ_{ij}(t)=-\left(P(t)^{\intercal}\circ dX(t)\circ
P(t)\right)_{ij}\circ\dfrac{1}{\lambda_{i}(t)-\lambda_{j}(t)},~{}i\neq j.$ (8)

On the other hand, the diagonal entries of (2.1) can be written as

$\displaystyle d\lambda_{i}(t)=$ $\displaystyle\left(P(t)^{\intercal}\circ
dX(t)\circ P(t)\right)_{ii}$ $\displaystyle=$
$\displaystyle\left(P(t)^{\intercal}dX(t)P(t)\right)_{ii}+\dfrac{1}{2}\Big{(}dP(t)^{\intercal}dX(t)P(t)+P(t)^{\intercal}dX(t)dP(t)\Big{)}_{ii}.$
(9)

Recalling Definition 1, one can see that $\\{\left(P(t)^{\intercal}\circ
dX(t)\circ P(t)\right)_{ii}\\}_{1\leq i\leq N}$ is a family of local
martingales with quadratic covariation

$\displaystyle\left(P(t)^{\intercal}\circ dX(t)\circ
P(t)\right)_{ii}\left(P(t)^{\intercal}\circ dX(t)\circ
P(t)\right)_{jj}=\left(P(t)^{\intercal}dX(t)P(t)\right)_{ii}\left(P(t)^{\intercal}dX(t)P(t)\right)_{jj}$
$\displaystyle=$
$\displaystyle\sum_{k,l=1}^{N}P_{ki}(t)P_{li}(t)dX_{kl}(t)\sum_{k^{\prime},l^{\prime}=1}^{N}P_{k^{\prime}j}(t)P_{l^{\prime}j}(t)dX_{k^{\prime}l^{\prime}}(t)=\sum_{k,l,k^{\prime},l^{\prime}=1}^{N}P_{ki}(t)P_{li}(t)P_{k^{\prime}j}(t)P_{l^{\prime}j}(t)\left(\text{1}_{[k=k^{\prime}]}\text{1}_{[l=l^{\prime}]}+\text{1}_{[k=l^{\prime}]}\text{1}_{[l=k^{\prime}]}\right)\dfrac{dt}{N}$
$\displaystyle=$
$\displaystyle\dfrac{2dt}{N}\left(\sum_{k=1}^{N}P_{ki}(t)P_{kj}(t)\right)^{2}=\dfrac{2}{N}\text{1}_{[i=j]}dt,$

where we use the orthogonality of the columns of the matrix $P(t)$. Thus, by
Knight’s theorem, there exists a family of independent standard
$1$-dimensional Brownian motions $\\{W_{1}(t),\ldots,W_{N}(t)\\}$, such that

$\displaystyle\left(P(t)^{\intercal}\circ dX(t)\circ
P(t)\right)_{ii}=\dfrac{\sqrt{2}}{\sqrt{N}}dW_{i}(t).$ (10)

Note that $X(t)$ is symmetric, by (6) and (8), we have

$\displaystyle\dfrac{1}{2}\Big{(}dP(t)^{\intercal}dX(t)P(t)+P(t)^{\intercal}dX(t)dP(t)\Big{)}_{ii}=\Big{(}dP(t)^{\intercal}dX(t)P(t)\Big{)}_{ii}$
$\displaystyle=~{}$
$\displaystyle\Big{(}dP(t)^{\intercal}P(t)P(t)^{\intercal}dX(t)P(t)\Big{)}_{ii}=\Big{(}dQ(t)^{\intercal}\left(P(t)^{\intercal}\circ
dX(t)\circ P(t)\right)\Big{)}_{ii}$ $\displaystyle=~{}$
$\displaystyle-\sum_{j=1}^{N}dQ_{ij}(t)\left(P(t)^{\intercal}\circ dX(t)\circ
P(t)\right)_{ji}=\sum_{j:j\neq
i}\dfrac{\left(P(t)^{\intercal}dX(t)P(t)\right)_{ij}\left(P(t)^{\intercal}dX(t)P(t)\right)_{ji}}{\lambda_{i}(t)-\lambda_{j}(t)}.$
(11)

For $i\neq j$, we have

$\displaystyle\left(P(t)^{\intercal}dX(t)P(t)\right)_{ij}\left(P(t)^{\intercal}dX(t)P(t)\right)_{ji}=\sum_{k,l=1}^{N}P_{ki}(t)P_{lj}(t)dX_{kl}(t)\sum_{k^{\prime},l^{\prime}=1}^{N}P_{k^{\prime}j}(t)P_{l^{\prime}i}(t)dX_{k^{\prime}l^{\prime}}(t)$
$\displaystyle=~{}$
$\displaystyle\sum_{k,l,k^{\prime},l^{\prime}=1}^{N}P_{ki}(t)P_{lj}(t)P_{k^{\prime}j}(t)P_{l^{\prime}i}(t)\left(\text{1}_{[k=k^{\prime}]}\text{1}_{[l=l^{\prime}]}+\text{1}_{[k=l^{\prime}]}\text{1}_{[l=k^{\prime}]}\right)\dfrac{dt}{N}$
$\displaystyle=~{}$
$\displaystyle\dfrac{dt}{N}\left(\sum_{k=1}^{N}P_{ki}(t)P_{kj}(t)\right)^{2}+\dfrac{dt}{N}\left(\sum_{k=1}^{N}P_{ki}(t)^{2}\right)\left(\sum_{l=1}^{N}P_{lj}(t)^{2}\right)=\dfrac{dt}{N}.$
(12)

Substituting (2.1) to (2.1), we have

$\displaystyle\dfrac{1}{2}\Big{(}dP(t)^{\intercal}dX(t)P(t)+P(t)^{\intercal}dX(t)dP(t)\Big{)}_{ii}=\dfrac{1}{N}\sum_{j:j\neq
i}\dfrac{dt}{\lambda_{i}(t)-\lambda_{j}(t)}.$ (13)

Therefore, (4) follows from (2.1), (10) and (13).

Step 2: We prove that the system of SDEs (4) has a unique strong solution
before its first collision time by approximating the singular drift with
regular functions. For the existence and uniqueness of SDE, we refer to [33].

For $R>0$, define

$\displaystyle\psi_{R}(x)=x^{-1}{\large 1}_{\\{|x|\geq
R^{-1}\\}}+R^{2}x{\large 1}_{\\{|x|<R^{-1}\\}}.$ (14)

One can easily check that $\psi_{R}(x)$ is continuous on $\mathbb{R}$
satisfying $|\psi_{R}(x)|\leq(1+R^{2})(1+|x|^{2}).$ Consider the following
system of SDEs

$\displaystyle
d\lambda_{i}^{R}(t)=\dfrac{\sqrt{2}}{\sqrt{N}}dW_{i}(t)+\dfrac{1}{N}\sum_{j:j\neq
i}\psi_{R}\left(\lambda_{i}^{R}(t)-\lambda_{j}^{R}(t)\right)dt,~{}i\in\\{1,\ldots,N\\},$
(15)

with initial condition $\lambda_{i}^{R}(0)=\lambda_{i}(0)$ for $1\leq i\leq
N$. Noting that for each $R>0$, the coefficient functions in (15) are global
Lipschitz and of linear growth, the existence of the strong solution of (15)
follows from [33, Theorem 2.9], and moreover, by [33, Theorem 2.5], we also
have the strong uniqueness.

For $R>0$, let

$\displaystyle\tau(R)=\inf\left\\{t>0:\min_{i\neq
j}\left|\lambda_{i}^{R}(t)-\lambda_{j}^{R}(t)\right|<R^{-1}\right\\}.$

Then $\tau(R)$ is a stopping time which is increasing with respect to $R$. We
denote $\tau(+\infty)=\lim_{R\to+\infty}\tau(R)$, which may be $+\infty$. Let
$R_{0}$ be a positive number such that $R_{0}^{-1}=\min_{i\neq
j}|\lambda_{i}(0)-\lambda_{j}(0)|$. For $R_{1}>R_{2}>R_{0}$, we have the
following observation

$\displaystyle\lambda_{i}^{R_{1}}(t)=\lambda_{i}^{R_{2}}(t),~{}\forall
t\leq\tau(R_{2}),~{}\forall 1\leq i\leq N.$

Thus, for $t<\tau(+\infty)$, we can define the processes
$\lambda_{i}^{\infty}(t)$ in a consistent way by

$\displaystyle\lambda_{i}^{\infty}(t)=\lambda_{i}^{R}(t),~{}\text{if}~{}t<\tau(R)$

for $1\leq i\leq N$. Then, recalling the definition (14) of $\psi_{R}$,
$(\lambda_{1}^{\infty}(t),\ldots,\lambda_{N}^{\infty}(t))$ solves (4) for
$t<\tau(+\infty)$. Note that for any strong solution of (4), it solves (15)
before the time when the least distance of its entries reaches $R^{-1}$ for
$R>R_{0}$. Thus, the strong uniqueness of (4) follows from the strong
uniqueness of (15) by letting $R\to\infty$.

Step 3: We prove the almost sure non-collision of the eigenvalue processes by
McKean’s argument ([46, Proposition 4.3], see also [47]).

From Step 1 and Step 2, we can see that the eigenvalue processes of
$H^{N,\beta}(t)$ is the unique strong solution to (4), and thus
$\tau_{N,\beta}$ given by (3) is also the collision time for the strong
solution to (4). For $t<\tau_{N,\beta}$, define

$\displaystyle U(t)=\sum_{i<j}\ln|\lambda_{i}(t)-\lambda_{j}(t)|,$ (16)

then by (4) and Itô formula, noting that
$d\langle\lambda_{i}(t),\lambda_{j}(t)\rangle=0$ for $i\neq j$, we have

$\displaystyle dU(t)=$ $\displaystyle\sum_{i\neq
j}\dfrac{d\lambda_{i}(t)}{\lambda_{i}(t)-\lambda_{j}(t)}-\dfrac{1}{2}\sum_{i\neq
j}\dfrac{d\langle\lambda_{i}(t)\rangle}{\left(\lambda_{i}(t)-\lambda_{j}(t)\right)^{2}}$
(17) $\displaystyle=$ $\displaystyle\dfrac{\sqrt{2}}{\sqrt{N}}\sum_{i\neq
j}\dfrac{dW_{i}}{\lambda_{i}(t)-\lambda_{j}(t)}+\dfrac{1}{N}\sum_{i\neq
j}\sum_{l:l\neq
i}\dfrac{dt}{\left(\lambda_{i}(t)-\lambda_{j}(t)\right)\left(\lambda_{i}(t)-\lambda_{l}(t)\right)}-\dfrac{1}{N}\sum_{i\neq
j}\dfrac{dt}{\left(\lambda_{i}(t)-\lambda_{j}(t)\right)^{2}}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{2}}{\sqrt{N}}\sum_{i\neq
j}\dfrac{dW_{i}}{\lambda_{i}(t)-\lambda_{j}(t)}+\dfrac{1}{N}\sum_{i\neq j\neq
l\neq
i}\dfrac{dt}{\left(\lambda_{i}(t)-\lambda_{j}(t)\right)\left(\lambda_{i}(t)-\lambda_{l}(t)\right)}$
$\displaystyle=$ $\displaystyle\dfrac{\sqrt{2}}{\sqrt{N}}\sum_{i\neq
j}\dfrac{dW_{i}}{\lambda_{i}(t)-\lambda_{j}(t)}+\dfrac{1}{N}\sum_{i\neq j\neq
l\neq
i}\dfrac{\lambda_{l}(t)-\lambda_{j}(t)}{\left(\lambda_{i}(t)-\lambda_{j}(t)\right)\left(\lambda_{j}(t)-\lambda_{l}(t)\right)\left(\lambda_{l}(t)-\lambda_{i}(t)\right)}dt$
$\displaystyle=$ $\displaystyle\dfrac{\sqrt{2}}{\sqrt{N}}\sum_{i\neq
j}\dfrac{dW_{i}}{\lambda_{i}(t)-\lambda_{j}(t)}+\dfrac{1}{3N}\sum_{i\neq j\neq
l\neq
i}\dfrac{\left(\lambda_{l}(t)-\lambda_{j}(t)\right)+\left(\lambda_{i}(t)-\lambda_{l}(t)\right)+\left(\lambda_{j}(t)-\lambda_{i}(t)\right)}{\left(\lambda_{i}(t)-\lambda_{j}(t)\right)\left(\lambda_{j}(t)-\lambda_{l}(t)\right)\left(\lambda_{l}(t)-\lambda_{i}(t)\right)}dt=\dfrac{\sqrt{2}}{\sqrt{N}}\sum_{i\neq
j}\dfrac{dW_{i}}{\lambda_{i}(t)-\lambda_{j}(t)}.$

Here, we use the symmetry to change the summation index in the fifth equality.
Therefore, by Lemma 1 below, we have $\tau_{N,\beta}=+\infty$ almost surely.
The proof of Theorem 1 is complete. ∎

The following lemma is used in the Step 3 of the proof and is known as the
McKean’s argument, which can be found in [46].

######  Lemma 1 (McKean’s argument, [46], Proposition 4.3).

Let $Z=\\{Z(t);0\leq t<\infty\\}$ be an adapted $\mathbb{R}_{+}$-valued
stochastic process that is right-continuous with finite left-hand limits
(RCLL) on a stochastic interval $[0,\tau_{0})$ with $Z_{0}>0$, where

$\tau_{0}=\inf\\{s>0:Z_{s-}=0\\}.$

Suppose that there exists a continuous function $h$ satisfying the following:

  1. (i)

For all $t\in[0,\tau_{0})$, we have $h(Z(t))=h(Z(0))+M(t)+P(t)$, where $M$ is
a continuous local martingale on $[0,\tau_{0})$ with $M(0)=0$, and $P$ is an
adapted RCLL process on $[0,\tau_{0})$ such that almost surely and for each
$T>0$, $\displaystyle\inf_{t\in[0,\tau_{0}\wedge T)}P(t)>-\infty.$

  2. (ii)

$\lim_{z\downarrow 0}h(z)=-\infty$.

Then $\tau_{0}=\infty$ almost surely.

######  Remark 1.

The argument for non-collision in [1] is different. For $M>0$, it is shown
that the first time for $U(t)$ with $\lambda_{i}(t)$ replaced by the
$\lambda_{i}^{R}(t)$ to exceed $M$ is greater than any positive number almost
surely via Markov inequality and Borel–Cantelli Lemma.

######  Remark 2.

The unique solution to (4) is known as Dyson Brownian motion.

######  Remark 3.

The process given in (4) with general $\beta\in(0,\infty)$ is known as
$\beta$-Dyson Brownian motion. By the same argument used in Step 2, one can
show that (4) with general $\beta\in(0,\infty)$ has a unique strong solution
before the first collision time. Moreover, applying Itô formula to $U(t)$
given in (16), we have

$\displaystyle dU(t)=\dfrac{\sqrt{2}}{\sqrt{\beta N}}\sum_{i\neq
j}\dfrac{dW_{i}}{\lambda_{i}^{N,\beta}(t)-\lambda_{j}^{N,\beta}(t)}+\left(1-\dfrac{1}{\beta}\right)\dfrac{1}{N}\sum_{i\neq
j}\dfrac{dt}{\big{(}\lambda_{i}^{N,\beta}(t)-\lambda_{j}(t)^{N,\beta}\big{)}^{2}}.$

Then the non-collision of the system of particles
$\lambda_{1}^{N,\beta}(t)\geq\cdots\geq\lambda_{N}^{N,\beta}(t)$ follows from
McKean’s argument for the case $\beta\geq 1$. It is well known that the
$\beta$-Dyson Brownian motion has collisions for $\beta\in(0,1)$ (see, e.g.,
[24, Remark 3]).

Real symmetric matrix whose entries are i.i.d. Ornstein-Uhlenbeck processes
(real symmetric matrix OU process) was considered in [12]. Let $X^{N}(t)$ be a
symmetric $N\times N$ matrix-valued process that solves the following matrix
SDE

$\displaystyle
dX^{N}(t)=\dfrac{1}{2\sqrt{N}}\left(dB(t)+dB(t)^{\intercal}\right)-\dfrac{1}{2}X^{N}(t)dt,$
(18)

where $B(t)$ is a $N\times N$ matrix Brownian motion. Then the entries
$\\{X^{N}_{i,j}(t)\\}_{1\leq i\leq j\leq N}$ are independent Ornstein-
Uhlenbeck processes with invariant distribution $N(0,(1+\delta_{ij})/(2N))$.
By Itô calculus and martingale theory, [12] derived the following system of
SDEs for the eigenvalue processes $\\{\lambda_{i}^{N}(t)\\}_{1\leq i\leq N}$
of $X^{N}(t)$ in (18)

$\displaystyle
d\lambda_{i}^{N}(t)=\dfrac{1}{\sqrt{N}}dB_{i}(t)+\left(-\dfrac{1}{2}\lambda_{i}^{N}(t)+\dfrac{1}{2N}\sum_{j:j\neq
i}\dfrac{1}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,~{}1\leq i\leq N.$
(19)

By assuming the non-collision of the initial state
$\lambda_{1}^{N}(0)>\cdots>\lambda_{N}^{N}(0)$, the non-collision of the
eigenvalue processes was also established in [12] by an argument similar to
the one used in the proof of Theorem 1.

###  2.2 High-dimensional limits

Let $\mathcal{P}(\mathbb{R})$ be the space of probability measures on
$\mathbb{R}$ equipped with the weak topology and corresponding metric
$d_{\mathcal{P}(\mathbb{R})}$. For $T>0$, let
$C([0,T],\mathcal{P}(\mathbb{R}))$ be the space of continuous processes with
values in $\mathcal{P}(\mathbb{R})$. Then the space
$C([0,T],\mathcal{P}(\mathbb{R}))$ endowed with the metric

$\displaystyle
d_{C([0,T],\mathcal{P}(\mathbb{R}))}\left(\mu^{(1)},\mu^{(2)}\right)=\sup_{t\in[0,T]}d_{\mathcal{P}(\mathbb{R})}\left(\mu^{(1)}(t),\mu^{(2)}(t)\right),$

is complete. For a test function $f(x)$ and a measure $\mu(dx)$ on
$\mathbb{R}$, we write $\langle f,\mu\rangle=\int_{\mathbb{R}}f(x)\mu(dx).$

Recall the definition of $X^{N,\beta}(t)$ in Theorem 1. Let $L_{N}^{\beta}(t)$
be the empirical measure of the eigenvalue processes
$\\{\lambda_{i}^{N,\beta}(t)\\}_{1\leq i\leq N}$ of $X^{N,\beta}(t)$, that is

$\displaystyle
L_{N}^{\beta}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}^{N,\beta}(t)}(dx).$
(20)

In connection with the theory of random matrices, it is of interest to
investigate possible limits of these empirical measures
$\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ when $N$ grows to
infinity.

Such high-dimensional limits are known in the literature only for some simple
cases. An early result for eigenvalue empirical measure processes can be found
in [12]: the exponential tightness of the sequence of corresponding eigenvalue
empirical measure processes was established, which implies the almost sure
convergence of the sequence. The equation satisfied by the limiting measure-
valued process was also obtained, which is known as McKean-Vlasov equation.
Moreover, [12] proved that the semi-circle law is the only equilibrium point
of the equation (with finite moments of all orders).

The high-dimensional limit results were later generalized in [59] to the
following system of symmetric matrix SDE

$\displaystyle
dX^{N}(t)=\sqrt{\dfrac{\alpha}{2N}}\left(dB(t)+dB(t)^{\intercal}\right)-\theta
X^{N}(t)dt,$ (21)

where $B(t)$ is a $N\times N$ matrix Brownian motion. Note that if we choose
$\alpha=1$ and $\theta=0$, then the $X^{N}(t)$ in (21) is the real symmetric
matrix Brownian motion appeared in Theorem 1. The real symmetric matrix OU
processes in (18) corresponds to the case $\alpha=1/2$ and $\theta=1/2$. The
eigenvalue processes $\\{\lambda_{i}^{N}(t)\\}_{1\leq i\leq N}$ of $X^{N}(t)$
in (21) are called the interacting Brownian particles in [59] and satisfy the
following system of SDEs

$\displaystyle
d\lambda_{i}^{N}(t)=\sqrt{\dfrac{2\alpha}{N}}dB_{i}(t)+\left(-\theta\lambda_{i}^{N}(t)+\dfrac{\alpha}{N}\sum_{j:j\neq
i}\dfrac{1}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,\quad 1\leq i\leq
N,~{}~{}t\geq 0.$ (22)

In [59], the non-collision and non-explosion of the particles (22) was
established assuming initial state
$\lambda_{1}^{N}(0)>\cdots>\lambda_{N}^{N}(0)$. Moreover, [59, Theorem 1]
proved the weak convergence in law of the sequence of eigenvalue empirical
measure processes by Itô calculus and a tightness argument that is similar to
Theorem 3. The equation that characterizes the limiting measure valued process
was also derived.

The family of eigenvalue processes given in (22) was further generalized in
[11]. More precisely, for some Lipschitz functions $b_{N}$, $\sigma_{N}$ and
positive constant $\gamma_{N}$, [11] proved that the following particle system

$\displaystyle
dx_{i}^{N}(t)=\sigma_{N}(x_{i}^{N}(t))dB_{i}(t)+\left(b_{N}(x_{i}^{N}(t))+\sum_{j:j\neq
i}\dfrac{\gamma_{N}}{x_{i}^{N}(t)-x_{j}^{N}(t)}\right)dt,\quad 1\leq i\leq
N,~{}~{}t\geq 0,$ (23)

has a unique strong solution for all the time, even with collision. For the
case $\gamma_{N}=2\gamma/N$, [11, Theorem 4.2] established the weak
convergence in law of the sequence of eigenvalue empirical measure processes
and derived the equation for all possible limits. In [11, Theorem 5.1], the
uniqueness of this equation was obtained if $b_{N}(x)$ is linear and
$\sigma_{N}(x)=\sigma_{N}>0$. The non-collision property was also established
in [11, Proposition 4.1] under the assumptions that the particles are distinct
at $t=0$, $b_{N}(x)$ is linear, and
$\sigma_{N}(x)=\sigma_{N}\in[0,\sqrt{2\gamma_{N}}]$. However, it is worth
pointing out that the high-dimensional results does not require the non-
collision of the particles.

Another generalization of the real symmetric matrix Brownian motion in Theorem
1 and real symmetric matrix OU process in (18) was introduced in [43] as the
solution of the following matrix SDE

$\displaystyle
dX^{N}(t)=\dfrac{1}{\sqrt{2N}}\left(dB(t)+dB(t)^{\intercal}\right)-\dfrac{1}{2}V^{\prime}(X^{N}(t))dt,$
(24)

whose ordered eigenvalue processes $\\{\lambda_{i}^{N}(t)\\}_{1\leq i\leq N}$
satisfy

$\displaystyle
d\lambda_{i}^{N}(t)=\sqrt{\dfrac{2}{N}}dB_{i}(t)+\left(-\dfrac{1}{2}V^{\prime}\left(\lambda_{i}^{N}(t)\right)+\dfrac{1}{N}\sum_{j:j\neq
i}\dfrac{1}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,\quad 1\leq i\leq
N,~{}t\geq 0.$ (25)

Here, $V$ is an external potential functions in $C^{1}(\mathbb{R})$ satisfying
certain convexity conditions. The weak convergence in law of the sequence of
eigenvalue empirical measure processes and the equation for the limiting
process were obtained in [43, Theorem 1.1].

For the real symmetric or complex Hermitian matrix Brownian motion
$X^{N,\beta}$ defined in Theorem 1 with null initial value $X^{N,\beta}(0)=0$,
the high-dimensional limits was investigated in [9] by studying large
deviation bounds. The exponential tightness of the sequence
$\\{L_{N}^{\beta}(t)\\}_{N\in\mathbb{N}}$ was established. In [9, Corollary
1.2], the almost sure convergence of the sequence
$\\{L_{N}^{\beta}(t)\\}_{N\in\mathbb{N}}$ was obtained and the equation for
the limit was derived. Moreover, the limit was proved to be the semi-circular
law. The complex case was also studied in [57, Proposition 3.1] where the
convergence in probability was obtained. We present [1, Proposition 4.3.10]
below, where the high-dimensional limit of the sequence
$\\{L_{N}^{\beta}(t)\\}_{N\in\mathbb{N}}$ was recovered without assuming the
null initial condition.

######  Theorem 2.

Let $T>0$ be a fixed number. Suppose that there exists a positive function
$\varphi\in C^{2}(\mathbb{R})$ with bounded first and second derivatives and
satisfying $\lim_{|x|\to\infty}\varphi(x)=+\infty$, such that $\displaystyle
C_{0}:=\sup_{N\in\mathbb{N}}\langle\varphi,L_{N}^{\beta}(0)\rangle<\infty.$
Assume that $L_{N}^{\beta}(0)$ converges weakly as $N$ tends to infinity
towards a probability measure $\mu_{0}$.

Then the sequence $\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$
converges almost surely in $C([0,T],\mathcal{P}(\mathbb{R}))$. Its limit $\mu$
is characterized by the following equation: for any $f\in
C_{b}^{2}(\mathbb{R})$,

$\displaystyle\langle f,\mu_{t}\rangle=\langle
f,\mu_{0}\rangle+\dfrac{1}{2}\int_{0}^{t}\iint_{\mathbb{R}^{2}}\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}\mu_{s}(dx)\mu_{s}(dy)ds,\quad\forall
t\in[0,T].$ (26)

###### Proof:.

The idea of the proof comes from [1, Proposition 4.3.10] (see also [65]). We
divide the proof into four steps.

Step 1: (Computation of $\langle f,L_{N}^{\beta}(t)\rangle$ by Itô calculus.)
By the definition (20) of $L_{N}^{\beta}(t)$, for $f\in C^{2}(\mathbb{R})$,

$\displaystyle\langle f,L_{N}^{\beta}(t)\rangle=\int
f(x)L_{N}^{\beta}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\int
f(x)\delta_{\lambda_{i}^{N,\beta}(t)}(dx)=\dfrac{1}{N}\sum_{i=1}^{N}f(\lambda_{i}^{N,\beta}(t)).$

By Itô’s formula and (4),

$\displaystyle f(\lambda_{i}^{N,\beta}(t))$
$\displaystyle=f(\lambda_{i}^{N,\beta}(0))+\int_{0}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(s))d\lambda_{i}^{N,\beta}(s)+\dfrac{1}{2}\int_{0}^{t}f^{\prime\prime}(\lambda_{i}^{N,\beta}(s))d\langle\lambda_{i}^{N,\beta}\rangle_{s}$
$\displaystyle=f(\lambda_{i}^{N,\beta}(0))+\dfrac{\sqrt{2}}{\sqrt{\beta
N}}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(s))dW_{i}(s)+\dfrac{1}{\beta
N}\int_{0}^{t}f^{\prime\prime}(\lambda_{i}^{N,\beta}(s))ds$
$\displaystyle\qquad+\dfrac{1}{N}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(s))\sum_{j:j\neq
i}\dfrac{1}{\lambda_{i}^{N,\beta}(s)-\lambda_{j}^{N,\beta}(s)}ds.$

Thus, using the convention
$\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}=f^{\prime\prime}(x)$ on $\\{x=y\\}$,
we have

$\displaystyle\langle f,L_{N}^{\beta}(t)\rangle=$
$\displaystyle\dfrac{1}{N}\sum_{i=1}^{N}f(\lambda_{i}^{N,\beta}(0))+\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(s))dW_{i}(s)+\dfrac{1}{\beta
N^{2}}\sum_{i=1}^{N}\int_{0}^{t}f^{\prime\prime}(\lambda_{i}^{N,\beta}(s))ds$
$\displaystyle+\dfrac{1}{N^{2}}\sum_{i\neq
j}\int_{0}^{t}\dfrac{f^{\prime}(\lambda_{i}^{N,\beta}(s))}{\lambda_{i}^{N,\beta}(s)-\lambda_{j}^{N,\beta}(s)}ds$
$\displaystyle=$
$\displaystyle\dfrac{1}{N}\sum_{i=1}^{N}f(\lambda_{i}^{N,\beta}(0))+\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(s))dW_{i}(s)+\dfrac{1}{\beta
N^{2}}\sum_{i=1}^{N}\int_{0}^{t}f^{\prime\prime}(\lambda_{i}^{N,\beta}(s))ds$
$\displaystyle+\dfrac{1}{2N^{2}}\sum_{i\neq
j}\int_{0}^{t}\dfrac{f^{\prime}(\lambda_{i}^{N,\beta}(s))-f^{\prime}(\lambda_{j}^{N,\beta}(s))}{\lambda_{i}^{N,\beta}(s)-\lambda_{j}^{N,\beta}(s)}ds$
$\displaystyle=$
$\displaystyle\dfrac{1}{N}\sum_{i=1}^{N}f(\lambda_{i}^{N,\beta}(0))+\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(s))dW_{i}(s)$
$\displaystyle+\left(\dfrac{1}{\beta}-\dfrac{1}{2}\right)\dfrac{1}{N^{2}}\sum_{i=1}^{N}\int_{0}^{t}f^{\prime\prime}(\lambda_{i}^{N,\beta}(s))ds+\dfrac{1}{2N^{2}}\sum_{i,j=1}^{N}\int_{0}^{t}\dfrac{f^{\prime}(\lambda_{i}^{N,\beta}(s))-f^{\prime}(\lambda_{j}^{N,\beta}(s))}{\lambda_{i}^{N,\beta}(s)-\lambda_{j}^{N,\beta}(s)}ds$
$\displaystyle=$ $\displaystyle\langle
f,L_{N}^{\beta}(0)\rangle+\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(s))dW_{i}(s)+\left(\dfrac{1}{\beta}-\dfrac{1}{2}\right)\dfrac{1}{N}\int_{0}^{t}\langle
f^{\prime\prime},L_{N}^{\beta}(s)\rangle ds$
$\displaystyle+\dfrac{1}{2}\int_{0}^{t}\iint_{\mathbb{R}^{2}}\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}L_{N}^{\beta}(s)(dx)L_{N}^{\beta}(s)(dy)ds.$
(27)

Step 2: We prove that the sequence
$\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ is almost surely
relatively compact in $C([0,T],\mathcal{P}(\mathbb{R}))$, that is, every
subsequence has a further subsequence that converges in
$C([0,T],\mathcal{P}(\mathbb{R}))$ almost surely, following the argument [1,
Lemma 4.3.13].

Note that for $f\in C^{2}(\mathbb{R})$ with bounded first and second
derivatives, by mean value theorem, one can show
$|\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}|\leq\|f^{\prime\prime}\|_{L^{\infty}}$.
Hence, by (2.2),

$\displaystyle|\langle f,L_{N}^{\beta}(t)\rangle-\langle
f,L_{N}^{\beta}(s)\rangle|\leq~{}$
$\displaystyle\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{s}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|+\left(\dfrac{1}{\beta}-\dfrac{1}{2}\right)\dfrac{1}{N}\left|\int_{s}^{t}\langle
f^{\prime\prime},L_{N}^{\beta}(r)\rangle dr\right|$
$\displaystyle~{}+\dfrac{1}{2}\left|\int_{s}^{t}\iint_{\mathbb{R}^{2}}\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}L_{N}^{\beta}(r)(dx)L_{N}^{\beta}(r)(dy)dr\right|$
$\displaystyle\leq~{}$ $\displaystyle\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{s}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|+\left(\dfrac{1}{2}+\dfrac{1}{\beta
N}-\dfrac{1}{2N}\right)\|f^{\prime\prime}\|_{L^{\infty}}|t-s|.$ (28)

Note that $[0,T]$ can be partitioned into small intervals of length
$\eta<\|f^{\prime\prime}\|_{L^{\infty}}^{-8/7}$ and the number of the
intervals is $J=[T\eta^{-1}]$. Then by Markov inequality and Burkholder-Davis-
Gundy inequality, we have, for $M>0$,

$\displaystyle\mathbb{P}\left(\sup_{|t-s|\leq\eta}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{s}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|\geq
M\eta^{1/8}\right)\leq\sum_{k=0}^{J-1}\mathbb{P}\left(\sup_{k\eta\leq
t\leq(k+1)\eta}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{k\eta}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|\geq\dfrac{M\eta^{1/8}}{3}\right)$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{J-1}\dfrac{81}{M^{4}\eta^{1/2}}\mathbb{E}\left[\sup_{k\eta\leq
t\leq(k+1)\eta}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{k\eta}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|^{4}\right]\leq\sum_{k=0}^{J-1}\dfrac{324\Lambda_{2}}{M^{4}\eta^{1/2}\beta^{2}N^{6}}\mathbb{E}\left[\left\langle\sum_{i=1}^{N}\int_{k\eta}^{k\eta+\cdot}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right\rangle_{\eta}^{2}\right]$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{J-1}\dfrac{324\Lambda_{2}}{M^{4}\eta^{1/2}\beta^{2}N^{6}}\mathbb{E}\left[\left(\sum_{i=1}^{N}\int_{k\eta}^{(k+1)\eta}\left|f^{\prime}(\lambda_{i}^{N,\beta}(r))\right|^{2}dr\right)^{2}\right]\leq\dfrac{324\Lambda_{2}J\eta^{3/2}}{M^{4}\beta^{2}N^{4}}\|f^{\prime}\|_{L^{\infty}}^{4}\leq\dfrac{324\Lambda_{2}T\eta^{1/2}}{M^{4}\beta^{2}N^{4}}\|f^{\prime}\|_{L^{\infty}}^{4}.$
(29)

Hence, noting that $\beta\in\\{1,2\\}$ and
$\eta^{1/8}>\eta\|f^{\prime}\|_{L^{\infty}}$, by (2.2) and (2.2), for $M>0$,
we have

$\displaystyle\mathbb{P}\left(\sup_{|t-s|\leq\eta}|\langle
f,L_{N}^{\beta}(t)\rangle-\langle
f,L_{N}^{\beta}(s)\rangle|\geq(M+1)\eta^{1/8}\right)$ $\displaystyle\leq~{}$
$\displaystyle\mathbb{P}\left(\sup_{|t-s|\leq\eta}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{s}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|\geq(M+1)\eta^{1/8}-\dfrac{\eta\|f^{\prime}\|_{L^{\infty}}}{2}\right)$
$\displaystyle\leq~{}$
$\displaystyle\mathbb{P}\left(\sup_{|t-s|\leq\eta}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{s}^{t}f^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|\geq
M\eta^{1/8}\right)\leq\dfrac{324\Lambda_{2}T\eta^{1/2}}{M^{4}N^{4}}\|f^{\prime}\|_{L^{\infty}}^{4}.$
(30)

Let $\\{\tilde{f}_{k}\\}_{k\in\mathbb{N}}$ be a family of
$C_{b}^{2}(\mathbb{R})$ functions that is dense in $C_{0}(\mathbb{R})$. Choose
$\varepsilon_{k}=\left(1+k\|\tilde{f}_{k}^{\prime}\|_{L^{\infty}}\right)^{-1}$
and define

$\displaystyle C_{T}(\tilde{f}_{k},\varepsilon_{k})$
$\displaystyle=\bigcap_{n=1}^{\infty}\left\\{\mu\in
C([0,T],\mathcal{P}(\mathbb{R})):\sup_{|t-s|\leq
n^{-4}}\left|\langle\tilde{f}_{k},\mu_{t}\rangle-\langle\tilde{f}_{k},\mu_{s}\rangle\right|\leq\dfrac{1}{\varepsilon_{k}\sqrt{n}}\right\\}$
$\displaystyle=\left\\{\mu\in C([0,T],\mathcal{P}(\mathbb{R})):\sup_{|t-s|\leq
n^{-4}}\left|\langle\tilde{f}_{k},\mu_{t}\rangle-\langle\tilde{f}_{k},\mu_{s}\rangle\right|\leq\dfrac{1}{\varepsilon_{k}\sqrt{n}},\forall
n\in\mathbb{N}\right\\}$ $\displaystyle=\left\\{\mu\in
C([0,T],\mathcal{P}(\mathbb{R})):t\rightarrow\langle\tilde{f}_{k},\mu_{t}\rangle\in
C_{\left\|\tilde{f}_{k}\right\|_{L^{\infty}}}(\\{(\varepsilon_{k}\sqrt{n})^{-1}\\},\\{n^{-4}\\})\right\\},$

where the set

$C_{M}(\\{(\varepsilon_{k}\sqrt{n})^{-1}\\},\\{n^{-4}\\})=\bigcap_{n=1}^{\infty}\left\\{g\in
C([0,T],\mathbb{R}):\sup_{|t-s|\leq
n^{-4}}|g(t)-g(s)|\leq(\varepsilon_{k}\sqrt{n})^{-1},\sup_{t\in[0,T]}|g(t)|\leq
M\right\\},$

is (sequentially) compact in $C([0,T],\mathbb{R})$ according to Arzela-Ascoli
Lemma. By (2.2),

$\displaystyle\sum_{N=1}^{\infty}\sum_{k\geq 1}\mathbb{P}(L_{N}^{\beta}\notin
C_{T}(\tilde{f}_{k},\varepsilon_{k}))\leq\sum_{N=1}^{\infty}\sum_{k\geq
1}\sum_{n=1}^{\infty}\mathbb{P}\left(\sup_{|t-s|\leq
n^{-4}}\left|\langle\tilde{f}_{k},L_{N}^{\beta}(t)\rangle-\langle\tilde{f}_{k},L_{N}^{\beta}(s)\rangle\right|>\dfrac{1}{\varepsilon_{k}\sqrt{n}}\right)$
$\displaystyle\leq~{}$ $\displaystyle\sum_{N=1}^{\infty}\sum_{k\geq
1}\sum_{n=1}^{\infty}\dfrac{324\Lambda_{2}T}{(\varepsilon_{k}^{-1}-1)^{4}n^{2}N^{4}}\|\tilde{f}_{k}^{\prime}\|_{L^{\infty}}^{4}=324\Lambda_{2}T\sum_{n=1}^{\infty}n^{-2}\sum_{k\geq
1}\dfrac{\|\tilde{f}_{k}^{\prime}\|_{L^{\infty}}^{4}}{(\varepsilon_{k}^{-1}-1)^{4}}\sum_{N=1}^{\infty}\dfrac{1}{N^{4}}$
$\displaystyle=~{}$ $\displaystyle
324\Lambda_{2}T\sum_{n=1}^{\infty}n^{-2}\sum_{k\geq
1}k^{-4}\sum_{N=1}^{\infty}\dfrac{1}{N^{4}}<\infty,$ (31)

Since the function $\varphi$ is positive and tends to infinity as
$|x|\rightarrow+\infty$, the set

$\displaystyle
K(\varphi)=\left\\{\mu\in\mathcal{P}(\mathbb{R}):\langle\varphi,\mu\rangle\leq
1+C_{0}+T\left\|\varphi^{\prime\prime}\right\|_{L^{\infty}}\right\\}$

is tight, i.e., it is (sequentially) compact in $\mathcal{P}(\mathbb{R})$. By
(2.2) for $f=\varphi$ and $s=0$, Markov inequality and Burkholder-Davis-Gundy
inequality, we have

$\displaystyle\sum_{N=1}^{\infty}\mathbb{P}\left(\exists t\in[0,T],\
\mathrm{s.t.}\ L_{N}^{\beta}(t)\notin
K(\varphi)\right)=\sum_{N=1}^{\infty}\mathbb{P}\left(\sup_{t\in[0,T]}\langle\varphi,L_{N}(t)\rangle>1+C_{0}+T\left\|\varphi^{\prime\prime}\right\|_{L^{\infty}}\right)$
$\displaystyle\leq~{}$
$\displaystyle\sum_{N=1}^{\infty}\mathbb{P}\left(\sup_{t\in[0,T]}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{0}^{t}\varphi^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|>1\right)\leq\sum_{N=1}^{\infty}\mathbb{E}\left[\sup_{t\in[0,T]}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{0}^{t}\varphi^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right|^{2}\right]$
$\displaystyle\leq~{}$
$\displaystyle\sum_{N=1}^{\infty}\Lambda_{1}\mathbb{E}\left[\left\langle\dfrac{\sqrt{2}}{\sqrt{\beta
N^{3}}}\sum_{i=1}^{N}\int_{0}^{\cdot}\varphi^{\prime}(\lambda_{i}^{N,\beta}(r))dW_{i}(r)\right\rangle_{T}\right]=\sum_{N=1}^{\infty}\dfrac{2\Lambda_{1}}{\beta
N^{3}}\mathbb{E}\left[\sum_{i=1}^{N}\int_{0}^{T}\left(\varphi^{\prime}(\lambda_{i}^{N,\beta}(r))\right)^{2}dr\right]$
$\displaystyle\leq~{}$
$\displaystyle\sum_{N=1}^{\infty}\dfrac{2\Lambda_{1}T\left\|\varphi^{\prime}\right\|_{L^{\infty}}^{2}}{\beta
N^{2}}<\infty.$ (32)

By [1, Lemma 4.3.13] (see also [64, Lemma B.4]), the set

$\displaystyle\mathcal{H}=\Bigl{\\{}\mu\in
C([0,T],\mathcal{P}(\mathbb{R})):\mu_{t}\in K(\varphi),\ \forall
t\in[0,T]\Bigr{\\}}\cap\bigcap_{k\geq 1}C_{T}(\tilde{f}_{k},\varepsilon_{k})$

is compact in $C([0,T],\mathcal{P}(\mathbb{R}))$. By (2.2) and (2.2), we have

$\sum_{N=1}^{\infty}\mathbb{P}(L_{N}^{\beta}\notin\mathcal{H})\leq\sum_{N=1}^{\infty}\mathbb{P}(\exists
t\in[0,T],\ \mathrm{s.t.}\ L_{N}^{\beta}(t)\notin
K(\varphi))+\sum_{N=1}^{\infty}\sum_{k\geq 1}\mathbb{P}(L_{N}^{\beta}\notin
C_{T}(\tilde{f}_{k},\varepsilon_{k}))<\infty.$

Therefore, the Borel-Cantelli Lemma implies that
$\displaystyle\mathbb{P}\left(\liminf_{N\rightarrow\infty}\\{L_{N}^{\beta}\in\mathcal{H}\\}\right)=1$.

Finally, the relative compactness of the family
$\\{L_{N}^{\beta}\\}_{N\in\mathbb{N}}$ follows from the compactness of
$\mathcal{H}$.

Step 3: We derive the equation (26) for any limit point $\mu$ of the sequence
$\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$.

Let $\\{N_{k}\\}_{k\in\mathbb{N}}$ be a subsequence such that
$L_{N_{k}}^{\beta}$ converges to $\mu$ in $C([0,T],\mathcal{P}(\mathbb{R}))$.
For any $\epsilon>0$, for any $f\in C_{b}^{2}(\mathbb{R})$, by Markov
inequality and Burkholder-Davis-Gundy inequality, we have

$\displaystyle\sum_{k=1}^{\infty}\mathbb{P}\left(\sup_{t\in[0,T]}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N_{k}^{3}}}\sum_{i=1}^{N_{k}}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N_{k},\beta}(r))dW_{i}(r)\right|\geq\epsilon\right)\leq\sum_{k=1}^{\infty}\epsilon^{-2}\mathbb{E}\left[\sup_{t\in[0,T]}\left|\dfrac{\sqrt{2}}{\sqrt{\beta
N_{k}^{3}}}\sum_{i=1}^{N_{k}}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N_{k},\beta}(r))dW_{i}(r)\right|^{2}\right]$
$\displaystyle\leq~{}$
$\displaystyle\sum_{k=1}^{\infty}\dfrac{\Lambda_{1}}{\epsilon^{2}}\mathbb{E}\left[\left\langle\dfrac{\sqrt{2}}{\sqrt{\beta
N_{k}^{3}}}\sum_{i=1}^{N_{k}}\int_{0}^{\cdot}f^{\prime}(\lambda_{i}^{N_{k},\beta}(r))dW_{i}(r)\right\rangle_{T}\right]=\sum_{k=1}^{\infty}\dfrac{2\Lambda_{1}}{\epsilon^{2}\beta
N_{k}^{3}}\mathbb{E}\left[\sum_{i=1}^{N_{k}}\int_{0}^{T}\left(f^{\prime}(\lambda_{i}^{N_{k},\beta}(r))\right)^{2}dr\right]\leq\sum_{k=1}^{\infty}\dfrac{2\Lambda_{1}T}{\epsilon^{2}\beta
N_{k}^{2}}\left\|f^{\prime}\right\|_{L^{\infty}}^{2},$

which is finite since $N_{k}\geq k$. By Borel–Cantelli Lemma,

$\displaystyle\dfrac{\sqrt{2}}{\sqrt{\beta
N_{k}^{3}}}\sum_{i=1}^{N_{k}}\int_{0}^{t}f^{\prime}(\lambda_{i}^{N_{k},\beta}(r))dW_{i}(r)\to
0,\quad k\to\infty,$ (33)

uniformly with respect to $t$ almost surely. Moreover, the boundedness of
$\|f^{\prime\prime}\|_{L^{\infty}}$ yields

$\displaystyle\left(\dfrac{1}{\beta}-\dfrac{1}{2}\right)\dfrac{1}{N_{k}}\int_{0}^{t}\langle
f^{\prime\prime},L_{N_{k}}^{\beta}(s)\rangle ds,\to 0,\quad k\to\infty,$ (34)

uniformly with respect to $t$ almost surely. Therefore, by considering the
subsequence $\\{N_{k}\\}$ in (2.2), and using (33) and (34), we have

$\displaystyle\langle f,\mu_{t}\rangle=\langle
f,\mu_{0}\rangle+\dfrac{1}{2}\int_{0}^{t}\iint_{\mathbb{R}^{2}}\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}\mu_{s}(dx)\mu_{s}(dy)ds.$

Step 4: We establish the uniqueness of the solution to the equation (26). For
simplicity, we only prove the uniqueness for the self-similar solution under
null initial condition $X^{N,\beta}(0)=0$. The idea can be found in [65] and
[1, Exercise 4.3.18]. For general case, we refer the interested readers to [1,
Lemma 4.3.15]. Note that under the null initial condition, the limit points of
the sequence $\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ inherit the
self-similarity property from the Brownian motions on the matrix entries, and
hence the uniqueness of the limit of
$\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$.

To prove the uniqueness, it is convenient to choose $f(x)=(z-x)^{-1}$ for
$z\in\mathbb{C}\setminus\mathbb{R}$. Denote

$\displaystyle G_{t}(z)=\int\dfrac{1}{z-x}\mu_{t}(dx),$

which is known as the Stieltjes transform of the measure $\mu_{t}$. We refer
to [67, Section 2.4.3] and [2, Section 1.3.2] for more details about Stieltjes
transform.

Recall the matrix Brownian motion $H^{N,\beta}(t)$ in Definition 1, by the
self-similarity of Brownian motion, under null initial condition, we have
$X^{N,\beta}(t)\overset{d}{=}\sqrt{t}X^{N,\beta}(1)$, where $\overset{d}{=}$
is the equality in distribution. Thus, we have the scaling property of the
Stieltjes transform

$\displaystyle
G_{t}(z)=\dfrac{1}{\sqrt{t}}G_{1}\left(\dfrac{z}{\sqrt{t}}\right).$ (35)

Hence,

$\displaystyle
G_{t}(z)\partial_{z}G_{t}(z)=\dfrac{1}{\sqrt{t}^{3}}G_{1}\left(\dfrac{z}{\sqrt{t}}\right)G_{1}^{\prime}\left(\dfrac{z}{\sqrt{t}}\right)=-\dfrac{1}{z}\dfrac{d}{dt}\left(G_{1}^{2}\left(\dfrac{z}{\sqrt{t}}\right)\right).$

Letting $f(X)=(z-x)^{-1}$ for $z\in\mathbb{C}\setminus\mathbb{R}$, (26) can be
written as

$\displaystyle G_{t}(z)=$ $\displaystyle
G_{0}(z)-\int_{0}^{t}G_{s}(z)\partial_{z}G_{s}(z)ds=G_{0}(z)+\dfrac{1}{z}G_{1}^{2}\left(\dfrac{z}{\sqrt{s}}\right)\Bigg{|}_{s=0}^{s=t}=G_{0}(z)+\dfrac{1}{z}G_{1}^{2}\left(\dfrac{z}{\sqrt{t}}\right),$
(36)

where we use

$\displaystyle
G_{1}\left(\dfrac{z}{\sqrt{s}}\right)\Bigg{|}_{s=0}=\int\dfrac{1}{z/\sqrt{s}-x}\mu_{t}(dx)\Bigg{|}_{s=0}=\int\dfrac{\sqrt{s}}{z-\sqrt{s}x}\mu_{t}(dx)\Bigg{|}_{s=0}=0.$

Letting $t=1$ in (36) and noting that $G_{0}(z)=1/z$, we have
$G_{1}^{2}(z)-zG_{1}(z)+1=0,$ of which the solution is

$\displaystyle G_{1}(z)=\dfrac{z-\sqrt{z^{2}-4}}{2}.$ (37)

Note that by definition,

$\displaystyle\Im\left(G_{t}(z)\right)=\Im\left(\int\dfrac{1}{z-x}\mu_{t}(dx)\right)=\Im\left(\int\dfrac{\bar{z}-x}{|z-x|^{2}}\mu_{t}(dx)\right)=-\Im(z)\int\dfrac{1}{|z-x|^{2}}\mu_{t}(dx).$

Here, we use the notation $\Im(w)$ for the imaginary part of $w$. Thus, for
all $t$, $G_{t}(z)$ maps $z\in\mathbb{C}_{+}$ to $\mathbb{C}_{-}$. Thus, the
square root in (37) should be the branch that maps from $\mathbb{C}_{+}$ to
$\mathbb{C}_{+}$.

Lastly, (35) and (37) yield

$\displaystyle G_{t}(z)=\dfrac{z-\sqrt{z^{2}-4t}}{2t},$ (38)

which is the unique self-similar solution to (26). ∎

######  Remark 4.

The equation (38) is consistent with the Stieltjes transform of the semi-
circle law [1, (2.4.6)], [67, (2.103)] and [2, Lemma 2.11].

######  Remark 5.

Note that under the null initial condition $X^{N,\beta}(0)=0$, at the time
$t=1$, the matrix $X^{N,\beta}(1)=H^{N,\beta}(1)$ is the Gaussian Orthogonal
Ensemble (GOE) when $\beta=1$, and the Gaussian Unitary Ensemble (GUE) when
$\beta=2$. We refer to [67, Section 2.3] for more details about GOE and GUE.
Moreover, (37) is the Stieltjes transform of the famous semi-circle law (see
[1, (2.4.6)], [67, (2.103)] and [2, Lemma 2.11]). Thus, Theorem 2 gives a
dynamical proof of the semi-circle law (see [67, Theorem 2.4.2] or [2, Theorem
2.5] for the statement of semi-circle law).

######  Remark 6.

The differential form of (36),

$\displaystyle\partial_{t}G_{t}(z)+G_{s}(z)\partial_{z}G_{s}(z)=0,$

is the complex version of inviscid Burgers’ equation.

In some literature (see for example [59, 31, 56, 55, 64]), some other
tightness argument was used to obtain the convergence in law of the eigenvalue
empirical measure processes in $C([0,T],\mathcal{P}(\mathbb{R}))$. To
illustrate this argument, we present the following weak version of Theorem 2.

######  Theorem 3 ([57], Proposition 3.1).

Assume that all the conditions in Theorem 2 hold. Then the sequence
$\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ converges in probability
in $C([0,T],\mathcal{P}(\mathbb{R}))$. Moreover, its limit $\mu$ is
characterized by the equation (26).

###### Proof:.

The equation (26) can be derived as in Theorem 2 by using Itô calculus and
martingale theory. We only prove the convergence in law of the sequence
$\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$, noting that the limit is
a deterministic measure. The key idea is to obtain the following moment
estimation

$\displaystyle\mathbb{E}\left[\left|\left\langle
f,L_{N}^{\beta}(t)\right\rangle-\left\langle
f,L_{N}^{\beta}(s)\right\rangle\right|^{1+a}\right]\leq
C_{f,T}|t-s|^{1+b},~{}\forall t,s\in[0,T],~{}\forall N\in\mathbb{N},$ (39)

for some positive constants $a$ and $b$, and for all $f\in C^{2}(\mathbb{R})$
with bounded first and second derivatives. Here $C_{f,T}$ is a positive
constant depending only on $f$ and $T$. Then the tightness of the sequence
$\\{L_{N}^{\beta}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ in
$C([0,T],\mathcal{P}(\mathbb{R}))$ follows from (39) and [64, Proposition
B.3].

To establish (39), one may first obtain (2.2) by Itô calculus and then apply
the Burkholder-Davis-Gundy inequality to get an upper bound for some even
moment of the martingale term. In the following, we provide another approach
to get (39), where the pathwise Hölder continuity of the matrix entries is
used. This idea can also be found in [31, 55, 56, 64].

It is well known that almost all the paths of Brownian motion are
$(1/2-\varepsilon)$-Hölder continuous for any $\varepsilon\in(0,1/2)$, and so
are the paths of the entries in $X^{N,\beta}$. Consider the Hölder norm of the
matrix entries $X_{i,j}^{N,\beta}(t)$, for $1\leq i\leq j\leq N$,

$\displaystyle\left\|X_{i,j}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}=\sup_{0\leq
s<t\leq
T}\dfrac{\left|X_{i,j}^{N,\beta}(t)-X_{i,j}^{N,\beta}(s)\right|}{|t-s|^{1/2-\varepsilon}}.$

By the Fernique Theorem ([17]), we have the following estimation

$\displaystyle\mathbb{E}\left[\exp\left(\alpha
N\left\|X_{i,j}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{2}\right)\right]<\infty,$
(40)

where $\alpha=\alpha(\varepsilon,T)$ is a positive constant depending on
$(\varepsilon,T)$ and $X$.

By mean value theorem and Hoffman-Wielandt inequality ([1, Lemma 2.1.19]), we
have

$\displaystyle\left|\left\langle f,L_{N}^{\beta}(t)\right\rangle-\left\langle
f,L_{N}^{\beta}(s)\right\rangle\right|^{2}=\left|\dfrac{1}{N}\sum_{i=1}^{N}\left(f\left(\lambda_{i}^{N,\beta}(t)\right)-f\left(\lambda_{i}^{N,\beta}(s)\right)\right)\right|^{2}$
$\displaystyle\leq~{}$
$\displaystyle\dfrac{\|f^{\prime}\|_{L^{\infty}}^{2}}{N}\sum_{i=1}^{N}\left|\lambda_{i}^{N,\beta}(t)-\lambda_{i}^{N,\beta}(s)\right|^{2}\leq\dfrac{\|f^{\prime}\|_{L^{\infty}}^{2}}{N}\sum_{i,j=1}^{N}\left|X^{N,\beta}(t)-X^{N,\beta}(s)\right|^{2}.$
(41)

This together with Minkowski inequality, the pathwise Hölder continuity of
$X_{i,j}^{N,\beta}$ and Cauchy-Schwarz inequality yields

$\displaystyle\mathbb{E}\left[\left|\left\langle
f,L_{N}^{\beta}(t)\right\rangle-\left\langle
f,L_{N}^{\beta}(s)\right\rangle\right|^{4}\right]\leq\dfrac{\|f^{\prime}\|_{L^{\infty}}^{4}}{N^{2}}\mathbb{E}\left[\left(\sum_{i,j=1}^{N}\left|X^{N,\beta}(t)-X^{N,\beta}(s)\right|^{2}\right)^{2}\right]\leq\dfrac{\|f^{\prime}\|_{L^{\infty}}^{4}}{N^{2}}\left(\sum_{i,j=1}^{N}\left(\mathbb{E}\left[\left|X^{N,\beta}(t)-X^{N,\beta}(s)\right|^{4}\right]\right)^{1/2}\right)^{2}$
$\displaystyle\leq~{}$
$\displaystyle\dfrac{\|f^{\prime}\|_{L^{\infty}}^{4}}{N^{2}}\left(\sum_{i,j=1}^{N}\left(\mathbb{E}\left[\left\|X_{i,j}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{4}|t-s|^{2-4\varepsilon}\right]\right)^{1/2}\right)^{2}=\dfrac{\|f^{\prime}\|_{L^{\infty}}^{4}|t-s|^{2-4\varepsilon}}{N^{4}}\left(\sum_{i,j=1}^{N}\left(\mathbb{E}\left[N^{2}\left\|X_{i,j}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{4}\right]\right)^{1/2}\right)^{2}$
$\displaystyle\leq$
$\displaystyle\dfrac{\|f^{\prime}\|_{L^{\infty}}^{4}|t-s|^{2-4\varepsilon}}{N^{2}}\sum_{i,j=1}^{N}\mathbb{E}\left[N^{2}\left\|X_{i,j}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{4}\right].$
(42)

Recall that the matrix $X^{N,\beta}$ is symmetric for $\beta=1$ and Hermitian
for $\beta=2$. Also note that the upper-diagonal entries are i.i.d., as well
as the diagonal entries. Thus,

$\displaystyle\dfrac{1}{N^{2}}\sum_{i,j=1}^{N}\mathbb{E}\left[N^{2}\left\|X_{i,j}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{4}\right]\leq~{}$
$\displaystyle\mathbb{E}\left[N^{2}\left\|X_{1,1}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{4}\right]+\mathbb{E}\left[N^{2}\left\|X_{1,2}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{4}\right]$
$\displaystyle\leq~{}$
$\displaystyle\dfrac{2}{\alpha^{2}}\mathbb{E}\left[\exp\left(\alpha
N\left\|X_{1,1}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{2}\right)\right]+\dfrac{2}{\alpha^{2}}\mathbb{E}\left[\exp\left(\alpha
N\left\|X_{1,2}^{N,\beta}\right\|_{0,T;1/2-\varepsilon}^{2}\right)\right],$
(43)

where we use the inequality $x^{2}/2\leq e^{x}$ for $x\geq 0$. Therefore, by
(2.2), (2.2) and (40), we obtain (39) with $a=3$ and $b=1-4\varepsilon$ for
$\varepsilon<1/4$. ∎

######  Remark 7.

The weak semi-circle law, which can be found in [1, Theorem 2.1.1], is
recovered when we choose $X^{N,\beta}(0)=0$ and $t=1$ in Theorem 3. Thus,
Theorem 3 gives a dynamical proof of the weak semi-circle law.

In the framework of free probability theory, [68, 69, 4] showed that
independent $N\times N$ random matrices converge to free random variables as
$N$ tends to infinity. In this sense, the large $N$ limit of Brownian motion
with values in the space of $N\times N$ Hermitian matrices is known as free
Brownian motion ([4, Theorem 1]). More precisely, a (one-side) free (additive)
Brownian motion $\\{S(t),t\geq 0\\}$ is a non-commutative stochastic process
that satisfies:

  * 1.

$S(0)=0$;

  * 2.

For $t_{2}>t_{1}\geq 0$, the law of $S(t_{2})-S(t_{1})$ is the semicircular
distribution with mean $0$ and variance $t_{2}-t_{1}$;

  * 3.

For all $n\in\mathbb{N}$, and $t_{n}>\cdots>t_{1}\geq 0$, the increments
$S(t_{1}),S(t_{2})-S(t_{1}),\ldots,S(t_{n})-S(t_{n-1})$ are freely
independent.

We refer the interested reader to [51, 27, 49] and the references therein for
this topic.

For the complex model ($\beta=2$), for $p\in\mathbb{N}$, the $p$-th moment of
the sequence of eigenvalue empirical measure processes
$\\{L_{N}^{2}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ was considered in [57], the
motivation of which came from the study of moments for the GUE in [48]. By
using Itô calculus and martingale theory, [57] established a recursive formula
for the sequence $\\{\langle x^{p},L_{N}^{2}(t)\rangle,t\geq
0\\}_{p\in\mathbb{N}}$ for $N\in\mathbb{N}$, and proved that for
$p\in\mathbb{N}$, the sequence $\\{\langle x^{2p},L_{N}^{2}(t)\rangle,t\geq
0\\}_{N\in\mathbb{N}}$ converges to $\langle x^{2p},\mu_{t}\rangle$ uniformly
in $t\in[0,T]$ almost surely and in $L^{2q}$ with $q\geq 1$. Moreover, [57]
also investigated the largest and least eigenvalue processes and showed that

$\displaystyle\max_{t\in[0,T]}\lambda_{1}^{N,\beta}(t)\to
2\sqrt{T},~{}\min_{t\in[0,T]}\lambda_{N}^{N,\beta}(t)\to-2\sqrt{T},~{}\text{as}~{}N\to\infty,$

almost surely.

It is natural to consider the fluctuation of the sequence
$\\{L_{N}^{\beta}(t)\\}_{N\in\mathbb{N}}$ around its limit $\mu$. Consider the
random fluctuations

$\displaystyle\mathcal{L}_{N}^{\beta}(f)(t)=N\big{(}\langle
f,L_{N}^{\beta}(t)\rangle-\langle f,\mu_{t}\rangle\big{)},$

for test function $f$ belonging to some proper function space. For the complex
Dyson Brownian motion (4), [8, Theorem 1.1] established the central limit
theorem (CLT) for Chebyshev polynomials with null initial condition. Note that
for monomials $f(x)=x^{p}$, $\mathcal{L}_{N}^{2}(x^{p})(t)$ is the fluctuation
of the $p$th moment processes $\langle x^{p},L_{N}^{2}(t)\rangle$ around the
$p$th moment of the corresponding limit measure. By martingale theory, [57,
Theorem 4.3] proved the convergence in distribution of
$\mathcal{L}_{N}^{2}(x^{p})(t)$ to a centred Gaussian process
$\mathcal{L}^{2}(x^{p})(t)$ characterized by a recursive formula. The CLT for
the sequence $\\{L_{N}^{\beta}(t)\\}_{N\in\mathbb{N}}$ with polynomial test
functions was obtained in [1, Theorem 4.3.20] and is presented below.

######  Theorem 4 ([1], Theorem 4.3.20).

Let $T>0$ be a fixed number. Assume
$\displaystyle\sup_{N\in\mathbb{N}}\max_{1\leq i\leq
N}\left|\lambda_{i}^{N,\beta}(0)\right|<\infty$, and for all $n\in\mathbb{N}$,
$p\geq 1$,

$\displaystyle\sup_{N\in\mathbb{N}}\mathbb{E}\left[\left|N\left(\left\langle
x^{n},L_{N}^{\beta}(0)\right\rangle-\left\langle
x^{n},\mu_{0}\right\rangle\right)\right|^{p}\right]<\infty.$

Furthermore, assume that for any $f(x)\in\mathbb{C}[x]$, the initial value
$\mathcal{L}_{N}^{\beta}(f)(0)$ converges in probability to a random variable
$\mathcal{L}^{\beta}(f)(0)$. Here, $\mathbb{C}[x]$ is the set of polynomials
with complex coefficients.

Then there exists a family of processes
$\\{\mathcal{L}^{\beta}(f)(t),t\in[0,T]\\}_{f\in\mathbb{C}[x]}$, such that for
any $n\in\mathbb{N}$ and any polynomials $P_{1},\ldots,P_{n}\in\mathbb{C}[x]$,
the vector-valued process
$\\{(\mathcal{L}_{N}^{\beta}(P_{1})(t),\ldots,\mathcal{L}_{N}^{\beta}(P_{n})(t)),t\in[0,T]\\}_{N\in\mathbb{N}}$
converges in distribution to
$\\{(\mathcal{L}^{\beta}(P_{1})(t),\ldots,\mathcal{L}^{\beta}(P_{n})(t)),t\in[0,T]\\}$.

The limit processes
$\\{\mathcal{L}^{\beta}(f)(t),t\in[0,T]\\}_{f\in\mathbb{C}[x]}$ are
characterized by the following properties:

  1. 1.

For $P_{1},P_{2}\in\mathbb{C}[x]$, $\alpha_{1},\alpha_{2}\in\mathbb{C}$,
$t\in[0,T]$,

$\displaystyle\mathcal{L}^{\beta}(\alpha_{1}P_{1}+\alpha_{2}P_{2})(t)=\alpha_{1}\mathcal{L}^{\beta}(P_{1})(t)+\alpha_{2}\mathcal{L}^{\beta}(P_{2})(t).$

  2. 2.

The basis $\\{\mathcal{L}^{\beta}(x^{n})(t),t\in[0,T]\\}_{n\in\mathbb{N}}$ of
$\\{\mathcal{L}^{\beta}(f)(t),t\in[0,T]\\}_{f\in\mathbb{C}[x]}$ is
characterised by

$\displaystyle\mathcal{L}^{\beta}(1)(t)=0,\quad\mathcal{L}^{\beta}(x)(t)=\mathcal{L}^{\beta}(x)(0)+G_{t}^{\beta}(x),$

and for $n\geq 0$,

$\displaystyle\mathcal{L}^{\beta}(x^{n+2})(t)=$
$\displaystyle\mathcal{L}^{\beta}(x^{n+2})(0)+\dfrac{2-\beta}{2\beta}(n+2)(n+1)\int_{0}^{t}\langle
x^{n},\mu_{s}\rangle
ds+(n+2)\sum_{k=0}^{n}\int_{0}^{t}\mathcal{L}^{\beta}(x^{n-k})(s)\langle
x^{k},\mu_{s}\rangle ds+G_{t}^{\beta}(x^{n+2}),$

where $\\{G_{t}^{\beta}(x^{n}),t\in[0,T]\\}_{n\in\mathbb{N}}$ is a family of
centred Gaussian processes with the covariance

$\displaystyle\mathbb{E}\left[G_{t}^{\beta}(x^{n})G_{s}^{\beta}(x^{m})\right]=\dfrac{2mn}{\beta}\int_{0}^{t\wedge
s}\langle x^{n+m-2},\mu_{u}\rangle du,\quad n,m\geq 1.$

##  3 Positive-definite symmetric matrix valued processes

###  3.1 Brownian motions of ellipsoids

The study of stochastic processes with values in the space of positive-
definite symmetric matrices, or the space of ellipsoids, can be dated back to
at least [15], where a class of Markov processes were studied by using
differential geometry. Later, the Brownian motions of ellipsoids were
considered in [50], and some of the results in [15, 54] were recovered without
using differential geometry.

Let $B(t)$ be a $N\times N$ matrix whose entries are i.i.d. standard Brownian
motions (matrix Brownian motion). Let $G^{N}(t)$ be a process on the group of
invertible $N\times N$ matrices that solves the following matrix SDE

$\displaystyle dG^{N}(t)=dB(t)\circ G^{N}(t).$

Then the process $\\{G^{N}(t+u)G^{N}(u)^{-1}:t\geq 0\\}$ is identical in law
to the process $G^{N}(t)$ and is independent of the process
$\\{G^{N}(r):r\in[0,u]\\}$ for all $u>0$. The process $G^{N}(t)$ is known as
the right-invariant Brownian motion. Let
$X^{N}(t)=G^{N}(t)G^{N}(t)^{\intercal}$ and
$Y^{N}(t)=G^{N}(t)^{\intercal}G^{N}(t)$, which are both Markov processes on
the space of ellipsoids. The process $Y^{N}(t)$ is known as Dynkin’s Brownian
motion.

Suppose that $G^{N}(0)$ is chosen such that $X^{N}(0)$ has distinct
eigenvalues.

######  Theorem 5 ([50], Theorem A).

The eigenvalue processes of $X^{N}(t)$ never collide and never hit $0$ for all
$t>0$ almost surely. The ordered eigenvalue processes
$\lambda_{1}^{N}(t)>\cdots>\lambda_{N}^{N}(t)(>0)$ satisfy the following
system of SDEs

$\displaystyle\dfrac{1}{2}d\left(\ln\lambda_{i}^{N}(t)\right)=dW_{i}(t)+\dfrac{1}{2}\sum_{j:j\neq
i}\dfrac{\lambda_{i}^{N}(t)+\lambda_{j}^{N}(t)}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}dt,$
(44)

where $\\{W_{1}(t),\ldots,W_{N}(t)\\}$ are independent standard Brownian
motions. Moreover,

$\displaystyle\lim_{t\to\infty}\dfrac{\ln\lambda_{i}^{N}(t)}{t}=N+1-2i.$

The system of SDEs for eigenvalue processes was derived in [50] by using Itô
calculus and martingale theory as in Theorem 1. The almost sure non-collision
of the eigenvalue processes was proved by the theorem of time-change for local
martingales (see [33, Chapter 3, Theorem 4.6]), which is of the same spirit as
the McKean’s argument (Lemma 1). The long time behavior of the eigenvalue
processes was studied by constructing auxiliary processes with a comparison
result.

The eigenvector processes of $X^{N}(t)$ and $Y^{N}(t)$ were also investigated
in [50] by Itô calculus and their behavior is very different. The eigenvector
matrix of $X^{N}(t)$ ultimately behaves like Brownian motion on $O(N)$, while
that of $Y^{N}(t)$ converges to a limiting value.

###  3.2 Wishart processes

Wishart process was introduced in [6] to perform principal component analysis
on a set of resistance data of Escherichia Coli to certain antibiotics. Let
$B(t)$ be a $N\times p$ matrix whose entries are i.i.d. standard real Brownian
motions (matrix Brownian motion). The $N\times N$ symmetric matrix
$X^{N}(t)=(B(t)+A)(B(t)+A)^{\intercal}$, where $A$ is a $N\times p$ real
deterministic matrix, is the Wishart process. By [7], the Wishart process
$X^{N}(t)$ solves the following matrix SDE

$\displaystyle
dX^{N}(t)=\sqrt{X^{N}(t)}dW(t)+dW(t)^{\intercal}\sqrt{X^{N}(t)}+pI_{N}dt,$
(45)

where $W(t)$ is a $N\times N$ matrix Brownian motion. The ordered eigenvalue
processes
$\lambda_{1}^{N}(t)\geq\lambda_{2}^{N}(t)\geq\cdots\geq\lambda_{N}^{N}(t)$ of
$X^{N}(t)$ was studied first in [6].

######  Theorem 6 ([6], Theorem 1).

Assume that $X^{N}(0)$ has $N$ distinct eigenvalues
$\lambda_{1}^{N}(0)>\lambda_{2}^{N}(0)>\cdots>\lambda_{N}^{N}(0)$. Denote the
first collision time of the eigenvalue processes by

$\displaystyle\tau_{N}=\inf\left\\{t>0:\exists\ i\neq
j,~{}\lambda_{i}^{N}(t)=\lambda_{j}^{N}(t)\right\\}.$

Then

$\displaystyle\mathbb{P}\left(\tau_{N}=+\infty\right)=1.$

Furthermore, the ordered eigenvalue processes
$\lambda_{1}^{N}(t),\cdots,\lambda_{N}^{N}(t)$ of $X^{N}(t)$ satisfy the
following system of SDEs

$\displaystyle
d\lambda_{i}^{N}(t)=2\sqrt{\lambda_{i}^{N}(t)}dW_{i}(t)+\left(p+\sum_{j:j\neq
i}\dfrac{\lambda_{i}^{N}(t)+\lambda_{j}^{N}(t)}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,~{}i\in\\{1,\ldots,N\\},$
(46)

where $\\{W_{1}(t),\ldots,W_{N}(t)\\}$ are independent standard Brownian
motions.

Theorem 6 can be proved following the idea of the proof of Theorem 1.
Similarly, the system of SDEs for eigenvalue processes can be derived by Itô
calculus and martingale theory. The almost sure non-collision of the
eigenvalue processes can also be proved by the McKean’s argument.

The eigenvector processes were also studied in [6] by Itô calculus. Under the
same assumption as in Theorem 6, with an appropriate choice of unit eigenvalue
vector processes, the system of SDEs for them were established in [6, Theorem
2].

The assumption that the eigenvalues of $X^{N}(0)$ are distinct in Theorem 6
automatically implies that $p\geq N-1$. For the case $p<N$, by [7, Corollary
1], Theorem 6 is still valid for the set of non-trivial eigenvalue processes
$\lambda_{1}^{N}(t),\cdots,\lambda_{p}^{N}(t)$. Note that the Wishart
processes $X^{N}(t)$ is positive semi-definite, and is degenerate when $p<N$
for all $t$. In some situations, it interesting to know whether the Wishart
processes is non-degenerated, which is equivalent to know the infiniteness of
the hitting time of the least eigenvalue processes on $0$. By using the
McKean’s argument, [7, Proposition 1] proved that $\lambda_{N}^{N}(t)>0$ for
all $t$ almost surely for the case $p>N$. For the critical case, $p=N$, the
set of hitting time on $0$ ($\\{t:\lambda_{N}^{N}(t)=0\\}$) has zero Lebesgue
measure almost surely. [7] also considered the matrix model (45) whenever
$p>0$ is not an integer, and proved that the conclusion of Theorem 6 holds for
the unique solution to (45) for $p>N-1$ ([7, Theorem 2]).

Let $Y^{N}(t)$ be the complex analogue of $X^{N}(t)$, that is,
$Y^{N}(t)=(\tilde{B}(t)+\tilde{A})(\tilde{B}(t)+\tilde{A})^{\intercal}$, where
$\tilde{B}(t)$ is a $N\times p$ complex matrix whose real and imaginary parts
are independent matrix Brownian motions, and $\tilde{A}$ is a $N\times p$
complex deterministic matrix. Then $Y^{N}(t)$ is known as Laguerre process
([41]). With minor modification to the Wishart case (Theorem 6), the non-
collision property of the eigenvalue processes can be established and the
following system of SDEs for eigenvalue processes can be obtained

$\displaystyle
d\lambda_{i}^{N}(t)=2\sqrt{\lambda_{i}^{N}(t)}dW_{i}(t)+2\left(p+\sum_{j:j\neq
i}\dfrac{\lambda_{i}^{N}(t)+\lambda_{j}^{N}(t)}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,~{}i\in\\{1,\ldots,N\\}.$
(47)

See for example [58, (2)], [37, (1.2)].

The eigenvalue processes (47) were treated as particle system in [41], and
they were proved to evolve like $N$ independent squared Bessel processes of
dimension $2(p-N+1)$ conditioned to no collision among each other, assuming
$p\geq N$. For more properties of particle systems related to Brownian
motions, we refer to [35, 36, 37].

The high-dimensional limits of the normalized eigenvalue processes
$\\{\lambda_{i}^{N}(t)/N\\}_{1\leq i\leq N}$ of (47) was studied in [9] by
proving large deviation bounds. Denote the empirical measure process by

$\displaystyle
L_{N}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}^{N}(t)/N}(dx).$

The almost sure weak convergence of the sequence
$\\{L_{N}(t)\\}_{N\in\mathbb{N}}$ as well as the differential equation
satisfied by the limiting measure-valued processes was established in [9,
Corollary 3.1], assuming that $p/N$ converges to a positive number $c$.
Moreover, the limit $\mu$ is the well-known Marčenko-Pastur law (free Poisson
distribution). The results were recovered in [58, Theorem 3.3].

For $p\in\mathbb{N}$, the $p$-th moment of the sequence
$\\{L_{N}(t),t\in[0,T]\\}$ of normalized Laguerre process was considered in
[58]. By using Itô calculus and martingale theory, [58] established a
recursive formula for the sequence $\\{\langle x^{p},L_{N}(t)\rangle:t\geq
0\\}_{p\in\mathbb{N}}$ for $N\in\mathbb{N}$, and proved that as $N\to\infty$,
$\langle x^{p},L_{N}(t)\rangle$ converges to $\langle x^{p},\mu_{t}\rangle$
uniformly in $t\in[0,T]$ almost surely and in $L^{q}$ with $q\geq 1$ for
$p\in\mathbb{N}$. Moreover, [57] also investigated the largest eigenvalue
processes $\lambda_{1}^{N}(t)$ and the least eigenvalue processes
$\lambda_{N}^{N}(t)$, and showed that

$\displaystyle\max_{t\in[0,T]}\lambda_{1}^{N}(t)\to(1+\sqrt{c})^{2}\sqrt{T},~{}\min_{t\in[0,T]}\lambda_{N}^{N}(t)\to(1-\sqrt{c})^{2}\sqrt{T},~{}\text{as}~{}N\to\infty,$

almost surely, where $c=\lim_{N\to\infty}p/N$.

The fluctuation of the sequence $\\{L_{N}(t),t\in[0,T]\\}$ around its limit
$\mu$ has also been studied. Denote the random fluctuation

$\displaystyle\mathcal{L}_{N}(f)(t)=N\left(\left\langle
f,L_{N}(t)\right\rangle-\left\langle f,\mu_{t}\right\rangle\right)$

for an appropriate test function $f$. [8, Theorem 2.5] established the CLT for
a class of polynomial functions with null initial condition. Note that for
monomials $f(x)=x^{p}$, the random fluctuation $\mathcal{L}_{N}(x^{p})(t)$ is
the fluctuation of the moment processes $\langle x^{p},L_{N}(t)\rangle$ around
the corresponding moment of the limit measure. By martingale theory, [57,
Theorem 4.3] proved the convergence in distribution of
$\mathcal{L}_{N}(x^{p})(t)$ to a centred Gaussian process
$\mathcal{L}(x^{p})(t)$ whose distribution is characterized by recursive
formulas.

The Wishart process (45) was generalized in [26] to a symmetric matrix valued
process that solves the following matrix SDE

$\displaystyle
dX^{N}(t)=\sqrt{\left|X^{N}(t)\right|}dW(t)+dW(t)^{\intercal}\sqrt{\left|X^{N}(t)\right|}+pI_{N}dt.$
(48)

Its ordered eigenvalue processes
$\lambda_{1}^{N}(t)\geq\cdots\geq\lambda_{N}^{N}(t)$ satisfy the following
system of SDEs

$\displaystyle
d\lambda_{i}^{N}(t)=2\sqrt{\left|\lambda_{i}^{N}(t)\right|}dW_{i}(t)+\left(p+\sum_{j:j\neq
i}\dfrac{\left|\lambda_{i}^{N}(t)\right|+\left|\lambda_{j}^{N}(t)\right|}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,~{}i\in\\{1,\ldots,N\\},$
(49)

which is known as squared Bessel particle system. [26, Theorem 1] proved the
existence and uniqueness of the non-colliding strong solution for all
$p\in\mathbb{R}$. The conditions for the uniqueness of the strong solution
were given in [26, Theorem 2]. Moreover, the necessary and sufficient
conditions for the existence of non-negative solutions were provided in [26,
Theorem 3].

##  4 Other matrix models and related particle systems driven by Brownian
motion

In [7], M.-F. Bru generalized her Wishart process to the following symmetric
matrix valued process.

######  Theorem 7 ([7], Theorem 2’).

Let $X^{N}(0)$ be a symmetric non-negative definite deterministic $N\times N$
matrix with distinct eigenvalues. Let $W(t)$ be a matrix Brownian motion, then
for $p,\beta,\gamma\in\mathbb{R}$, the following matrix SDE

$\displaystyle
dX^{N}(t)=\gamma\left(\sqrt{X^{N}(t)}dW(t)+dW(t)^{\intercal}\sqrt{X^{N}(t)}\right)+2\beta
X^{N}(t)dt+p\gamma^{2}I_{N}dt,$ (50)

has a unique weak solution in the set of symmetric $N\times N$ matrices if
$p\in(N-1,N+1)$, and has a unique strong solution that is symmetric positive-
definite if $p\geq N+1$.

The ordered eigenvalue processes
$\lambda_{1}^{N}(t)\geq\cdots\geq\lambda_{N}^{N}(t)$ of the unique solution
never collide almost surely, and satisfy the following system of SDEs

$\displaystyle
d\lambda_{i}^{N}(t)=2\gamma\sqrt{\lambda_{i}^{N}(t)}dW_{i}(t)+\left(p\gamma^{2}+2\beta\lambda_{i}^{N}(t)+\gamma^{2}\sum_{j:j\neq
i}\dfrac{\lambda_{i}^{N}(t)+\lambda_{j}^{N}(t)}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,~{}i\in\\{1,\ldots,N\\},$
(51)

where $W_{1}(t),\ldots,W_{N}(t)$ are independent standard Brownian motions.

If $p\geq N+1$, $\lambda_{N}^{N}(t)>0$ for all $t>0$ almost surely.
Furthermore, if $p\in\\{1,2,\ldots,N-1\\}$, then the same results hold for the
largest $p$ eigenvalue processes
$\lambda_{1}^{N}(t),\ldots,\lambda_{p}^{N}(t)$.

######  Remark 8.

  1. 1.

The matrix model (50) reduces to the Wishart process (45) when $\beta=0$ and
$\gamma=1$.

  2. 2.

When $p\in\mathbb{Z}$ and $\gamma=1$, the system of SDEs for eigenvalue
processes was derived in [38] in shape theory.

  3. 3.

The system of SDEs (51) reduces to (47) when $\beta=0$ and $\gamma^{2}=2$.

For $\beta,\gamma\geq 0$, the singular value of $X^{N}(t)$ given by (50) was
studied in [32]. Let
$s_{i}^{N}(t)=\sqrt{\lambda_{i}^{N}(t)},i\in\\{1,\ldots,N\\}$ be the singular
value processes of $X^{N}(t)$. Then

$\displaystyle ds_{i}^{N}(t)=\gamma
dW_{i}(t)+\left(\dfrac{(p-1)\gamma^{2}}{2s_{i}^{N}(t)}-\beta
s_{i}^{N}(t)+\dfrac{\gamma^{2}}{2s_{i}^{N}(t)}\sum_{j:j\neq
i}\dfrac{\big{(}s_{i}^{N}(t)\big{)}^{2}+\big{(}s_{j}^{N}(t)\big{)}^{2}}{\big{(}s_{i}^{N}(t)\big{)}^{2}-\big{(}s_{j}^{N}(t)\big{)}^{2}}\right)dt.$

It was obtained in [32, Theorem 1.1] the convergence in probability of the
sequence of empirical measure processes

$\displaystyle\widetilde{L}_{N}(t)(dx)=\dfrac{1}{2N}\sum_{i=1}^{N}\Big{(}\delta_{s_{i}^{N}(t)/\sqrt{p}}(dx)+\delta_{-s_{i}^{N}(t)/\sqrt{p}}(dx)\Big{)}$

under general initial conditions. Moreover, the long time behavior of the
empirical measure process $\\{\lambda_{i}^{N}(t)/p,1\leq i\leq N\\}$ was also
characterized in [32, Theorem 1.2].

A more general class of real symmetric matrix valued processes was introduced
in [24], which is the solution to

$\displaystyle
dX_{t}^{N}=g_{N}(X_{t}^{N})dB_{t}h_{N}(X_{t}^{N})+h_{N}(X_{t}^{N})dB_{t}^{\intercal}g_{N}(X_{t}^{N})+b_{N}(X_{t}^{N})dt,\quad
t\geq 0,$ (52)

in the space of real symmetric $N\times N$ matrices. Here, $B_{t}$ is a
$N\times N$ matrix Brownian motion, and the functions
$g_{N},h_{N},b_{N}:\mathbb{R}\rightarrow\mathbb{R}$ act on the spectrum of
$X_{t}^{N}$. (Note. For a real-valued function $f$ and a real symmetric (or
complex Hermitian) matrix $X$ that has spectral decomposition
$X=\sum_{j=1}^{N}\alpha_{j}u_{j}u_{j}^{*}$ with eigenvalues
$\\{\alpha_{j}\\}_{1\leq j\leq N}$ and eigenvectors $\\{u_{j}\\}_{1\leq j\leq
N}$, $f(X)=\sum_{j=1}^{N}f(\alpha_{j})u_{j}u_{j}^{*}$ is the matrix obtained
by acting $f$ on the spectrum of $X$.)

The symmetric matrix valued process (52) extends the previous matrix models in
the following aspects:

  1. 1.

If we take $g_{N}(x)=\frac{1}{\sqrt{2N}}$, $h_{N}(x)=1$ and $b_{N}(x)=0$ in
(52), then $X^{N}$ becomes the real symmetric matrix Brownian motion
$X^{N,\beta}$ considered in Theorem 1..

  2. 2.

If we take $g_{N}(x)=\sqrt{\frac{\alpha}{2N}}$, $h_{N}(x)=1$ and
$b_{N}(x)=-\theta x$ in (52), then $X^{N}$ becomes the real symmetric matrix
given in (21). In particular, if $g_{N}(x)=\frac{1}{2\sqrt{N}}$, $h_{N}(x)=1$
and $b_{N}(x)=-\frac{1}{2}x$, it is the real symmetric matrix OU process given
in (18).

  3. 3.

If we take $g_{N}(x)=\sqrt{x}$, $h_{N}(x)=\gamma$ and $b_{N}(x)=2\beta
x+p\gamma^{2}$ in (52), then $X^{N}$ becomes the real symmetric matrix given
in (50). In particular, if $g_{N}(x)=\sqrt{x}$, $h_{N}(x)=1/\sqrt{N}$, and
$b_{N}(x)=p/N$, then the random matrix $Y^{N}=NX^{N}$ is the Wishart process
$B^{\intercal}B$, where $B$ is a $p\times N$ Brownian matrix.

  4. 4.

If we take $g_{N}(x)=\frac{1}{\sqrt{2N}}$, $h_{N}(x)=1$ and
$b_{N}(x)=-\frac{1}{2}V^{\prime}(x)$ in (52), then $X^{N}$ becomes the real
symmetric matrix given in (24).

  5. 5.

If we take $g_{N}(x)=\sqrt{x}$, $h_{N}(x)=\sqrt{1-x}$ and $b_{N}(x)=q-(q+r)x$
in (52) with $q,r>p-1$, then $X^{N}$ becomes matrix Jacobi processes. See [24,
(4.4)].

In [24], the non-collision property of the eigenvalue processes was
established and the system of SDEs for ordered eigenvalue processes was
derived. The results are presented below where
$\mathcal{C}^{1,1}(\mathbb{R})=\\{f\in\mathcal{C}^{1}(\mathbb{R}):|f^{\prime}(x)-f^{\prime}(y)|/|x-y|<\infty\\}$.

######  Theorem 8 ([24], Theorems 3 and 5).

Let $X_{t}^{N}$ be a real symmetric matrix valued stochastic process that
solves (52). Let $\lambda_{1}^{N}(t)\geq\cdots\geq\lambda_{N}^{N}(t)$ be the
ordered eigenvalue processes of $X_{t}^{N}$ and denote the first collision
time by

$\displaystyle\tau_{N}=\inf\\{t>0:\exists\ i\neq
j,~{}\lambda_{i}(t)=\lambda_{j}(t)\\}.$

Suppose that the functions $b_{N}(x),g_{N}^{2}(x),h_{N}^{2}(x)$ are Lipschitz
continuous. Besides, assume that $g_{N}^{2}(x)h_{N}^{2}(x)$ is convex or in
$\mathcal{C}^{1,1}(\mathbb{R})$. Furthermore, we assume
$\lambda_{1}^{N}(0)>\cdots>\lambda_{N}^{N}(0)$.

Then we have $\mathbb{P}\left(\tau_{N}=+\infty\right)=1$. Moreover, the
eigenvalue processes satisfy the following SDEs: for $1\leq i\leq N$,

$\displaystyle d\lambda_{i}^{N}(t)$
$\displaystyle=2g_{N}(\lambda_{i}^{N}(t))h_{N}(\lambda_{i}^{N}(t))dW_{i}(t)+\left(b_{N}(\lambda_{i}^{N}(t))+\sum_{j:j\neq
i}\dfrac{G_{N}(\lambda_{i}^{N}(t),\lambda_{j}^{N}(t))}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,$
(53)

where $\\{W_{i}(t)\\}_{1\leq i\leq N}$ are independent Brownian motions and

$G_{N}(x,y)=g_{N}^{2}(x)h_{N}^{2}(y)+g_{N}^{2}(y)h_{N}^{2}(x).$ (54)

######  Remark 9.

Similar results hold for the complex version of (52). Namely, under the same
conditions in Theorem 8, the eigenvalue processes of the complex Hermitian
matrix that solves the matrix SDE

$\displaystyle
dX_{t}^{N}=g_{N}(X_{t}^{N})dW_{t}h_{N}(X_{t}^{N})+h_{N}(X_{t}^{N})dW_{t}^{*}g_{N}(X_{t}^{N})+b_{N}(X_{t}^{N})dt,\quad
t\geq 0,$

where $W_{t}$ is a complex $N\times N$ matrix Brownian motion, never collide
almost surely and satisfy the following system of SDEs

$\displaystyle d\lambda_{i}^{N}(t)$
$\displaystyle=2g_{N}(\lambda_{i}^{N}(t))h_{N}(\lambda_{i}^{N}(t))dW_{i}(t)+\left(b_{N}(\lambda_{i}^{N}(t))+2\sum_{j:j\neq
i}\dfrac{G_{N}(\lambda_{i}^{N}(t),\lambda_{j}^{N}(t))}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt,$
(55)

where $\\{W_{i}(t)\\}_{1\leq i\leq N}$ are independent Brownian motions.

Theorem 8 (and its complex analogous) can be proved following the idea used in
the proof of Theorem 1: the SDEs (53) (and (55)) for eigenvalue processes can
be derived by Itô calculus and martingale theory, and the almost sure non-
collision of the eigenvalue processes can be proved by the McKean’s argument.

######  Remark 10.

The system of SDEs for eigenvector processes were also derived in [24, Theorem
3]. It was also shown in [24, Corollary 3] that the system of SDEs for
eigenvalue processes and eigenvector processes admits a unique strong solution
on $[0,\infty)$ if assuming that $G_{N}(x,y)$ is strictly positive on the set
$\\{(x,y)\in\mathbb{R}^{2}:x\neq y\\}$ together with all conditions in Theorem
8. Another set of conditions for the existence and uniqueness of strong
solution (before colliding/exploding) can be found in [65]. However, whether
the pathwise uniqueness holds for the matrix SDE (52) is still unknown.

Let $L_{N}(t)$ be the empirical measure process of the eigenvalue processes
$\\{\lambda_{i}^{N}(t)\\}_{1\leq i\leq N}$ of the symmetric matrix-valued
processes $X^{N}(t)$ given in (52), that is

$\displaystyle
L_{N}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}^{N}(t)}(dx).$

The almost sure compactness of the sequence
$\\{L_{N}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ was obtained in [65] by using the
compactness argument presented in the proof of Theorem 2, and the equation for
the limit measures was derived as well. Note that similar problems were also
investigated in [45] independently.

######  Theorem 9 ([65], Theorems 2.1 and 2.2).

Let $T>0$ be a fixed number. Suppose that (53) has a strong solution which
does not explode or collide for $t\in[0,T]$. Assume the following conditions
hold:

  1. 1.

There exists a positive function $\varphi(x)\in C^{2}(\mathbb{R})$ such that
$\lim\limits_{|x|\rightarrow+\infty}\varphi(x)=+\infty,$
$\varphi^{\prime}(x)b_{N}(x)$ is bounded with respect to $(x,N)$, and
$\varphi^{\prime}(x)g_{N}(x)h_{N}(x)$ satisfies, for some positive integer
$l_{1}$,

$\displaystyle\sum_{N=1}^{\infty}\left(\dfrac{\|\varphi^{\prime}g_{N}h_{N}\|_{L^{\infty}(dx)}^{2}}{N}\right)^{l_{1}}<\infty.$

  2. 2.

The function $NG_{N}(x,y)\dfrac{\varphi^{\prime}(x)-\varphi^{\prime}(y)}{x-y}$
is bounded with respect to $(x,y,N)$.

  3. 3.

The empirical measure $L_{N}(0)$ converges weakly to a measure $\mu_{0}$ as
$N$ goes to infinity almost surely, and

$\displaystyle
C_{0}=\sup_{N>0}\langle\varphi,L_{N}(0)\rangle=\sup_{N>0}\dfrac{1}{N}\sum_{i=1}^{N}\varphi\left(\lambda_{i}^{N}(0)\right)<\infty.$
(56)

  4. 4.

There exists a sequence $\\{\tilde{f}_{k}\\}_{k\in\mathbb{N}}$ of
$C^{2}(\mathbb{R})$ functions such that it is dense in the space
$C_{0}(\mathbb{R})$ of continuous functions vanishing at infinity and that
$\tilde{f}_{k}^{\prime}(x)g_{N}(x)h_{N}(x)$ satisfies, for some positive
integer $l_{2}\geq 2$,

$\displaystyle\psi(k)=\sum_{N=1}^{\infty}\left(\dfrac{\|\tilde{f}_{k}^{\prime}g_{N}h_{N}\|_{L^{\infty}(dx)}^{2}}{N}\right)^{l_{2}}<\infty.$
(57)

  5. 5.

There exist continuous functions $b(x)$ and $G(x,y)$, such that $b_{N}(x)$
converges to $b(x)$ and $NG_{N}(x,y)$ converges to $G(x,y)$ uniformly as $N$
tends to infinity.

Then the sequence $\\{L_{N}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ is relatively
compact in $C([0,T],M_{1}(\mathbb{R}))$ almost surely, i.e., every subsequence
has a further subsequence that converges in $C([0,T],M_{1}(\mathbb{R}))$
almost surely. Furthermore, any limit measure $\mu$ in
$C([0,T],M_{1}(\mathbb{R}))$ satisfies the equation

$\displaystyle\langle f,\mu_{t}\rangle=$ $\displaystyle\langle
f,\mu_{0}\rangle+\int_{0}^{t}\langle bf^{\prime},\mu_{s}\rangle
ds+\dfrac{1}{2}\int_{0}^{t}\left[\iint_{\mathbb{R}^{2}}\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}G(x,y)\mu_{s}(dx)\mu_{s}(dy)\right]ds,$
(58)

for all $f\in C_{b}^{2}(\mathbb{R})$ such that $f^{\prime}(x)b(x)$ and
$\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}G(x,y)$ are bounded.

######  Remark 11.

The equation (58) with the test function $f=(z-x)^{-1}$ for
$z\in\mathbb{C}_{+}$ was derived in [65]. Indeed, the computation therein is
valid for all $f\in C_{b}^{2}(\mathbb{R})$.

By using the symmetric polynomials and the tightness argument presented in the
proof of Theorem 3, [45] obtained the tightness of the sequence
$\\{L_{N}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ for both the real case (53) and
the complex case (55) with general test functions in $C_{b}^{2}(\mathbb{R})$.
The equation for the limit measures in law was also derived. The results are
presented below.

######  Theorem 10 ([45], Theorem 1).

Assume that $g_{N}$, $h_{N}$ and $b_{N}$ are continuous and satisfy

$\displaystyle g_{N}^{2}(x)+h_{N}^{2}(x)\leq K(1+|x|),\ |b_{N}(x)|\leq
KN(1+|x|),\ \forall x\in\mathbb{R},\ \forall N\in\mathbb{N},$

for some positive constant $K$. Suppose that

$\displaystyle\sup_{N\in\mathbb{N}}\int_{\mathbb{R}}x^{8}L_{N}(0)(dx)<\infty,$

then the sequence of the measure-valued processes
$\\{L_{N}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ related to (53) (resp. (55)) is
tight. Furthermore, assuming that $g_{N}^{2}(x)\to g^{2}(x)$, $h_{N}^{2}(x)\to
h^{2}(x)$ and $b_{N}(x)/N\to b(x)$ locally uniformly on $\mathbb{R}$ as
$N\to\infty$, then any limit measure $\mu$ of a weakly convergent subsequence
in law is an element in $C([0,T],\mathcal{P}(\mathbb{R}))$ that satisfies

$\displaystyle\langle f,\mu_{t}\rangle=$ $\displaystyle\langle
f,\mu_{0}\rangle+\int_{0}^{t}\langle bf^{\prime},\mu_{s}\rangle
ds+\dfrac{\beta}{2}\int_{0}^{t}\left[\iint_{\mathbb{R}^{2}}\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}G(x,y)\mu_{s}(dx)\mu_{s}(dy)\right]ds,$

for all $t>0$, for all $f\in C_{b}^{2}(\mathbb{R})$, where $\beta=1$
corresponds to the real case (53) while $\beta=2$ corresponds to the complex
case (55).

######  Remark 12.

It is worth pointing out that the almost sure compactness obtained in Theorem
9 is stronger than the tightness established in Theorem 10. However, in
comparison with Theorem 9, Theorem 10 does not require the non-colliding
property of the strong solution to (53). Hence, Theorem 10 is applicable to
the $\beta$ version of Dyson Brownian motion (4) with $\beta\in(0,\infty)$,
while Theorem 9 is only valid for $\beta\in[1,\infty)$.

The system (53) for eigenvalues of matrix-valued process (52) was further
generalized in [25] to the following particle system: for $1\leq i\leq N$,

$\displaystyle\begin{cases}&dx_{i}^{N}(t)=\sigma_{i}^{N}(x_{i}^{N}(t))dW_{i}(t)+\left(b_{i}^{N}(x_{i}^{N}(t))+\sum_{j:j\neq
i}\dfrac{H_{ij}^{N}(x_{i}^{N}(t),x_{j}^{N}(t))}{x_{i}^{N}(t)-x_{j}^{N}(t)}\right)dt,\\\
&x_{1}(t)\leq\cdots\leq x_{N}(t),~{}t\geq 0,\end{cases}$ (59)

where $\\{W_{i}(t)\\}_{1\leq i\leq N}$ is a family of independent Brownian
motions.

######  Remark 13.

  * 1.

If we take $\sigma_{i}^{N}(x)=\sigma_{N}(x),b_{i}^{N}(x)=b_{N}(x)$ and
$H_{ij}^{N}(x,y)=\gamma_{N}$, then the particle system (59) reduces to the
system (23).

  * 2.

If we take $\sigma_{i}^{N}(x)=2x,b_{i}^{N}(x)=(N+1)x$ and
$H_{ij}^{N}(x,y)=2xy$, then the particle system (59) reduces to the system
(44). To see this, we apply Itô’s formula to (44) to obtain

$\displaystyle
d\lambda_{i}^{N}(t)=d\left(e^{\ln\lambda_{i}^{N}(t)}\right)=e^{\ln\lambda_{i}^{N}(t)}d\left(\ln\lambda_{i}^{N}(t)\right)+\dfrac{1}{2}e^{\ln\lambda_{i}^{N}(t)}d\langle\ln\lambda_{i}^{N}(t)\rangle$
$\displaystyle=~{}$ $\displaystyle
2\lambda_{i}^{N}(t)dW_{i}(t)+\lambda_{i}^{N}(t)\sum_{j:j\neq
i}\dfrac{\lambda_{i}^{N}(t)+\lambda_{j}^{N}(t)}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}dt+2\lambda_{i}^{N}(t)dt=2\lambda_{i}^{N}(t)dW_{i}(t)+\left((N+1)\lambda_{i}^{N}(t)+\sum_{j:j\neq
i}\dfrac{2\lambda_{i}^{N}(t)\lambda_{j}^{N}(t)}{\lambda_{i}^{N}(t)-\lambda_{j}^{N}(t)}\right)dt.$

The following theorem guarantees the existence and uniqueness of the strong
non-exploding and non-colliding solution to (59)

######  Theorem 11 ([25], Theorem 2.2).

Consider the system (59) with initial condition $x_{1}(0)\leq\cdots\leq
x_{N}(0)$. Assume the following conditions hold:

  1. 1.

The coefficient functions $\sigma_{i}^{N}(x)$, $b_{i}^{N}(x)$ are continuous
for $1\leq i\leq N$ while $H_{ij}^{N}(x,y)$ is non-negative, continuous and
satisfies the symmetric condition $H_{ij}^{N}(x,y)=H_{ji}^{N}(y,x)$ for $1\leq
i\neq j\leq N$.

  2. 2.

There exists a function $\rho:\mathbb{R}_{+}\to\mathbb{R}_{+}$ satisfying
$\int_{0+}\rho^{-1}(x)dx=\infty$, such that for $1\leq i\leq N$,

$\displaystyle\left|\sigma_{i}^{N}(x)-\sigma_{i}^{N}(y)\right|^{2}\leq\rho(|x-y|),~{}\forall
x,y\in\mathbb{R}.$

  3. 3.

There exists a positive constant $C$ that may depends on $N$, such that for
all $1\leq i\neq j\leq N$,

$\sigma_{i}^{N}(x)^{2}+xb_{i}^{N}(x)\leq C(1+x^{2}),~{}\forall
x\in\mathbb{R};\qquad H_{ij}^{N}(x,y)\leq C(1+|xy|),~{}\forall
x,y\in\mathbb{R}.$

  4. 4.

For $1\leq i\neq j\leq N$,

$\displaystyle\dfrac{H_{ij}^{N}(w,z)}{z-w}\leq\dfrac{H_{ij}^{N}(x,y)}{y-x},~{}\forall
w<x<y<z.$

  5. 5.

There exists a positive constant $C$ that may depends on $N$, such that for
all $1\leq i\neq j\leq N$,

$\displaystyle\sigma_{i}^{N}(x)^{2}+\sigma_{j}^{N}(y)^{2}\leq
C(x-y)^{2}+4H_{ij}^{N}(x,y),~{}\forall x,y\in\mathbb{R}.$

  6. 6.

There exists a positive constant $C$ that may depends on $N$, such that for
all $1\leq i<j<k\leq N$, for all $x<y<z$,

$\displaystyle(y-x)H_{ij}^{N}(x,y)+(z-y)H_{jk}^{N}(y,z)\leq
C(z-y)(z-x)(y-x)+(z-x)H_{ik}^{N}(x,z).$

  7. 7.

For $1\leq k<l\leq N$, the set $\displaystyle
G_{kl}=\bigcap_{k<i<j<l}\left\\{x\in\mathbb{R}:\sigma_{i}^{N}(x)^{2}+\sigma_{j}^{N}(x)^{2}+H_{ij}^{N}(x,x)=0\right\\}$
consists of isolated points and for every $x\in G_{kl}$,

$\displaystyle\sum_{i=k}^{l}\left(b_{i}^{N}(x)+\sum_{j=1}^{N-2}\dfrac{H_{ij}^{N}(x,y_{j})}{x-y_{j}}\text{1}_{\mathbb{R}\setminus\\{x\\}}(y_{j})\right)\neq
0,~{}\forall y_{1},\ldots,y_{N-2}\in\mathbb{R}.$

  8. 8.

The function $b_{i}^{N}(x)$ is Lipschitz continuous or non-increasing for
$1\leq i\leq N$. Moreover, for $1\leq i<j\leq N$, for all $x\in\mathbb{R}$,
$b_{i}^{N}(x)\leq b_{j}^{N}(x)$.

Then there exists a unique strong non-exploding solution of (59), such that
the first collision time

$\displaystyle\tau_{N}=\inf\\{t>0:\exists\ i\neq
j,~{}x_{i}^{N}(t)=x_{j}^{N}(t)\\}$

is infinite almost surely.

######  Remark 14.

The initial values for the particles in Theorem 11 are allowed to collide.

Let $L_{N}(t)$ be the empirical measure process of the particles
$\\{x_{i}^{N}(t)\\}_{1\leq i\leq N}$ given in (59), that is

$\displaystyle
L_{N}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{N}(t)}(dx).$

The convergence of the sequence $\\{L_{N}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$
for $T>0$ was studied in [65] for the case that the family of functions
$\\{b_{i}^{N}(x)\\}_{1\leq i\leq N}$ and $\\{H_{ij}^{N}(x,y)\\}_{1\leq i\neq
j\leq N}$ are identical respectively. For simplicity, we assume
$b_{i}^{N}(x)=b_{N}(x)$, $\sigma_{i}^{N}(x)=\sigma_{N}(x)$ for all $1\leq
i\leq N$ and $H_{ij}^{N}(x,y)=H_{N}(x,y)$ for $1\leq i\neq j\leq N$, and then
the particle system (59) becomes

$\displaystyle
dx_{i}^{N}(t)=\sigma^{N}(x_{i}^{N}(t))dW_{i}(t)+\left(b_{N}(x_{i}^{N}(t))+\sum_{j:j\neq
i}\dfrac{H_{N}(x_{i}^{N}(t),x_{j}^{N}(t))}{x_{i}^{N}(t)-x_{j}^{N}(t)}\right)dt,\
t\geq 0,$ (60)

for $1\leq i\leq N$.

######  Theorem 12 ([65], Theorems 3.1 and 3.2).

Let $T>0$ be a fixed number. Suppose that (60) has a strong solution that is
non-exploding and non-colliding for $t\in[0,T]$. Assume the following
conditions hold:

  1. 1.

There exists a positive function $\varphi(x)\in C^{2}(\mathbb{R})$ such that
$\lim\limits_{|x|\rightarrow+\infty}\varphi(x)=+\infty$,
$\varphi^{\prime}(x)b_{N}(x)$ and $\varphi^{\prime\prime}(x)\sigma^{N}(x)^{2}$
are bounded with respect to $(x,N)$, and $\varphi^{\prime}(x)\sigma^{N}(x)$
satisfies, for some positive integer $l_{1}$.

$\displaystyle\sum_{N=1}^{\infty}\left(\dfrac{\|\varphi^{\prime}\sigma^{N}\|_{L^{\infty}(dx)}^{2}}{N}\right)^{l_{1}}<\infty.$

  2. 2.

The function $NH_{N}(x,y)\dfrac{\varphi^{\prime}(x)-\varphi^{\prime}(y)}{x-y}$
is bounded with respect to $(x,y,N)$.

  3. 3.

The empirical measure $L_{N}(0)$ converges weakly to a measure $\mu_{0}$ as
$N$ goes to infinity almost surely, and

$\displaystyle
C_{0}=\sup_{N>0}\langle\varphi,L_{N}(0)\rangle=\sup_{N>0}\dfrac{1}{N}\sum_{i=1}^{N}\varphi\left(x_{i}^{N}(0)\right)<\infty.$

  4. 4.

There exists a sequence $\\{\tilde{f}_{k}\\}_{k\in\mathbb{N}}$ of
$C^{2}(\mathbb{R})$ functions such that it is dense in the space
$C_{0}(\mathbb{R})$ of continuous functions vanishing at infinity and that
$\tilde{f}_{k}^{\prime}(x)\sigma^{N}(x)$ satisfies, for some positive integer
$l_{2}\geq 2$,

$\displaystyle\psi(k)=\sum_{N=1}^{\infty}\left(\dfrac{\|\tilde{f}_{k}^{\prime}\sigma^{N}\|_{L^{\infty}(dx)}^{2}}{N}\right)^{l_{2}}<\infty.$

  5. 5.

There exist continuous functions $b(x)$, $\sigma(x)$ and $H(x,y)$, such that
$b_{N}(x)$ converges to $b(x)$, $\sigma^{N}(x)$ converges to $\sigma(x)$ and
$NH_{N}(x,y)$ converges to $H(x,y)$ uniformly as $N$ tends to infinity.

Then the sequence $\\{L_{N}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ is relatively
compact in $C([0,T],M_{1}(\mathbb{R}))$, i.e., every subsequence has a further
subsequence that converges in $C([0,T],M_{1}(\mathbb{R}))$ almost surely.
Furthermore, any limit measure $\mu$ in $C([0,T],M_{1}(\mathbb{R}))$ satisfies
the equation

$\displaystyle\langle f,\mu_{t}\rangle=$ $\displaystyle\langle
f,\mu_{0}\rangle+\int_{0}^{t}\langle f^{\prime}b,\mu_{s}\rangle
ds+\dfrac{1}{2}\int_{0}^{t}\langle f^{\prime\prime}\sigma^{2},\mu_{s}\rangle
ds+\dfrac{1}{2}\int_{0}^{t}\left[\iint_{\mathbb{R}^{2}}\dfrac{f^{\prime}(x)-f^{\prime}(y)}{x-y}H(x,y)\mu_{s}(dx)\mu_{s}(dy)\right]ds,$
(61)

for all $f\in C_{b}^{2}(\mathbb{R})$ such that $f^{\prime}(x)b(x)$,
$f^{\prime}(x)\sigma(x)$, $f^{\prime\prime}(x)(\sigma(x))^{2}$ and
$\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}H(x,y)$ are bounded as well as
$\|f^{\prime\prime}(\cdot)(\sigma^{N}(\cdot))^{2}-f^{\prime\prime}(\cdot)(\sigma(\cdot))^{2}\|_{L^{\infty}}\to
0$ as $N\to\infty$.

######  Remark 15.

The equation (61) for limit measures with the test function $f=(z-x)^{-1}$ for
$z\in\mathbb{C}_{+}$ was derived in [65]. Indeed, the computation there is
valid for all $f\in C_{b}^{2}(\mathbb{R})$.

We would like to point out that the conditions for the uniqueness of the
solution to (58) and (61) are still unknown. The fluctuations of the sequence
$\\{L_{N}(t),t\in[0,T]\\}_{N\in\mathbb{N}}$ around its limits was studied in
[66]. We refer the CLT in [66, Theorem 2.1] for details.

There is a huge literature on related interacting particle systems,
particularly on those related to Bessel processes. For more details, we refer
to the survey papers [21, 71] and the recent book [34].

##  5 Matrix-valued stochastic processes driven by fractional Brownian motion

A common feature of the matrix-valued stochastic processes discussed so far is
that they are all driven by independent Brownian motions. In contrast, the
study of matrix-valued SDEs driven by fractional Brownian motions has a
shorter history and is relatively limited.

Recall that a centred Gaussian process $B=\\{B(t),t\geq 0\\}$ is called
fractional Brownian motion with Hurst parameter $H\in(0,1)$ if it has the
covariance function

$\displaystyle\mathbb{E}[B(t)B(s)]=\dfrac{1}{2}\big{(}s^{2H}+t^{2H}-|t-s|^{2H}\big{)}.$

We refer the reader to [52] for more details.

To our best knowledge, the first paper in this area is [53], where the real
symmetric matrix fractional Brownian motion was introduced and studied.

######  Definition 2.

Let $\\{B_{i,j}(t),1\leq i\leq j\leq N\\}$ be a family of i.i.d. fractional
Brownian motions with Hurst parameter $H\in(0,1)$. Let
$H^{N}(t)=\left(H_{k,l}^{N}(t)\right)_{1\leq k\leq l\leq N}$ be a real
symmetric $N\times N$ matrix-valued process whose entries are

$H_{k,l}^{N}(t)=B_{k,l}(t){\large 1}_{\\{k<l\\}}+\sqrt{2}B_{l,l}(t){\large
1}_{\\{k=l\\}}.$

Then $B(t)$ is called the real symmetric matrix fractional Brownian motion
with Hurst parameter $H$.

It is natural to consider the eigenvalue processes as we have done for the
matrix Brownian motion. For $1\leq i\leq N$, denote by $\varPhi_{i}$ the
function that maps a $N\times N$ real symmetric matrix to its $i$-th largest
eigenvalue. The following results can be found in [53].

######  Theorem 13 ([53], Theorems 4.1 and 5.2).

Let $X^{N}(0)$ be a real symmetric $N\times N$ deterministic matrix and let
$X^{N}(t)=X^{N}(0)+H^{N}(t)$, where $H^{N}(t)$ is defined in Definition 2 with
Hurst parameter $H\in(1/2,1)$. Let
$\lambda_{1}^{N}(t)\geq\lambda_{2}^{N}(t)\geq\cdots\geq\lambda_{N}^{N}(t)$ be
the ordered eigenvalue processes of $X^{N}(t)$. Denote the first collision
time of the eigenvalue processes by

$\displaystyle\tau_{N}=\inf\left\\{t>0:\exists\ i\neq
j,~{}\lambda_{i}^{N}(t)=\lambda_{j}^{N}(t)\right\\}.$

Then $\mathbb{P}\left(\tau_{N}=+\infty\right)=1$. Furthermore, the ordered
eigenvalue processes $\lambda_{1}^{N}(t),\cdots,\lambda_{N}^{N}(t)$ satisfy

$\displaystyle\lambda_{i}^{N}(t)=\lambda_{i}^{N}(0)+\left(\sum_{k\leq
l}\int_{0}^{t}\dfrac{\partial\varPhi_{i}(X^{N}(s))}{\partial B_{k,h}}\delta
B_{k,h}(s)\right)+2H\sum_{j:j\neq
i}\int_{0}^{t}\dfrac{s^{2H-1}}{\lambda_{i}^{N}(s)-\lambda_{j}^{N}(s)}ds.$ (62)

The main tool used in [53] is the fractional calculus and Malliavin calculus,
for which we refer to [60] and [52] respectively. The non-collision property
was obtained in [53, Theorem 4.1] by establishing an upper bound for negative
moments of the difference of eigenvalue processes. The equation (62) for
eigenvalue processes was derived in [53, Theorem 5.2] by employing a
multidimensional version of the Itô’s formula for the Skorohod integral ([53,
Theorem 3.1], see also [56, Theorem 2]).

######  Remark 16.

The complex version of Theorem 13 can be obtained by the same argument.

######  Remark 17.

Unlike the Brownian motion case ($H=1/2$), the Lévy characterization theorem
for fractional Brownian motion [28, Theorem 3.1] is not applicable here, and
it is conjectured that the second term in the right hand side of (62) is even
not be Gaussian, see [53, Remark 5.3].

For the sequence of eigenvalue processes $\\{\lambda_{i}^{N}(t)\\}_{1\ lei\leq
N}$ in Theorem 13, denote the empirical measure of the normalized eigenvalue
processes by

$\displaystyle
L_{N}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}^{N}(t)/\sqrt{N}}(dx).$

For the case $H\in(1/2,1)$, the convergence in probability of the sequence
$\\{L_{N}(t)\\}_{N\in\mathbb{N}}$ to the semi-circle law was established in
[55] under null initial condition $X^{N}(0)=0$ by using Malliavin calculus and
the tightness argument used in the proof of Theorem 3.

Similar to the free Brownian motion, the non-commutative fractional Brownian
motion with Hurst parameter $H\in(0,1)$ was introduced in [51, Definition 3.1]
as a centred semicircular process $\\{S^{H}(t),t\geq 0\\}$ with covariance
function

$\displaystyle\tau\big{(}S^{H}(t)S^{H}(s)\big{)}=\dfrac{1}{2}\big{(}t^{2H}+s^{2H}-|t-s|^{2H}\big{)},$

where $\tau$ is the trace on the non-commutative probability space. The semi-
circle law $\\{\mu_{t}^{H},t\geq 0\\}$ established in [55] is the law of a
non-commutative fractional Brownian motion with Hurst parameter $H$.

The results of [55] were extended to normalized real symmetric matrix Gaussian
processes with general initial condition in [31]. In particular, the real
symmetric matrix Gaussian processes considered in [31] include the real
symmetric matrix fractional Brownian motion with $H\in(0,1)$.

The almost sure convergence of the sequence of eigenvalue empirical measure
valued processes of Wigner-type matrices, whose entries are generated from the
solution of $1$-dimensional Stratonovich SDE

$\displaystyle dX_{t}=\sigma(X_{t})\circ dB_{t}^{H}+b(X_{t})dt,~{}t\geq 0,$
(63)

to the semi-circle law was established in [64, Theorem 3.1, 3.2] for
$H\in(1/2,1)$ by using fractional calculus and the argument used in the proof
of Theorem 2. [64] also studied the convergence of the sequence of eigenvalue
empirical measure-valued processes of the complex analogue and the real
symmetric matrix with local dependent entries.

For any test function $f\in C^{4}(\mathbb{R})$ whose fourth derivative has
polynomial growth, the random fluctuation

$\displaystyle\mathcal{L}_{N}^{\circ}(f)(t)=N\big{(}\langle
f,L_{N}(t)\rangle-\mathbb{E}\left[\langle f,L_{N}(t)\rangle\right]\big{)}.$

of the sequence $\\{L_{N}(t)\\}_{N\in\mathbb{N}}$ of normalized real symmetric
matrix Gaussian process around its expectation was studied in [14, Theorem
2.3].

The fractional version of Wishart process was studied in [56] for
$H\in(1/2,1)$. Let $B(t)$ be a $N\times p$ matrix whose entries are i.i.d.
standard real fractional Brownian motion (matrix fractional Brownian motion)
with Hurst parameter $H\in(1/2,1)$. Let
$X^{N}(t)=(B(t)+A)(B(t)+A)^{\intercal}$, where $A$ is a $N\times p$ real
deterministic matrix, is the fractional Wishart process. The following result
provides are the non-collision property of eigenvalue processes as well as the
equations satisfied by the eigenvalue processes.

######  Theorem 14 ([56], Theorems 3 and 4).

Let $X^{N}(t)$ be a fractional Wishart process with $H\in(1/2,1)$. Let
$\lambda_{1}^{N}(t)\geq\lambda_{2}^{N}(t)\geq\cdots\geq\lambda_{N}^{N}(t)$ be
the ordered eigenvalue processes of $X^{N}(t)$. Denote the first collision
time of the eigenvalue processes by

$\displaystyle\tau_{N}=\inf\left\\{t>0:\exists\ i\neq
j,~{}\lambda_{i}^{N}(t)=\lambda_{j}^{N}(t)\right\\}.$

Then $\mathbb{P}\left(\tau_{N}=+\infty\right)=1$. Furthermore, the ordered
eigenvalue processes $\lambda_{1}^{N}(t),\ldots,\lambda_{N}^{N}(t)$ satisfy

$\lambda_{i}^{N}(t)=\lambda_{i}^{N}(0)+\left(\sum_{k=1}^{N}\sum_{l=1}^{p}\int_{0}^{t}\dfrac{\partial\varPhi_{i}(X^{N}(s))}{\partial
B_{k,l}}\delta B_{k,l}(s)\right)+2H\int_{0}^{t}\left(p+\sum_{j:j\neq
i}\dfrac{\lambda_{i}^{N}(s)+\lambda_{j}^{N}(s)}{\lambda_{i}^{N}(s)-\lambda_{j}^{N}(s)}\right)s^{2H-1}ds.$

For the sequence of eigenvalue processes $\\{\lambda_{i}^{N}(t)\\}_{1\leq
i\leq N}$ in Theorem 14, denote the empirical measure of the normalized
eigenvalue processes by

$\displaystyle
L_{N}(t)(dx)=\dfrac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}^{N}(t)/N}(dx).$

For the case $H\in(1/2,1)$, the convergence in probability of the sequence
$\\{L_{N}(t)\\}_{N\in\mathbb{N}}$ to the Marčenko-Pastur law was established
in [56] under null initial condition $X^{N}(0)=0$ by using Malliavin calculus
and the tightness argument used in the proof of Theorem 3. As an extension,
the convergence in probability of the sequence of eigenvalue empirical measure
valued processes of Wishart-type matrices, whose entries are generated from
the solution of (63), to the Marčenko-Pastur law was established in [64,
Theorems 3.1, 3.2] by using fractional calculus and the argument used in the
proof of Theorem 2.

##  6 Matrix-valued stochastic processes driven by Brownian sheet

In stochastic analysis, multi-parameter processes, which are also known as
random fields, are a natural extension of one-parameter processes. Various
interactions exist between the theory of multi-parameter processes and other
disciplines, such as analysis, algebra, mathematical statistics and
statistical mechanics. The most important multi-parameter process is the
Brownian sheet. Recall that the standard $1$-dimensional ($2$-parameter)
Brownian sheet $\\{B(s,t),(s,t)\in\mathbb{R}_{+}^{2}\\}$ is a centred Gaussian
random field with covariance function

$\displaystyle\mathbb{E}\left[B(s_{1},t_{1})B(s_{2},t_{2})\right]=(s_{1}\wedge
s_{2})(t_{1}\wedge t_{2}).$

We refer to [39] for more details on multi-parameter processes, in particular
for properties of the Brownian sheet.

As shown in the review, there is a rich literature on eigenvalue processes of
matrix-valued one-parameter processes. In contrast, the study on matrix-valued
multi-parameter processes is just beginning. To our best knowledge, [63] is
the only reference on this topic.

Let $\\{B_{i,j}(s,t),s,t\in\mathbb{R}_{+}\\}_{i,j\geq 1}$ be a family of
independent standard $1$-dimensional Brownian sheets. The $N\times N$ real
symmetric matrix-valued stochastic process
$H^{N}(s,t)=\left(H_{i,j}^{N}(s,t)\right)_{1\leq i,j\leq N}$ with entries

$\displaystyle H_{i,j}^{N}(s,t)=\begin{cases}B_{i,j}(s,t),&i<j,\\\
\sqrt{2}B_{i,i}(s,t),&i=j,\end{cases}$ (64)

is known as the real symmetric matrix Brownian sheet. Let $A^{N}$ be a
$N\times N$ real symmetric deterministic matrix with distinct eigenvalues, and
let

$\displaystyle X^{N}(s,t)=H^{N}(s,t)+A^{N}.$ (65)

In the case of symmetric matrix Brownian motion, multi-dimensional Itô’s
formula for Brownian motions plays a key role when deriving the system of SDEs
(4) for eigenvalue processes (see [1, Theorem 4.3.2]). The system of SDEs (62)
for eigenvalue processes of real symmetric matrix fractional Brownian motion
also heavily relies on the multi-dimensional Itô’s formula for the Skorohod
integral with respect to fractional Brownian motion (see [53]). For the case
of Brownian sheet, though the one-dimensional Itô’s formula has been
established in [10], the multi-dimensional version was not available until it
was obtained in [63] by using the stochastic calculus on the plane developed
in [10]. Using the multi-dimensional Itô’s formula, [63] derived the system of
SPDEs for the ordered eigenvalue processes of $X^{N}(s,t)$. Moreover, the
convergence of the sequence of the eigenvalue empirical measure processes of
$\frac{1}{\sqrt{d}}X^{N}(s,t)$ was also studied in [63].

##  7 Open problems

For the system of eigenvalue processes (53), the conditions of the uniqueness
to (58) are still unknown. The uniqueness can be obtained under proper
conditions for some special matrix-valued processes. It was established in [9]
for real symmetric and complex Hermitian matrix Brownian motion and complex
Wishart process using large deviation technique. For real symmetric and
complex Hermitian matrix Brownian motion, [1] obtained the uniqueness by
analyzing the Stieltjes transform of the limit measure process. For real
symmetric matrix OU process, the uniqueness was established by computing the
moments of limiting measure in [12]. In [59], the uniqueness was established
also by analyzing the Stieltjes transform of the limit measure process. For
the Dyson’s Brownian motion with general drift, the uniqueness was established
in [43] by the entropy technique. In [11], the uniqueness was established by
transferring the equation of the Stieltjes transform of the limit measure to a
PDE, which was solved via Fourier transform. It was established in [65] the
uniqueness of self-similar solution for real symmetric and complex Hermitian
matrix Brownian motion, Wishart process and Laguerre process. However, [45,
Proposition 5, 6] provided an example of (58) which have at least two
solutions. It is also of interest to know the conditions under which the
equation (58) has unique solution that is supported on $[0,+\infty)$.
Similarly, the conditions for the uniqueness of the limiting equation (61) are
still unknown.

For the fractional Wishart process considered in Theorem 14, the fluctuation
of the sequence of eigenvalue empirical measure processes around its limiting
measure process is still unknown.

For the symmetric matrix valued Brownian sheet, the fluctuation of the
sequence of eigenvalue empirical measure processes around its limiting measure
process is also unknown.

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