Functional Equations Solving Initial-Value Problems of Complex Burgers-Type
Equations

Functional Equations Solving Initial-Value Problems
of Complex Burgers-Type Equations
for One-Dimensional Log-Gases††This paper is a contribution to the Special
Issue on Non-Commutative Algebra, Probability and Analysis in Action. The full
collection is available at https://www.emis.de/journals/SIGMA/non-commutative-
probability.html

Taiki ENDO a, Makoto KATORI a and Noriyoshi SAKUMA b

T. Endo, M. Katori and N. Sakuma

a) Department of Physics, Faculty of Science and Engineering, Chuo University,
a) Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
b) Graduate School of Natural Sciences, Nagoya City University,
b) Mizuho-ku, Nagoya, Aichi 467-8501, Japan<EMAIL_ADDRESS>
Received February 24, 2022, in final form June 23, 2022; Published online July
02, 2022

We study the hydrodynamic limits of three kinds of one-dimensional stochastic
log-gases known as Dyson’s Brownian motion model, its chiral version, and the
Bru–Wishart process studied in dynamical random matrix theory. We define the
measure-valued processes so that their Cauchy transforms solve the complex
Burgers-type equations. We show that applications of the method of
characteristic curves to these partial differential equations provide the
functional equations relating the Cauchy transforms of measures at an
arbitrary time with those at the initial time. We transform the functional
equations for the Cauchy transforms to those for the $R$-transforms and the
$S$-transforms of the measures, which play central roles in free probability
theory. The obtained functional equations for the $R$-transforms and the
$S$-transforms are simpler than those for the Cauchy transforms and useful for
explicit calculations including the computation of free cumulant sequences.
Some of the results are argued using the notion of free convolutions.

stochastic log-gases; complex Burgers-type equations; functional equations;
Cauchy transforms; $R$-transforms; $S$-transforms; free probability and free
convolutions

82C22; 60B20; 44A15; 46L54

##  1 Introduction and results

###  1.1 Transformations of measures and complex Burgers-type equations

Among a variety of recent developments in random matrix theory [1, 2, 35, 55],
we study in this paper an intersection of two important topics; time-dependent
random matrix models [19, 33, 46, 50, 52] and free probability theory [8, 11,
12, 56, 61, 69, 71].

The cases of the Gaussian unitary ensemble (GUE) and the chiral GUE are
typical eigenvalue distributions on $\mathbb{R}$ of Hermitian random matrices,
and their dynamical extensions are described by systems of stochastic
differential equations (SDEs) called Dyson’s Brownian motion model [33] and
its chiral version [20, 50, 52]. (Compare the SDEs (2.4) for the chiral
version with (2.1) for the original one. The two terms in the parentheses in
the last term of the r.h.s. in (2.4) are chiral to each other.)

For $S\subset\mathbb{R}$, let ${\cal P}^{0}(S)$ be a set of all Borel
probability measures on $S$ with bounded supports equipped with the weak
topology. For an arbitrary but fixed $T>0$, ${\cal C}\big{(}[0,T]\to{\cal
P}^{0}(S)\big{)}$ denotes the space of continuous processes defined in the
time period $[0,T]$ realized in ${\cal P}^{0}(S)$.

In this manuscript, we consider the hydrodynamic limits of Dyson’s Brownian
motion model and its chiral version with an additional parameter
$\lambda\in\mathbb{R}_{+}:=[0,\infty)$ as elements of ${\cal
C}\big{(}[0,\infty)\allowbreak\to{\cal P}^{0}(\mathbb{R})\big{)}$, which are
denoted by $({\rm w}_{t})_{t\geq 0}$ and $({\rm w}_{\lambda,t})_{t\geq 0}$,
respectively. First, we assume that the initial probability measure ${\rm
w}_{0}$ and ${\rm w}_{\lambda,0}$ are in ${\cal P}^{0}(\mathbb{R})$ and the
Cauchy transforms of measures,

$\displaystyle G_{\mu}(z):=\int_{S}\frac{\mu({\rm d}x)}{z-x},\qquad
z\in\mathbb{C}^{+}:=\\{z\in\mathbb{C}\colon\operatorname{Im}z>0\\},$ (1.1)

are well defined satisfying the condition
$\lim_{y\uparrow\infty}\sqrt{-1}yG_{\mu}\big{(}\sqrt{-1}y\big{)}=1$ for
$\mu={\rm w}_{t}$ and ${\rm w}_{\lambda,t}$, $t\geq 0$. It is known that [14,
16, 17, 59, 60, 63] given $G_{{\rm w}_{0}}$ and $G_{{\rm w}_{\lambda,0}}$
obtained by ${\rm w}_{0}$ and ${\rm w}_{\lambda,0}$, respectively, $(G_{{\rm
w}_{t}})_{t\geq 0}$ and $(G_{{\rm w}_{\lambda,t}})_{t\geq 0}$ are uniquely
determined by the solutions of the following partial differential equations
(PDEs):

$\displaystyle\frac{\partial G_{{\rm w}_{t}}(z)}{\partial t}+G_{{\rm
w}_{t}}(z)\frac{\partial G_{{\rm w}_{t}}(z)}{\partial z}=0,\qquad t\geq 0,$
(1.2) $\displaystyle\frac{\partial G_{{\rm w}_{\lambda,t}}(z)}{\partial
t}+\bigg{(}G_{{\rm
w}_{\lambda,t}}(z)-\frac{1-\lambda}{2z}\bigg{)}\frac{\partial G_{{\rm
w}_{\lambda,t}}(z)}{\partial z}+\frac{1-\lambda}{2z^{2}}G_{{\rm
w}_{\lambda,t}}(z)=0,\qquad t\geq 0.$ (1.3)

Equation (1.2) is known as the complex Burgers equation in the inviscid limit.
It is obvious that when $\lambda=1$ (1.3) is reduced to (1.2) and hence, if
${\rm w}_{0}={\rm w}_{1,0}$, then

$\displaystyle{\rm w}_{t}={\rm w}_{1,t},\qquad t\geq 0.$ (1.4)

In other words, the process $({\rm w}_{\lambda,t})_{t\geq 0}$ is the one-
parameter ($\lambda\in\mathbb{R}_{+}$) extension of $({\rm w}_{t})_{t\geq 0}$.

We denote by ${\bf 1}_{(E)}$ the indicator function of an event $E$; ${\bf
1}_{(E)}=1$ if $E$ occurs, and ${\bf 1}_{(E)}=0$ otherwise. As a special case,
Kronecker’s delta is defined by $\delta^{ij}:={\bf 1}_{(i=j)}$,
$i,j\in\mathbb{N}$. Moreover, the $\sigma$-algebra of Borel sets on $S$ is
denoted by ${\cal B}(S)$ and we define $1_{B}(x):={\bf 1}_{(x\in B)}$ for
$B\in{\cal B}(S)$.

For $p>0$ and $\mu\in{\cal P}^{0}(\mathbb{R})$, the $p$-th push-forward
measure $\mu^{(p)}$ is defined by [62]

$\displaystyle\mu^{(p)}(B):=\int_{\mathbb{R}}1_{B}(|x|^{p})\mu({\rm
d}x),\qquad B\in{\cal B}((0,\infty)).$ (1.5)

We define

$\displaystyle{\rm m}_{\lambda,t}:={\rm w}_{\lambda,t}^{(2)}\in{\cal
P}^{0}(\mathbb{R}_{+}),\qquad t\geq 0,$ (1.6)

provided the matching of initial measures ${\rm m}_{\lambda,0}={\rm
w}_{\lambda,0}^{(2)}$. Note that combining this definition with (1.4) we have
the equality

$\displaystyle{\rm w}_{t}^{(2)}={\rm m}_{1,t},\qquad t\geq 0.$ (1.7)

We can show that by (1.6) the PDE (1.3) is transformed to the following
equation for the Cauchy transform $G_{{\rm m}_{\lambda,t}}(z)$ of ${\rm
m}_{\lambda,t}$, $\lambda\in\mathbb{R}_{+}$,

$\displaystyle\frac{\partial G_{{\rm m}_{\lambda,t}}(z)}{\partial
t}+\big{\\{}2zG_{{\rm m}_{\lambda,t}}(z)-(1-\lambda)\big{\\}}\frac{\partial
G_{{\rm m}_{\lambda,t}}(z)}{\partial z}+G_{{\rm
m}_{\lambda,t}}(z)^{2}=0,\qquad t\geq 0.$ (1.8)

It should be noted that this PDE is obtained when we consider the hydrodynamic
limit of the system of SDEs [21] known as the Bru–Wishart process in
multivariate stochastic calculus [20, 50] and as the Laguerre process in
dynamical random matrix theory [52]. The PDEs (1.3) and (1.8) describe the
large-number limits of colors of the systems with the chiral symmetry in the
quantum chromodynamics (QCD) in high energy physics [14, 15, 45, 53, 59, 60,
64, 67, 68]. (See also [1, 35].) We call (1.3) and (1.8) complex Burgers-type
equations [16, 17, 34, 36], in which drift terms are modified and “external-
force terms” are added compared with (1.2). (See Section 2.1 below.)

The simplest initial probability measure in ${\cal P}^{0}(S)$, $S=\mathbb{R}$
or $\mathbb{R}_{+}$, is the single delta measure $\delta_{0}$ at the origin.
We regard the solution of the complex Burgers-type equation starting from
$G_{\delta_{0}}(z)=1/z$ as the fundamental solution and denote the obtained
measure-valued process as $\big{(}\mu_{t}^{0}\big{)}_{t\geq 0}$ with a
superscript 0. We can show that (see, for instance, [56])

$\displaystyle G_{{\rm
w}^{0}_{t}}(z)=\frac{1}{2t}\Big{[}z-\sqrt{z^{2}-4t}\Big{]},\qquad t\geq 0,$
$\displaystyle G_{{\rm
m}^{0}_{\lambda,t}}(z)=\frac{1}{2tz}\Big{[}z\\!+t(1\\!-\lambda)\\!-\sqrt{(z\\!-x^{+}_{\lambda,t})(z\\!-x^{-}_{\lambda,t})}\,\Big{]}\qquad\text{with}\quad
x^{\pm}_{\lambda,t}=t\big{(}1\pm\sqrt{\lambda}\big{)}^{2},\quad t\geq 0,$

and they determine the time-dependent measures for $t\geq 0$ as

$\displaystyle{\rm w}^{0}_{t}({\rm d}x)=\frac{1}{2\pi
t}\sqrt{4t-x^{2}}\,1_{[-2\sqrt{t},2\sqrt{t}]}(x)\,{\rm d}x,$ (1.9)
$\displaystyle{\rm m}^{0}_{\lambda,t}({\rm
d}x)=\max(0,1-\lambda)\delta_{0}({\rm d}x)+\frac{1}{2\pi
tx}\sqrt{(x-x^{-}_{\lambda,t})(x^{+}_{\lambda,t}-x)}\,1_{[x^{-}_{\lambda,t},x^{+}_{\lambda,t}]}(x)\,{\rm
d}x,$ (1.10)

respectively. The measure (1.9) is known as the centered Wigner’s semicircle
distribution with variance $t$ and (1.10) is as the two-parametric
Marcenko–Pastur distribution with parameters $\lambda$ and $t$ [16, 17, 34].
For the fundamental solutions explicitly given by (1.9) and (1.10), it is easy
to verify the well-known equality $\big{(}{\rm w}_{t}^{0}\big{)}^{(2)}={\rm
m}^{0}_{1,t}$, $t\geq 0$. The equality (1.7) mentioned above generalizes it
for any initial probability measure with bounded support satisfying ${\rm
w}_{0}^{(2)}={\rm m}_{1,0}$.

Let ${\cal P}^{0}_{{\rm s}}(\mathbb{R})$ be the set of all symmetric Borel
probability measures on $\mathbb{R}$ (i.e., $\mu(B)=\mu(-B)$, $B\in{\cal
B}((0,\infty))$ for $\mu\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$) and define the
symmetric Bernoulli delta measure with displacement $2a>0$ as

$\displaystyle{\rm d}_{a}:=\frac{1}{2}(\delta_{-a}+\delta_{a})\in{\cal
P}^{0}_{{\rm s}}(\mathbb{R}).$ (1.11)

The processes $({\rm w}_{t})_{t\geq 0}$ and $({\rm w}_{\lambda,t})_{t\geq 0}$
starting from ${\rm d}_{a}$, which are defined as the time-dependent
probability measures so that their Cauchy transforms solve the PDEs (1.2) and
(1.3) under the initial condition $G_{{\rm
d}_{a}}(z)=z/\big{(}z^{2}-a^{2}\big{)}$, were reported in [3, 57, 72] and in
[34], respectively. These solutions show a dynamical phase transition, in
which the positive parameter $a$ controls the transition observed at a
critical time. The singularity associated with this phase transition is very
interesting and important, since it gives the mean-field description of the
spontaneous chiral symmetry breaking [34]. But, due to this singularity, the
solutions are much more complicated compared with the fundamental solutions
(1.9) and $({\rm w}_{\lambda,t}^{0})_{t\geq 0}$ obtained from (1.10) by (1.6)
(see [34, Remark 3]). It had seemed to be difficult to argue general
properties of these measure-valued processes. (See [38] for the exact
solutions for other initial probability measures.)

In the present paper, we solve the initial-value problem for these measure-
valued processes $(\mu_{t})_{t\geq 0}=({\rm w}_{t})_{t\geq 0}$, $({\rm
w}_{\lambda,t})_{t\geq 0}$, and $({\rm m}_{\lambda,t})_{t\geq 0}$. We found
that, by the method of characteristic curves, each of the PDEs of the Cauchy
transforms $(G_{\mu_{t}})_{t\geq 0}$, given by (1.2), (1.3), and (1.8), is
transformed to a functional equation which relates $G_{\mu_{t}}(z)$ at an
arbitrary time $t>0$ with the initial function $G_{\mu_{0}}(z)$. The result
for $(G_{{\rm w}_{t}})_{t\geq 0}$ given as Proposition 3.1$(i)$ is well known
and found in literature. The results for $(G_{{\rm m}_{\lambda,t}})_{t\geq 0}$
and $(G_{{\rm w}_{\lambda,t}})_{t\geq 0}$ given as Proposition 3.1$(ii)$ and
$(iii)$, respectively, are new, but they are complicated and do not seem to be
useful for explicit calculations. On the other hand, in free probability
theory [4, 8, 11, 12, 40, 56, 61, 62, 69], we learn other importance
transformations of probability measures different from the Cauchy transform;
the $R$-transform and the $S$-transform. They define new types of convolutions
of probability measures called free convolutions. (A brief review is given
shortly.) We applied these transformations to our functional equations. The
results for the $R$-transforms denoted by $(R_{\mu_{t}})_{t\geq 0}$ and the
$S$-transforms by $(S_{\mu_{t}})_{t\geq 0}$ are given by Theorems 1.1 and 1.5,
respectively. The obtained functional equations are expressed using the
$R$-transforms and the $S$-transforms of the fundamental solutions and much
simplified. In particular, as shown in Theorem 1.1$(i)$, the functional
equation (1.18) implies the decomposition formula (1.19) with respect to the
free additive convolution (see also Remarks 1.2 and 1.3). Theorem 1.5$(i)$
gives the simple but useful relationship among $({\rm w}_{\lambda,t})_{t\geq
0}$, $({\rm m}_{\lambda,t})_{t\geq 0}$, and ${\rm d}$ (see Remark 1.6). The
functional equations are also useful for explicit calculations as demonstrated
by Propositions 1.7 and 1.8. There the solutions for the processes $({\rm
w}_{\lambda,t})_{t\geq 0}$ starting from ${\rm d}_{a}$, $a>0$ and $({\rm
m}_{\lambda,t})_{t\geq 0}$ from $\delta_{b}$, $b>0$ are shown, which are much
simpler than the corresponding solutions of the Cauchy transformations
reported in [3, 57, 72] and [34]. Comparing the results for $({\rm
w}_{\lambda,t})_{t\geq 0}$ and those for $({\rm m}_{\lambda,t})_{t\geq 0}$,
the latter is simpler than the former. In the original complex Burgers-type
equations, it is obvious at the PDE (1.3) for $({\rm w}_{\lambda,t})_{t\geq
0}$ that $({\rm w}_{\lambda,t})_{t\geq 0}$ is a one-parameter extension
$(\lambda\in\mathbb{R}_{+})$ of $({\rm w}_{t})_{t\geq 0}$, but it is not at
the PDE (1.8) for $({\rm m}_{\lambda,t})_{t\geq 0}$. Our results shows,
however, that the transformation from $({\rm w}_{\lambda,t})_{t\geq 0}$ to
$({\rm m}_{\lambda,t})_{t\geq 0}$ by the second push-forward transform is
effective to make practical problems be simpler and solvable. Further study of
the $p$-th push-forward transforms (1.5) will be important.

Given a probability measure $\mu\in{\cal P}^{0}(S)$, let $\tau_{n}(\mu)$
represent the $n$-th moment of $\mu$, $n\in\mathbb{N}$ and define the moment
generating function by

$\displaystyle\Psi_{\mu}(z):=\int_{S}\frac{xz}{1-xz}\mu({\rm
d}x)=\sum_{n=1}^{\infty}\tau_{n}(\mu)z^{n}.$ (1.12)

The Cauchy transform $G_{\mu}$ of $\mu$ is related to $\Psi_{\mu}$ by

$\displaystyle\Psi_{\mu}(z)=\frac{G_{\mu}(1/z)}{z}-1\Longleftrightarrow
G_{\mu}(1/z)=z(\Psi_{\mu}(z)+1).$ (1.13)

We write the inverse function of $G_{\mu}$ as $G_{\mu}^{\langle-1\rangle}$.
The $R$-transform of $\mu$ is then defined by

$\displaystyle R_{\mu}(z):=zG_{\mu}^{\langle-1\rangle}(z)-1\Longleftrightarrow
G_{\mu}^{\langle-1\rangle}(z)=\frac{R_{\mu}(z)+1}{z}.$ (1.14)

This is the generating function of free cumulants
$\kappa_{n}=\kappa_{n}(\mu)$, $n\in\mathbb{N}$,

$\displaystyle R_{\mu}(z)=\sum_{n=1}^{\infty}\kappa_{n}(\mu)z^{n}.$

(Notice that in the free probability literature the function ${\cal
R}_{\mu}(z):=R_{\mu}(z)/z$ is also used and referred to as “the $R$-transform
of $\mu$”. See, for instance, [61, equation (16.8)].) The relations between
the moments $\\{\tau_{n}(\mu)\\}_{n\in\mathbb{N}}$ in (1.12) and the free
cumulants $\\{\kappa_{n}(\mu)\\}_{n\in\mathbb{N}}$ are given by

$\displaystyle\kappa_{1}(\mu)=\tau_{1}(\mu),$
$\displaystyle\kappa_{2}(\mu)=\tau_{2}(\mu)-\tau_{1}(\mu)^{2},$
$\displaystyle\kappa_{3}(\mu)=\tau_{3}(\mu)-3\tau_{2}(\mu)\tau_{1}(\mu)+2\tau_{1}(\mu)^{3},$
$\displaystyle\kappa_{4}(\mu)=\tau_{4}(\mu)-4\tau_{3}(\mu)\tau_{1}(\mu)-2\tau_{2}(\mu)^{2}+10\tau_{2}(\mu)\tau_{1}(\mu)^{2}-5\tau_{1}(\mu)^{4},$
$\displaystyle\dots\dots\dots\dots,$

which are generally different from the relations satisfied in the “classical
probability theory”. For two probability measures $\mu$, $\nu$, the free
additive convolution of them is denoted by $\mu\boxplus\nu$ and defined by [8]

$\displaystyle R_{\mu\boxplus\nu}(z)=R_{\mu}(z)+R_{\nu}(z),$

which implies

$\displaystyle\kappa_{n}(\mu\boxplus\nu)=\kappa_{n}(\mu)+\kappa_{n}(\nu),\qquad
n\in\mathbb{N}.$

For $\mu\in{\cal P}^{0}(\mathbb{R}_{+})$ with $\mu(\\{0\\})<1$, the moment
generating function $\Psi_{\mu}(z)$ defined by (1.13) has a unique inverse

$\displaystyle\chi_{\mu}(z):=\Psi_{\mu}^{\langle-1\rangle}(z)$ (1.15)

on the left-half plane $\sqrt{-1}\mathbb{C}^{+}$ [8]. In this case the
$S$-transform of $\mu$ is defined by

$\displaystyle S_{\mu}(z):=\frac{1+z}{z}\chi_{\mu}(z),\qquad
z\in\Psi_{\mu}\big{(}\sqrt{-1}\mathbb{C}^{+}\big{)}.$ (1.16)

For two probability measures $\mu$, $\nu$ having $S$-transforms, the free
multiplicative convolution [8] is defined as the the probability measure
$\mu\boxtimes\nu$ such that

$\displaystyle S_{\mu\boxtimes\nu}(z)=S_{\mu}(z)S_{\nu}(z)$

for $z$ in a common region of
$\Psi_{\mu}\big{(}\sqrt{-1}\mathbb{C}^{+}\big{)}\cup\Psi_{\nu}\big{(}\sqrt{-1}\mathbb{C}^{+}\big{)}$.

The definition of the $S$-transform can be extended for symmetric probability
measures $\mu\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$ [8, 62]. When $\mu\in{\cal
P}^{0}_{{\rm s}}(\mathbb{R})$ with $\mu(\\{0\\})<1$, $\Psi_{\mu}(z)$ has a
unique inverse on
$H:=\\{z\in\mathbb{C}^{-}\colon|\operatorname{Re}z|<|\operatorname{Im}z|\\}$,
$\chi_{\mu}\colon\Psi_{\mu}(H)\to H$ and a unique inverse on
$\widetilde{H}:=\\{z\in\mathbb{C}^{+}\colon|\operatorname{Re}z|<|\operatorname{Im}z|\\}$,
$\widetilde{\chi}_{\mu}\colon\Psi_{\mu}(\widetilde{H})\to\widetilde{H}$, where
$\mathbb{C}^{-}:=\\{z\in\mathbb{C}\colon\operatorname{Im}z<0\\}$. Therefore,
there are two $S$-transforms for $\mu$ given by

$\displaystyle
S_{\mu}(z)=\frac{1+z}{z}\chi_{\mu}(z)\qquad\text{and}\qquad\widetilde{S}_{\mu}(z)=\frac{1+z}{z}\widetilde{\chi}_{\mu}(z),$

and they satisfy

$\displaystyle
S_{\mu}(z)^{2}=\frac{1+z}{z}S_{\mu^{(2)}}(z)\qquad\text{and}\qquad\widetilde{S}_{\mu}(z)^{2}=\frac{1+z}{z}S_{\mu^{(2)}}(z).$

It is known that for a probability measure $\mu\in{\cal
P}^{0}(\mathbb{R}_{+})$, there exists a unique symmetric probability measure
$\mu^{\bf s}\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$ such that

$\displaystyle\int_{\mathbb{R}}f\big{(}x^{2}\big{)}\mu^{\bf s}({\rm
d}x)=\int_{\mathbb{R}}f(x)\mu^{(1/2)}({\rm d}x)$

for every compactly supported continuous function $f$. The probability measure
$\mu^{\bf s}$ is called symmetrization of a probability measure $\mu$. For
details, see [41, p. 134]. We note that the $S$-transform of ${\rm d}_{a}$
defined by (1.11) is given by

$\displaystyle S_{{\rm d}_{a}}(z)=\frac{1}{a}\sqrt{\frac{1+z}{z}},\qquad a>0.$
(1.17)

If the parameter $a=1$, we use the notation ${\rm d}$ for ${\rm d}_{1}$ [62].

The PDFs (1.2), (1.3) and (1.8) for the Cauchy transforms
$(G_{\mu_{t}})_{t\geq 0}$ are transforms as follows: For $(R_{\mu_{t}})_{t\geq
0}$,

$\displaystyle\frac{\partial R_{{\rm w}_{t}}(z)}{\partial t}-z^{2}=0,$
$\displaystyle\frac{\partial R_{{\rm w}_{\lambda,t}}(z)}{\partial
t}-\frac{(1-\lambda)z^{3}}{2(R_{{\rm w}_{\lambda,t}}(z)+1)^{2}}\frac{\partial
R_{{\rm w}_{\lambda,t}}(z)}{\partial z}-z^{2}+\frac{(1-\lambda)z^{2}}{R_{{\rm
w}_{\lambda,t}}(z)+1}=0,$ $\displaystyle\frac{\partial R_{{\rm
m}_{\lambda,t}}(z)}{\partial t}-z^{2}\frac{\partial R_{{\rm
m}_{\lambda,t}}(z)}{\partial z}-z(R_{{\rm m}_{\lambda,t}}(z)+\lambda)=0,$

and for $(S_{\mu_{t}})_{t\geq 0}$,

$\displaystyle\frac{\partial S_{{\rm w}_{t}}(z)}{\partial t}+z^{2}S_{{\rm
w}_{t}}(z)^{2}\frac{\partial S_{{\rm w}_{t}}(z)}{\partial z}+zS_{{\rm
w}_{t}}(z)^{3}=0,$ $\displaystyle\frac{\partial S_{{\rm
w}_{\lambda,t}}(z)}{\partial
t}+z^{2}\bigg{(}1-\frac{1-\lambda}{1+z}\bigg{)}S_{{\rm
w}_{\lambda,t}}(z)^{2}\frac{\partial S_{{\rm w}_{\lambda,t}}(z)}{\partial
z}+z\bigg{\\{}1-\frac{(1-\lambda)(z+2)}{2(1+z)^{2}}\bigg{\\}}S_{{\rm
w}_{\lambda,t}}(z)^{3}=0,$ $\displaystyle\frac{\partial S_{{\rm
m}_{\lambda,t}}(z)}{\partial t}+z(z+\lambda)S_{{\rm
m}_{\lambda,t}}(z)\frac{\partial S_{{\rm m}_{\lambda,t}}(z)}{\partial
z}+(2z+\lambda)S_{{\rm m}_{\lambda,t}}(z)^{2}=0.$

We find that the equation of $(R_{{\rm w}_{t}})_{t\geq 0}$ is extremely simple
and it corresponds to the asymptotic freeness of a random matrix in the GUE
and a deterministic Hermitian matrix (see Theorem 1.1$(i)$, Remarks 1.2 and
1.3 below). On the other hand, the “external-force terms” include both of the
coordinate $z$ and the “fields” in other equations and seem to be more
complicated than the equation (1.8) for $(G_{{\rm m}_{\lambda,t}})_{t\geq 0}$
whose external-force term is simply given by $G_{m_{\lambda,t}}(z)^{2}$.
Theorems 1.1 and 1.5 given below solve the initial-value problems for all of
them.

###  1.2 Results for the $\bm{R}$-transforms

The following are the results for the $R$-transforms.

######  Theorem 1.1.

  1. $(i)$

Assume that ${\rm w}_{0}\in{\cal P}^{0}(\mathbb{R})$. Then,

$\displaystyle R_{{\rm w}_{t}}(z)=R_{{\rm w}_{0}}(z)+R_{{\rm
w}^{0}_{t}}(z),\qquad t\geq 0,$ (1.18)

where $R_{{\rm w}_{t}^{0}}(z)=tz^{2}$, $t\geq 0$. It means that $({\rm
w}_{t})_{t\geq 0}$ is decomposed into the initial probability measure ${\rm
w}_{0}$ and the fundamental solution $\big{(}{\rm w}_{t}^{0}\big{)}_{t\geq 0}$
with respect to the free additive convolution as follows,

$\displaystyle{\rm w}_{t}={\rm w}_{0}\boxplus{\rm w}^{0}_{t},\qquad t\geq 0.$
(1.19)

  2. $(ii)$

Assume that ${\rm m}_{\lambda,0}\in{\cal P}^{0}(\mathbb{R}_{+})$,
$\lambda\in\mathbb{R}_{+}$. Then,

$\displaystyle R_{{\rm m}_{\lambda,t}}(z)=\frac{1}{1-tz}R_{{\rm
m}_{\lambda,0}}\bigg{(}\frac{z}{1-tz}\bigg{)}+R_{{\rm
m}^{0}_{\lambda,t}}(z),\qquad t\geq 0,$ (1.20)

where $R_{{\rm m}_{\lambda,t}^{0}}(z)=\lambda tz/(1-tz)$, $t\geq 0$.

  3. $(iii)$

Assume that ${\rm w}_{\lambda,0}\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$,
$\lambda\in\mathbb{R}_{+}$. Then,

$\displaystyle R_{{\rm w}_{\lambda,t}}(z)+\frac{(1-\lambda)tz^{2}}{R_{{\rm
w}_{\lambda,t}}(z)+1}$ $\displaystyle\qquad{}=R_{{\rm
w}_{\lambda,t}^{0}}(z)+\frac{(1-\lambda)tz^{2}}{R_{{\rm
w}_{\lambda,t}^{0}}(z)+1}$ $\displaystyle\qquad\hphantom{=}{}+R_{{\rm
w}_{\lambda,0}}\bigg{(}z\sqrt{1\\!-(1\\!-\lambda)\bigg{\\{}\frac{1}{R_{{\rm
w}_{\lambda,t}}(z)\\!+1}-\frac{1}{R_{{\rm
w}_{\lambda,t}}(z)\\!+1\\!-tz^{2}}\bigg{\\}}}\bigg{)},\quad t\geq 0,\\!\\!$
(1.21)

where $R_{{\rm
w}_{\lambda,t}^{0}}(z)=\big{\\{}{-}1+tz^{2}+\sqrt{1+2(2\lambda-1)tz^{2}+t^{2}z^{4}}\big{\\}}/2$.

######  Remark 1.2.

Consider a matrix-valued Brownian motion $(M_{t})_{t\geq 0}$ which is given by
a time-evolution of a Hermitian $N\times N$ matrix starting from a Hermitian
matrix $M_{0}$. When $M_{0}$ is a null matrix, we write this process as
$\big{(}M_{t}^{0}\big{)}_{t\geq 0}$. Then we have

$\displaystyle M_{t}=M_{0}+M_{t}^{0},\qquad t\geq 0.$ (1.22)

We assume that the empirical eigenvalue distribution of $M_{0}$ converges to
${\rm w}_{0}$ as $N\to\infty$. As an eigenvalue process of (1.22), we can
obtain Dyson’s Brownian motion model with $\beta=2$ starting from the
eigenvalues of $M_{0}$. (See Section 2.1 below.) Moreover, for any $\beta>0$,
we can obtain the same Cauchy transform. The process $({\rm w}_{t})_{t\geq 0}$
is obtained as the time-evolution of the limit empirical measure of Dyson’s
Brownian motion model. We can show that $M_{0}$ and $M_{t}^{0}$ are
asymptotically free for any $t\geq 0$. Thus at each time $t\geq 0$, the
limiting eigenvalue distribution ${\rm w}_{t}$ of $M_{t}$ converges to ${\rm
w}_{0}\boxplus{\rm w}_{t}^{0}$. That is, the assertion $(i)$ of Theorem 1.1 is
consistent with the asymptotic freeness of a random matrix in the GUE and an
arbitrary deterministic Hermitian matrix. Such an interpretation of the
assertion $(ii)$ of Theorem 1.1 is not yet known and is left as a challenging
future problem.

######  Remark 1.3.

With respect to the process $({\rm w}_{t})_{t\geq 0}$, it is pointed out that
the functional equations (1.18) for the $R$-transform and (3.4) for the Cauchy
transform given below are consequences of the Markov property of freeness and
its relation to analytic subordination [5, 13, 70]. As shown in Sections 3 and
4.1, we will prove (1.18) from (3.4). In the context of analytic
subordination, the functional equation (3.4) shall be considered to prove
unique existence of the one-parameter ($t\in[0,\infty)$) family of functions
$\omega_{t}$ such that

$\displaystyle\lim_{y\uparrow\infty}\frac{\omega_{t}\big{(}\sqrt{-1}y\big{)}}{\sqrt{-1}y}=1,\qquad\text{and}\qquad
G_{{\rm w}_{t}}=G_{{\rm w}_{0}}\circ\omega_{t},\quad t\in[0,\infty),$

by explicitly showing that $\omega_{t}(z)=z-tG_{{\rm w}_{t}}(z)$. Here the
unique existence of $(G_{{\rm w}_{t}}(z))_{t\geq 0}$ with appropriate
properties is guaranteed by the fact that $(G_{{\rm w}_{t}}(z))_{t\geq 0}$
solves the complex Burgers equation (1.2). For the other processes $({\rm
m}_{\lambda,t})_{t\geq 0}$ and $({\rm w}_{\lambda,t})_{t\geq 0}$, the
connections of our results with analytic subordination properties should be
studied in the future.

######  Remark 1.4.

The process $\big{(}{\rm w}_{t}^{0}\big{)}_{t\geq 0}$ given by (1.9) is
identified with the free Brownian motion studied in free probability theory
[11, 65]. Here we would like to consider the process $({\rm w}_{t})_{t\geq 0}$
determined by the Burgers equation (1.2) as a generalization of the free
Brownian motion, since initial probability measure ${\rm w}_{0}$ is now
arbitrary in ${\cal P}^{0}(\mathbb{R})$. See also [12, 69] for the important
connection between the complex Burgers equation (1.2) and free probability
theory. Capitaine and Donati-Martin [22] introduce the free Wishart processes
based on the two-parametric Marcenko–Pastur distribution (1.10). Our process
$({\rm m}_{\lambda,t})_{t\geq 0}$ is defined as a solution of the complex
Burgers-type equation (1.8) specified by its initial probability measure ${\rm
m}_{\lambda,0}\in{\cal P}^{0}(\mathbb{R}_{+})$. In other words, $({\rm
m}_{\lambda,t})_{t\geq 0}$ is a family of processes parameterized by an
initial measure ${\rm m}_{\lambda,0}$, and hence it is different from the free
Wishart process. In the functional equation (1.20) in Theorem 1.1$(ii)$, the
parameterization by an initial probability measure is realized by the first
term in the r.h.s., which is added to the fundamental solution for the
$R$-transform, $R_{{\rm m}_{\lambda,t}^{0}}(z)$, of the two-parametric
Marcenko–Pastur distribution (1.10). The rectangular free convolutions studied
by Benaych-Georges [6, 7] are very interesting and important extensions of the
square free convolutions. They are based on the original Marcenko–Pastur
distribution (i.e., the special case of (1.10) at $t=1$). By this reason, it
is not easy to discuss the present study from the viewpoint of the rectangular
free convolutions. We want to leave this topic as a future problem. The
equality (1.21) in Theorem 1.1$(iii)$ seems to be so complicated, but it
clearly shows that if $\lambda=1$, this equality is reduced to (1.18) as
expected from (1.4). To the best of our knowledge, the chiral GUE and its time
evolution $({\rm w}_{\lambda,t})_{t\geq 0}$ determined by (1.3) have not been
systematically studied in free probability theory.

###  1.3 Results for the $\bm{S}$-transforms

For $\big{(}{\rm w}^{0}_{t}\big{)}_{t\geq 0}$ starting from $\delta_{0}$, it
is easy to verify that (see, for instance, [62])

$\displaystyle S_{{\rm w}^{0}_{t}}(z)=\sqrt{\frac{1}{tz}},\qquad t\geq 0.$
(1.23)

For the process $\big{(}{\rm m}^{0}_{\lambda,t}\big{)}_{t\geq 0}$ starting
from $\delta_{0}$, we have [62]

$\displaystyle S_{{\rm m}^{0}_{\lambda,t}}(z)=\frac{1}{t(z+\lambda)},\qquad
t\geq 0.$

######  Theorem 1.5.

  1. $(i)$

Provided that ${\rm w}_{\lambda,0}={\rm m}_{\lambda,0}^{\bf s}$,
$\lambda\in\mathbb{R}_{+}$, the following equality holds,

$\displaystyle S_{{\rm w}_{\lambda,t}}(z)=S_{{\rm d}}(z)\sqrt{S_{{\rm
m}_{\lambda,t}}(z)},\qquad t\geq 0.$ (1.24)

  2. $(ii)$

Assume that the $S$-transform of the initial probability measure $S_{{\rm
w}_{0}}(z)$ is well defined. Then,

$\displaystyle\frac{S_{{\rm w}_{t}}(z)}{1-(S_{{\rm w}_{t}}(z)/S_{{\rm
w}^{0}_{t}}(z))^{2}}=S_{{\rm w}_{0}}\big{(}z\big{\\{}1-\big{(}S_{{\rm
w}_{t}}(z)/S_{{\rm w}^{0}_{t}}(z)\big{)}^{2}\big{\\}}\big{)},\qquad t\geq 0.$
(1.25)

  3. $(iii)$

Assume that the $S$-transform of the initial measure $S_{{\rm
m}_{\lambda,0}}(z)$, $\lambda\in\mathbb{R}_{+}$ is well defined. Then,

$\displaystyle\frac{S_{{\rm m}_{\lambda,t}}(z)}{\big{\\{}1-S_{{\rm
m}_{\lambda,t}}(z)/S_{{\rm w}^{0}_{t}}(z)^{2}\big{\\}}\big{\\{}1-S_{{\rm
m}_{\lambda,t}}(z)/S_{{\rm m}^{0}_{\lambda,t}}(z)\big{\\}}}$
$\displaystyle\qquad{}=S_{{\rm m}_{\lambda,0}}\big{(}z\big{\\{}1-S_{{\rm
m}_{\lambda,t}}(z)/S_{{\rm m}^{0}_{\lambda,t}}(z)\big{\\}}\big{)},\qquad t\geq
0.$ (1.26)

  4. $(iv)$

Assume that the $S$-transform of the initial measure $S_{{\rm
w}_{\lambda,0}}(z)$, $\lambda\in\mathbb{R}_{+}$ is well defined. Then,

$\displaystyle\sqrt{\frac{1-\frac{z}{1+z}\big{(}S_{{\rm
w}_{\lambda,t}}(z)/S_{{\rm
w}_{\lambda,t}^{0}}(z)\big{)}^{2}}{1-\frac{z}{1+z}\big{(}S_{{\rm
w}_{\lambda,t}}(z)/S_{{\rm w}_{t}^{0}}(z)\big{)}^{2}}}\frac{S_{{\rm
w}_{\lambda,t}}(z)}{1-\big{(}S_{{\rm w}_{\lambda,t}}(z)/S_{{\rm
w}_{\lambda,t}^{0}}(z)\big{)}^{2}}$ $\displaystyle\qquad{}=S_{{\rm
w}_{\lambda,0}}\big{(}z\big{\\{}1-(S_{{\rm w}_{\lambda,t}}(z)/S_{{\rm
w}_{\lambda,t}^{0}}(z))^{2}\big{\\}}\big{)},\qquad t\geq 0.$ (1.27)

Since ${\rm w}_{1,t}={\rm w}_{t}$, if $\lambda=1$ (1.27) is reduced to (1.25).

######  Remark 1.6.

The equality (1.24) in the assertion $(i)$ of Theorem 1.5 can be regarded as a
“push-back” representation of the equality (1.6) expressed using the second
push-forward measure. This is very simple, but reveals an important role of
the symmetric Bernoulli delta measure ${\rm d}_{a}$ with $a=1$ defined by
(1.11), whose $S$-transform is given by (1.17). If $\sqrt{S_{{\rm
m}_{\lambda,t}}(z)}$, $t\geq 0$ is realized as an $S$-transform of a
probability measure, say $(\nu_{\lambda,t})_{t\geq 0}$, then (1.24) implies
the equality ${\rm w}_{\lambda,t}={\rm d}\boxtimes\nu_{\lambda,t}$, $t\geq 0$.

###  1.4 Applications

First we apply the above theorems to the process $({\rm w}_{t})_{t\geq 0}$
starting from the symmetric Bernoulli delta measure ${\rm d}_{a}$ with
displacement $2a>0$ given by (1.11). Here we write this process as $({\rm
w}_{t}^{a})_{t\geq 0}$.

######  Proposition 1.7.

The $R$-transform and the $S$-transform of $({\rm w}_{t}^{a})_{t\geq 0}$ are
given by

$\displaystyle R_{{\rm
w}_{t}^{a}}(z)=\frac{1}{2}\Big{[}2tz^{2}-1+\sqrt{1+4a^{2}z^{2}}\Big{]},\qquad
t\geq 0,$ (1.28) $\displaystyle S_{{\rm
w}_{t}^{a}}(z)=\frac{1}{(tz)^{1/2}}\bigg{[}1+\frac{1}{2z}+\frac{a^{2}}{2tz}-\frac{1}{2z}\bigg{(}1+\frac{a^{2}}{t}\bigg{)}\sqrt{1+\frac{4a^{2}/t}{(1+a^{2}/t)^{2}}z}\,\bigg{]}^{1/2},\qquad
t\geq 0.$ (1.29)

Note that (1.28) determines the free cumulants of $({\rm w}_{t}^{a})_{t\geq
0}$ as

$\displaystyle\kappa_{n}({\rm
w}_{t}^{a})=\begin{cases}t+a^{2},&n=2,\cr\displaystyle{-(-1)^{n/2}\frac{(n-3)!!}{(n/2)!}2^{n/2-1}a^{n}},&n\in\\{4,6,8,\dots\\},\cr
0,&\mbox{otherwise},\end{cases}\qquad t\geq 0.$

As well known, for the fundamental solution $\big{(}{\rm
w}_{t}^{0}\big{)}_{t\geq 0}$, $R_{{\rm w}_{t}^{0}}(z)=tz^{2}$, and hence
$\kappa_{n}\big{(}{\rm w}_{t}^{0}\big{)}=t\delta_{n2}$, $n\in\mathbb{N}$,
$t\geq 0$. The complexity of the solution $({\rm w}_{t}^{a})_{t\geq 0}$ with
$a>0$ reported in [3, 57, 72] is simply expressed here by the emergence of
free cumulants $\kappa_{n}({\rm w}_{t}^{a})$ for all $n\in\mathbb{N}$, $t\geq
0$.

Next we apply the theorems to the process $({\rm m}_{\lambda,t})_{t\geq 0}$
starting from $\delta_{b}$ with $b>0$. Here we write this process as
$\big{(}{\rm m}_{\lambda,t}^{b}\big{)}_{t\geq 0}$. If the variable $x$ is
replaced by $x/\lambda$, the parameters $r$ by $1/\lambda$, and $a$ by
$b/\lambda$ in the three parametric Marcenko–Pastur measure studied in [34],
we obtain the present probability measure ${\rm m}_{\lambda,t}^{b}$, $t\geq
0$.

######  Proposition 1.8.

The $R$-transform and the $S$-transform of $\big{(}{\rm
m}_{\lambda,t}^{b}\big{)}_{t\geq 0}$ are given by

$\displaystyle R_{{\rm m}_{\lambda,t}^{b}}(z)=\frac{z\big{\\{}(\lambda
t+b)-\lambda t^{2}z\big{\\}}}{(1-tz)^{2}},\qquad t\geq 0,$ (1.30)
$\displaystyle S_{{\rm
m}_{\lambda,t}^{b}}(z)\frac{1}{t(\lambda+z)}\bigg{[}1+\frac{\lambda}{2z}+\frac{b}{2tz}-\frac{1}{2z}\bigg{(}\lambda+\frac{b}{t}\bigg{)}\sqrt{1+\frac{4b/t}{(\lambda+b/t)^{2}}z}\bigg{]},\qquad
t\geq 0.$ (1.31)

Comparing (1.29) and (1.31), we obtain the equality,

$\displaystyle S_{{\rm w}_{t}^{a}}(z)=\sqrt{\frac{1+z}{z}}\sqrt{S_{{\rm
m}_{1,t}^{a^{2}}}(z)}.$ (1.32)

It is readily confirmed by the definition of the second push-forward measure
(1.5) with $p=2$ that ${\rm d}_{a}^{(2)}=\delta_{a^{2}}$, $a>0$. Hence, the
matching of initial measures is established, and as a special case of (1.7),
$({\rm w}_{t}^{a})^{(2)}=m_{1,t}^{a^{2}}$, $t\geq 0$, $a>0$. Therefore, (1.32)
can be regarded as a special case of the assertion $(i)$ of Theorem 1.5.

Note that (1.30) determines the free cumulants of ${\rm m}_{\lambda,t}^{b}$ as

$\displaystyle\kappa_{n}\big{(}{\rm m}_{\lambda,t}^{b}\big{)}=(\lambda
t+bn)t^{n-1},\qquad n\in\mathbb{N},\quad t\geq 0.$

The complexity of $\big{(}{\rm m}_{\lambda,t}^{b}\big{)}_{t\geq 0}$ with $b>0$
reported in [34] is simply expressed by a shift $\lambda t\to\lambda t+bn$ in
the above formulas for $\kappa_{n}\big{(}{\rm m}_{\lambda,t}^{b}\big{)}$,
$n\in\mathbb{N}$, $t\geq 0$. The dynamical phase transitions studied by [3,
34, 57, 72] seem to be hidden in the above solutions for the $R$-transforms
and the $S$-transforms. Extracting the singularity at the transition point
from the above results will be a future problem.

The present paper is organized as follows. In Section 2 we explain how the
complex Burgers-type equations are derived in the hydrodynamic limits of
stochastic log-gases. Then we prove fundamental relations between a measure
$\mu\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$ and its second push-forward measure
$\mu^{(2)}\in{\cal P}^{0}(\mathbb{R}_{+})$. Section 3 is devoted to solving
the present three kinds of complex Burgers equations (1.2), (1.3), and (1.8)
by the method of characteristic curves [26, 29]. Proofs of Theorems 1.1, 1.5
and Propositions 1.7, 1.8 are given in Section 4. Concluding remarks are given
in Section 5.

##  2 Preliminaries

###  2.1 From matrix-valued processes to complex Burgers-type equations
through hydrodynamic limit

For $N\in\mathbb{N}:=\\{1,2,\dots\\}$, let ${\sf H}_{N}$ and ${\sf U}_{N}$ be
the space of $N\times N$ Hermitian matrices and the group of $N\times N$
unitary matrices, respectively. We consider complex-valued continuous semi-
martingale processes $\big{(}M^{ij}_{t}\big{)}_{t\geq 0},1\leq i,j\leq N$ with
the condition $\overline{M^{ji}_{t}}=M^{ij}_{t}$, where $\overline{z}$ denotes
the complex conjugate of $z\in\mathbb{C}$, and define an ${\sf H}_{N}$-valued
process by $M_{t}=\big{(}M^{ij}_{t}\big{)}_{1\leq i,j\leq N}$. For
$S=\mathbb{R}$ or $\mathbb{R}_{+}$, define the Weyl chambers as
$\mathbb{W}_{N}(S):=\big{\\{}\bm{x}=\big{(}x^{1},\dots,x^{N}\big{)}\in
S^{N}\colon x^{1}<\cdots<x^{N}\big{\\}}$, and write their closures as
$\overline{\mathbb{W}_{N}(S)}=\big{\\{}\bm{x}\in\overline{S}^{N}\colon
x^{1}\leq\cdots\leq x^{N}\big{\\}}$. For each $t\geq 0$, there exists
$U_{t}=\big{(}U^{ij}_{t}\big{)}_{1\leq i,j\leq N}\in{\sf U}_{N}$ such that it
diagonalizes $M_{t}$ as $U^{\dagger}_{t}M_{t}U_{t}={\rm
diag}\big{(}\Lambda^{1}_{t},\dots,\Lambda^{N}_{t}\big{)}$ with the eigenvalues
$\\{\Lambda^{i}_{t}\\}_{i=1}^{N}$ of $M_{t}$, where $U^{\dagger}_{t}$ is the
Hermitian conjugate of $U_{t}$;
$\big{(}U^{ij}_{t}\big{)}^{\dagger}=\overline{U^{ji}_{t}}$, $1\leq i,j\leq N$,
and we assume
$\Lambda_{t}:=\big{(}\Lambda^{1}_{t},\dots,\Lambda^{N}_{t}\big{)}\in\overline{\mathbb{W}_{N}(\mathbb{R})}$,
$t\geq 0$. For ${\rm d}M_{t}:=\big{(}{\rm d}M^{ij}_{t}\big{)}_{1\leq i,j\leq
N}$, define a set of quadratic variations,

$\displaystyle\Gamma^{ij,k\ell}_{t}:=\big{\langle}\big{(}U^{\dagger}{\rm
d}MU\big{)}^{ij},\big{(}U^{\dagger}{\rm
d}MU\big{)}^{k\ell}\big{\rangle}_{t},\qquad 1\leq i,j,k,\ell\leq N,\quad t\geq
0.$

The following is proved [19, 46, 50]. See [2, Section 4.3] for details of
proof.

######  Proposition 2.1.

The eigenvalue process $(\Lambda_{t})_{t\geq 0}$ satisfies the following
system of SDEs,

$\displaystyle{\rm d}\Lambda^{i}_{t}={\rm d}{\cal M}^{i}_{t}+{\rm
d}J^{i}_{t},\qquad t\geq 0,\quad 1\leq i\leq N,$

where $({\cal M}^{i}_{t})_{t\geq 0},1\leq i\leq N$ are martingales with
quadratic variations $\langle{\rm d}{\cal M}^{i},{\rm d}{\cal
M}^{j}\rangle_{t}=\Gamma^{ii,jj}_{t}{\rm d}t$, $t\geq 0$, and
$(J^{i}_{t})_{t\geq 0},1\leq i\leq N$ are the processes with finite variations
given by

$\displaystyle{\rm d}J^{i}_{t}=\sum_{j=1}^{N}\frac{{\bf
1}_{(\Lambda^{i}_{t}\not=\Lambda^{j}_{t})}}{\Lambda^{i}_{t}-\Lambda^{j}_{t}}\Gamma^{ij,ji}_{t}{\rm
d}t+{\rm d}\Upsilon^{i}_{t}.$

Here ${\rm d}\Upsilon^{i}_{t}$ denotes the finite-variation part of
$\big{(}U^{\dagger}_{t}{\rm d}M_{t}U_{t}\big{)}^{ii}$, $t\geq 0$, $1\leq i\leq
N$.

We will show two basic examples of $M_{t}\in{\sf H}_{N},t\geq 0$ and
applications of Proposition 2.1, see [50]. Let
$\nu\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$ and
$\big{(}B^{ij}_{t}\big{)}_{t\geq 0}$,
$\big{(}\widetilde{B}^{ij}_{t}\big{)}_{t\geq 0}$, $1\leq i\leq N+\nu$, $1\leq
j\leq N$ be independent one-dimensional standard Brownian motions. For $1\leq
i\leq j\leq N$, put

$\displaystyle S^{ij}_{t}=\begin{cases}B^{ij}_{t}/\sqrt{2},&i<j,\cr
B^{ii}_{t},&i=j,\end{cases}\qquad
A^{ij}_{t}=\begin{cases}\widetilde{B}^{ij}_{t}/\sqrt{2},&i<j,\cr
0,&i=j,\end{cases}$

and let $S^{ij}_{t}=S^{ji}_{t}$ (symmetric) and $A^{ij}_{t}=-A^{ji}_{t}$
(anti-symmetric), $t\geq 0$ for $1\leq j<i\leq N$.

######  Example 2.2.

Put

$\displaystyle
M_{t}=\big{(}M^{ij}_{t}\big{)}:=\big{(}S^{ij}_{t}+\sqrt{-1}A^{ij}_{t}\big{)}_{1\leq
i,j\leq N},\qquad t\geq 0.$

By definition $\langle{\rm d}M^{ij},{\rm
d}M^{k\ell}\rangle_{t}=\delta^{i\ell}\delta^{jk}{\rm d}t$, $t\geq 0$, $1\leq
i,j,k,\ell\leq N$. Hence, by unitarity of $U_{t}\in{\sf U}_{N}$, $t\geq 0$, we
see that $\Gamma^{ij,k\ell}_{t}=\delta^{i\ell}\delta^{jk}$, which gives
$\langle{\rm d}{\cal M}^{i},{\rm d}{\cal
M}^{j}\rangle_{t}=\Gamma^{ii,jj}_{t}{\rm d}t=\delta^{ij}{\rm d}t$ and
$\Gamma^{ij,ji}_{t}\equiv 1$, $t\geq 0$, $1\leq i,j\leq N$. Then Proposition
2.1 proves that the eigenvalue process $(\Lambda_{t})_{t\geq 0}$, satisfies
the following system of SDEs with $\beta=2$,

$\displaystyle{\rm d}\Lambda^{i}_{t}={\rm
d}B^{i}_{t}+\frac{\beta}{2}\sum_{1\leq j\leq N,j\not=i}\frac{{\rm
d}t}{\Lambda^{i}_{t}-\Lambda^{j}_{t}},\qquad t\geq 0,\quad 1\leq i\leq N.$
(2.1)

Here $\big{(}B^{i}_{t}\big{)}_{t\geq 0},1\leq i\leq N$ are independent one-
dimensional standard Brownian motions, which are different from
$(B^{ij}_{t})_{t\geq 0}$ and $\big{(}\widetilde{B}^{ij}_{t}\big{)}_{t\geq 0}$
used to define $\big{(}S^{ij}_{t}\big{)}_{t\geq 0}$ and
$\big{(}A^{ij}_{t}\big{)}_{t\geq 0}$, $1\leq i,j\leq N$. If $\beta=2$ and the
initial configuration is $N\delta_{0}$, that is, all $N$ particles are at the
origin, then at each time $t>0$,
$\Lambda_{t}=\big{(}\Lambda_{t}^{1},\dots,\Lambda_{t}^{N}\big{)}$ gives a
point process on $\mathbb{R}$ which is equal in distribution with the GUE
eigenvalue point process with variance $t$ [33, 50]. For $\beta>0$, we call
the solution of (2.1) the $N$-particle system of Dyson’s Brownian motion model
with parameter $\beta$ [33], and write it as $\big{(}\Lambda^{{\rm
D}(N,\beta)}_{t}\big{)}_{t\geq 0}$.

######  Example 2.3.

Consider an $(N+\nu)\times N$ rectangular-matrix-valued process given by

$\displaystyle
K_{t}:=\big{(}B^{ij}_{t}+\sqrt{-1}\widetilde{B}^{ij}_{t}\big{)}_{1\leq i\leq
N+\nu,1\leq j\leq N},\qquad t\geq 0,$

and define an ${\sf H}_{N}$-valued process by

$\displaystyle M_{t}=K^{\dagger}(t)K(t),\qquad t\geq 0.$

The matrix $M_{t}$ is positive semi-definite and hence the eigenvalues are
non-negative; $\Lambda^{i}_{t}\in\mathbb{R}_{+}$, $t\geq 0$, $1\leq i\leq N$.
We see that the finite-variation part of $dM^{ij}_{t}$ is equal to
$2(N+\nu)\delta^{ij}{\rm d}t$, $t\geq 0$, and $\langle{\rm d}M^{ij},{\rm
d}M^{k\ell}\rangle_{t}=2\big{(}M^{i\ell}_{t}\delta^{jk}+M^{kj}_{t}\delta^{i\ell}\big{)}{\rm
d}t$, $t\geq 0$, $1\leq i,j,k,\ell\leq N$, which implies that ${\rm
d}\Upsilon^{i}_{t}=2(N+\nu){\rm d}t$,
$\Gamma^{ij,ji}_{t}=2\big{(}\Lambda^{i}_{t}+\Lambda^{j}_{t}\big{)}$, and
$\langle{\rm d}{\cal M}^{i},{\rm d}{\cal
M}^{j}\rangle_{t}=\Gamma^{ii,jj}_{t}{\rm d}t=4\Lambda^{i}_{t}\delta^{ij}{\rm
d}t$, $t\geq 0$, $1\leq i,j\leq N$. Then we have the following SDEs with
$\beta=2$ for the eigenvalue process of $(M_{t})_{t\geq 0}$,

$\displaystyle{\rm d}\Lambda^{i}_{t}=2\sqrt{\Lambda^{i}_{t}}{\rm
d}\widetilde{B}^{i}_{t}+\beta\Bigg{[}(\nu+1)+2\Lambda^{i}_{t}\sum_{1\leq j\leq
N,j\not=i}\frac{1}{\Lambda^{i}_{t}-\Lambda^{j}_{t}}\Bigg{]}{\rm d}t,\qquad
t\geq 0,\quad 1\leq i\leq N,$ (2.2)

where $\big{(}\widetilde{B}^{i}_{t}\big{)}_{t\geq 0},1\leq i\leq N$ are
independent one-dimensional standard Brownian motions, which are different
from $\big{(}B^{ij}_{t}\big{)}_{t\geq 0}$ and
$\big{(}\widetilde{B}^{ij}_{t}\big{)}_{t\geq 0}$, $1\leq i,j\leq N$, used
above to define the rectangular-matrix-valued process $(K_{t})_{t\geq 0}$. The
parameter $\nu$ can be extended to $\nu>-1$, in which if $\nu\in(-1,0)$, a
reflecting wall is put at the origin [50]. We call the solution of (2.2) the
$N$-particle system of the Bru–Wishart process with parameters $(\beta,\nu)$
[20], and write it as $\big{(}\Lambda_{t}^{{\rm
BW}(N,\beta,\nu)}\big{)}_{t\geq 0}$.

The positive roots of eigenvalues of $M_{t}$ give the singular values of the
rectangular matrix $K_{t}$, which are denoted by

$\displaystyle{\cal S}^{i}_{t}:=\sqrt{\Lambda^{i}_{t}},\qquad t\geq 0,\quad
1\leq i\leq N.$ (2.3)

The system of SDEs for them is readily obtained from (2.2) as

$\displaystyle{\rm d}{\cal S}^{i}_{t}={\rm
d}\widetilde{B}^{i}_{t}+\frac{\beta(\nu+1)-1}{2{\cal S}^{i}_{t}}{\rm
d}t+\frac{\beta}{2}\sum_{1\leq j\leq N,j\not=i}\bigg{(}\frac{1}{{\cal
S}^{i}_{t}-{\cal S}^{j}_{t}}+\frac{1}{{\cal S}^{i}_{t}+{\cal
S}^{j}_{t}}\bigg{)}{\rm d}t,$ (2.4)

$t\geq 0,1\leq i\leq N$ with $\beta=2$ and $\nu>-1$. If $\beta=2$ and the
initial configuration is $N\delta_{0}$, then at each time $t>0$, ${\cal
S}_{t}=\big{(}{\cal S}_{t}^{1},\dots,{\cal S}_{t}^{N}\big{)}$ on
$\mathbb{R}_{+}$ gives the chiral GUE point process with parameter $\nu$ and
variance $t$ studied in random matrix theory for high energy physics [1, 35,
64, 67, 68]. For $\beta>0$, we call the solution of (2.4) the chiral version
of Dyson’s Brownian motion model with parameters $(\beta,\nu)$, and write it
as $\big{(}{\cal S}_{t}^{{\rm chD}(N,\beta,\nu)}\big{)}_{t\geq 0}$.

At each time $t>0$, the point processes $\Lambda_{t}^{{\rm D}(N,\beta)}$,
$\Lambda_{t}^{{\rm BW}(N,\beta,\nu)}$, and ${\cal S}_{t}^{{\rm
chD}(N,\beta,\nu)}$ are known as typical examples of one-dimensional log-gases
[35]. Therefore, we will call the solutions of the SDEs (2.1), (2.2), and
(2.4) stochastic log-gases. Note that, when $\beta=2$, (2.2) and (2.4) can be
regarded as the $N$-variable extensions of the $2(\nu+1)$-dimensional squared
Bessel process and the Bessel process with parameter $\nu>-1$, respectively
[46].

For $\Lambda^{{\rm D}(N,\beta)}_{t}=\big{(}\Lambda^{{\rm
D}(N,\beta)\,1}_{t},\dots,\Lambda^{{\rm D}(N,\beta)\,N}_{t}\big{)}$, $t\geq
0$, we regard the time evolution of empirical measures

$\displaystyle\Xi^{{\rm
D}(N,\beta)}_{t}(\cdot):=\frac{1}{N}\sum_{i=1}^{N}\delta_{\Lambda^{{\rm
D}(N,\beta)\,i}_{t}}(\cdot),\qquad t\in[0,T],$

as an element of ${\cal C}\big{(}[0,T]\to{\cal P}^{0}(\mathbb{R})\big{)}$. For
$\big{(}\Lambda^{{\rm BW}(N,\beta,\nu)}_{t}\big{)}_{t\geq 0}$ and
$\big{(}{\cal S}^{{\rm D}(N,\beta,\nu)}_{t}\big{)}_{t\geq 0}$, let

$\displaystyle\lambda:=\frac{N+\nu}{N}\Longleftrightarrow\nu=(\lambda-1)N,$

and consider

$\displaystyle\Xi^{{\rm
BW}(N,\beta,(\lambda-1)N)}_{t}(\cdot):=\frac{1}{N}\sum_{i=1}^{N}\delta_{\Lambda^{{\rm
BW}(N,\beta,(\lambda-1)N)\,i}_{t}}(\cdot),\qquad t\in[0,T]$

as an element of ${\cal C}\big{(}[0,T]\to{\cal P}^{0}(\mathbb{R}_{+})\big{)}$,
and with (2.3) consider

$\displaystyle\Sigma^{{\rm
chD}(N,\beta,(\lambda-1)N)}_{t}(\cdot):=\frac{1}{2N}\sum_{i=1}^{N}\big{\\{}\delta_{{\cal
S}^{{\rm chD}(N,\beta,(\lambda-1)N)\,i}_{t}}(\cdot)+\delta_{-{\cal S}^{{\rm
chD}(N,\beta,(\lambda-1)N)\,i}_{t}}(\cdot)\big{\\}},\qquad t\in[0,T],$

as an element of ${\cal C}\big{(}[0,T]\to{\cal P}^{0}_{{\rm
s}}(\mathbb{R})\big{)}$. The following is proved [16, 21, 25, 63].

######  Theorem 2.4.

Assume that for any $N\in\mathbb{N}$, the initial measures $\Xi^{{\rm
D}(N,\beta)}_{0}$, $\Xi^{{\rm BW}(N,\beta,(\lambda-1)N)}_{0}$, and
$\Sigma^{{\rm chD}(N,\beta,(\lambda-1)N)}_{0}$ have bounded supports, where
$\big{(}\Sigma^{{\rm chD}(N,\beta,(\lambda-1)N)}_{0}\big{)}^{(2)}=\Xi^{{\rm
BW}(N,\beta,(\lambda-1)N)}_{0}$ is satisfied, and in $N\to\infty$ they
converge weakly to the measures ${\rm w}_{0}\in{\cal P}^{0}(\mathbb{R})$,
${\rm m}_{\lambda,0}\in{\cal P}^{0}(\mathbb{R}_{+})$, and ${\rm
w}_{\lambda,0}\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$, respectively. Then for
any fixed $T<\infty$,

$\displaystyle\big{(}\Xi^{{\rm
D}(N,\beta)}_{t}(\cdot)\big{)}_{t\in[0,T]}\Longrightarrow({\rm
w}_{t}(\cdot))_{t\in[0,T]}\qquad\text{a.s. in}\quad{\cal
C}\big{(}[0,T]\to{\cal P}^{0}(\mathbb{R})\big{)},$
$\displaystyle\big{(}\Xi^{{\rm
BW}(N,\beta,(\lambda-1)N)}_{t}(\cdot)\big{)}_{t\in[0,T]}\Longrightarrow({\rm
m}_{\lambda,t}(\cdot))_{t\in[0,T]}\qquad\text{a.s. in}\quad{\cal
C}\big{(}[0,T]\to{\cal P}^{0}(\mathbb{R}_{+})\big{)},$
$\displaystyle\big{(}\Sigma^{{\rm
chD}(N,\beta,(\lambda-1)N)}_{t}(\cdot)\big{)}_{t\in[0,T]}\Longrightarrow({\rm
w}_{\lambda,t}(\cdot))_{t\in[0,T]}\qquad\text{a.s. in}\quad{\cal
C}\big{(}[0,T]\to{\cal P}^{0}_{{\rm s}}(\mathbb{R})\big{)},$

where $({\rm w}_{t})_{t\geq 0}$, $({\rm m}_{\lambda,t})_{t\geq 0}$, and $({\rm
w}_{\lambda,t})_{t\geq 0}$ are the time-dependent probability measures defined
so that their Cauchy transforms solve the PDEs (1.2), (1.8), and (1.3) under
the initial probability measures ${\rm w}_{0}$, ${\rm m}_{\lambda,0}$, and
${\rm w}_{\lambda,0}$, respectively.

Note that dependence on the parameter $\beta$ vanishes in the limit
$N\to\infty$.

By the construction mentioned above, the relation

$\displaystyle\big{(}\Sigma^{{\rm
chD}(N,\beta,(\lambda-1)N)}_{t}\big{)}^{(2)}\stackrel{{\scriptstyle\rm
d}}{{=}}\Xi^{{\rm BW}(N,\beta,(\lambda-1)N)}_{t},\qquad t\geq 0,$

holds, and then Theorem 2.4 proves the equality ${\rm
w}_{\lambda,t}^{(2)}={\rm m}_{\lambda,t}$, $t\geq 0$. This is consistent with
the definition (1.6).

###  2.2 Expressions of second push-forward measures

We will prove the following.

######  Lemma 2.5.

For $\mu\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$ and $\nu\in{\cal
P}^{0}(\mathbb{R}_{+})$, the following four statements are equivalent with
each other,

$\displaystyle(i)\quad\mu^{(2)}=\nu,$ $\displaystyle(ii)\quad
G_{\mu}(z)=zG_{\nu}\big{(}z^{2}\big{)},$ $\displaystyle(iii)\quad
R_{\mu}(z)=R_{\nu}\bigg{(}\dfrac{z^{2}}{R_{\mu}(z)+1}\bigg{)},$
$\displaystyle(iv)\quad S_{\mu}(z)=S_{{\rm d}}(z)\sqrt{S_{\nu}(z)}.$

###### Proof.

Assume that $\mu({\rm d}x)$ (resp. $\nu({\rm d}x)$) has a probability density
function $\rho_{\mu}(x)$ (resp. $\rho_{\nu}(x)$). Let $B\in{\cal
B}((0,\infty))$. Then

$\displaystyle\nu(B)=\int_{\mathbb{R}}1_{B}(x)\rho_{\nu}(x)\,{\rm
d}x=\int_{\mathbb{R}_{+}}1_{B}\big{(}x^{2}\big{)}\rho_{\nu}\big{(}x^{2}\big{)}\,{\rm
d}x^{2}=\int_{\mathbb{R}}1_{B}\big{(}x^{2}\big{)}\rho_{\nu}\big{(}x^{2}\big{)}|x|{\rm
d}x.$

On the other hand, by definition (1.5),
$\mu^{(2)}(B)=\int_{\mathbb{R}}1_{B}\big{(}x^{2}\big{)}\rho_{\mu}(x)\,{\rm
d}x$. Hence

$\displaystyle(i)\iff\rho_{\mu}(x)=\rho_{\nu}\big{(}x^{2}\big{)}|x|,\qquad
x\in\mathbb{R}.$ (2.5)

Since (2.5) implies the symmetry $\rho_{\mu}(-x)=\rho_{\mu}(x)$,
$x\in\mathbb{R}$, we see that

$\displaystyle G_{\mu}(z)$
$\displaystyle=\int_{\mathbb{R}}\frac{\rho_{\mu}(x)}{z-x}{\rm
d}x=\frac{1}{2}\bigg{\\{}\int_{\mathbb{R}}\frac{\rho_{\mu}(-x)}{z-x}{\rm
d}x+\int_{\mathbb{R}}\frac{\rho_{\mu}(x)}{z-x}{\rm d}x\bigg{\\}}$
$\displaystyle=\frac{1}{2}\bigg{\\{}\int_{\mathbb{R}}\frac{\rho_{\mu}(x)}{z+x}{\rm
d}x+\int_{\mathbb{R}}\frac{\rho_{\mu}(x)}{z-x}{\rm
d}x\bigg{\\}}=z\int_{\mathbb{R}}\frac{\rho_{\mu}(x)}{z^{2}-x^{2}}{\rm d}x.$

Then when (2.5) is satisfied,

$\displaystyle
G_{\mu}(z)=z\int_{\mathbb{R}}\frac{\rho_{\nu}\big{(}x^{2}\big{)}|x|}{z^{2}-x^{2}}{\rm
d}x=z\int_{\mathbb{R}_{+}}\frac{\rho_{\nu}\big{(}x^{2}\big{)}}{z^{2}-x^{2}}{\rm
d}x^{2}=zG_{\nu}\big{(}z^{2}\big{)}\iff(ii).$

When $(ii)$ is satisfied,

$\displaystyle
G_{\mu}\bigg{(}\frac{R_{\mu}(z)+1}{z}\bigg{)}=\frac{R_{\mu}(z)+1}{z}G_{\nu}\bigg{(}\bigg{(}\frac{R_{\mu}(z)+1}{z}\bigg{)}^{2}\bigg{)}$

holds. By (1.14), this implies

$\displaystyle z$
$\displaystyle=\frac{R_{\mu}(z)+1}{z}G_{\nu}\bigg{(}\bigg{(}\frac{R_{\mu}(z)+1}{z}\bigg{)}^{2}\bigg{)}\iff
G_{\nu}\bigg{(}\bigg{(}\frac{R_{\mu}(z)+1}{z}\bigg{)}^{2}\bigg{)}=\frac{z^{2}}{R_{\mu}(z)+1}$
$\displaystyle\iff\bigg{(}\frac{R_{\mu}(z)+1}{z}\bigg{)}^{2}=G_{\nu}^{\langle-1\rangle}\bigg{(}\frac{z^{2}}{R_{\mu}(z)+1}\bigg{)}=\frac{R_{\mu}(z)+1}{z^{2}}\bigg{\\{}R_{\nu}\bigg{(}\frac{z^{2}}{R_{\mu}(z)+1}\bigg{)}+1\bigg{\\}},$

where we used (1.14) again. This is equivalent with $(iii)$.

When $(ii)$ is satisfied,

$\displaystyle
G_{\mu}\bigg{(}\frac{z+1}{zS_{\mu}(z)}\bigg{)}=\frac{z+1}{zS_{\mu}(z)}G_{\nu}\bigg{(}\bigg{(}\frac{z+1}{zS_{\mu}(z)}\bigg{)}^{2}\bigg{)}$
(2.6)

holds. By (1.13), (1.15), and (1.16), we can prove the equality

$\displaystyle G_{\mu}\bigg{(}\frac{z+1}{zS_{\mu}(z)}\bigg{)}=zS_{\mu}(z).$

Then (2.6) gives

$\displaystyle
G_{\nu}\bigg{(}\bigg{(}\frac{z+1}{zS_{\mu}(z)}\bigg{)}^{2}\bigg{)}=(z+1)\bigg{(}\frac{zS_{\mu}(z)}{z+1}\bigg{)}^{2}.$
(2.7)

By (1.13), the l.h.s. of (2.7) is equal to
$\big{[}\Psi_{\nu}\big{(}\\{zS_{\mu}(z)/(z+1)\\}^{2}\big{)}+1\big{]}\\{zS_{\mu}(z)/(z+1)\\}^{2}$.
Hence we obtain the equalities
$z=\Psi_{\nu}\big{(}\\{zS_{\mu}(z)/(z+1)\\}^{2}\big{)}$
$\iff\chi_{\nu}(z)=\\{zS_{\mu}(z)/(z+1)\\}^{2}$. By (1.16), this gives

$\displaystyle S_{\mu}(z)^{2}=\frac{1+z}{z}S_{\nu}(z).$

We use (1.17) with $a=1$ and then $(iv)$ is obtained. Hence the proof is
complete. ∎

##  3 General solutions of complex Burgers-type equations

Let $t\in[0,\infty)$ and $z\in\mathbb{C}^{+}$ be independent variables and
consider a PDE for a complex function $g=g(t,z)\in\mathbb{C}$ in the form,

$\displaystyle A(t,z,g)\frac{\partial g}{\partial t}+B(t,z,g)\frac{\partial
g}{\partial z}=C(t,z,g).$ (3.1)

We regard the solution of (3.1) as a surface $g=g(t,z)$ in the space
$[0,\infty)\times\mathbb{C}^{+}\times\mathbb{C}$. Then (3.1) is interpreted as
a geometrical statement that the vector field $(A(t,z,g),B(t,z,g),C(t,z,g))$
is tangent to the surface at every point. This statement means that the graph
of solution is given by a union of integral curves of this vector field. They
are called the characteristic curves [26] of (3.1) and satisfy the
Lagrange–Charpit equatoin (see [29] and references therein),

$\displaystyle\frac{{\rm d}t}{A(t,z,g)}=\frac{{\rm d}z}{B(t,z,g)}=\frac{{\rm
d}g}{C(t,z,g)}.$ (3.2)

Here we consider the special case such that $A(t,z,g)\equiv 1$. Then (3.2) is
written as

$\displaystyle\begin{cases}\displaystyle{\frac{{\rm d}z}{{\rm
d}t}=B(t,z,g)},\\\\[8.61108pt] \displaystyle{\frac{{\rm d}g}{{\rm
d}t}=C(t,z,g)}.\end{cases}$ (3.3)

We will show that the solutions of (3.3) for (1.2), (1.3) and (1.8) are
obtained in the forms of functional equations.

######  Proposition 3.1.

  1. $(i)$

Given the Cauchy transform $G_{{\rm w}_{0}}(z)$ of the initial measure ${\rm
w}_{0}\in{\cal P}^{0}(\mathbb{R})$, the solution of (1.2) satisfies the
functional equation,

$\displaystyle G_{{\rm w}_{t}}(z)=G_{{\rm w}_{0}}(z-tG_{{\rm
w}_{t}}(z)),\qquad t\geq 0.$ (3.4)

  2. $(ii)$

Given the Cauchy transform $G_{{\rm m}_{\lambda,0}}(z)$ of the initial measure
${\rm m}_{\lambda,0}\in{\cal P}^{0}(\mathbb{R}_{+})$, the solution of (1.8)
satisfies the functional equation,

$\displaystyle\frac{1}{G_{{\rm m}_{\lambda,t}}(z)}=t+\frac{1}{G_{{\rm
m}_{\lambda,0}}((1-tG_{{\rm m}_{\lambda,t}}(z))\\{(1-\lambda)t+(1-tG_{{\rm
m}_{\lambda,t}}(z))z\\})},\quad t\geq 0.$ (3.5)

  3. $(iii)$

Given the Cauchy transform $G_{{\rm w}_{\lambda,0}}(z)$ of the initial measure
${\rm w}_{\lambda,0}\in{\cal P}^{0}_{{\rm s}}(\mathbb{R})$, the solution of
(1.3) satisfies the functional equation,

$\displaystyle\frac{1}{G_{{\rm
w}_{\lambda,t}}(z)}\\!=\\!\frac{t}{z}\\!+\\!\frac{\sqrt{(1\\!-\\!\frac{t}{z}G_{{\rm
w}_{\lambda,t}}(z))\\{(1\\!-\\!\lambda)t\\!+\\!(1\\!-\\!\frac{t}{z}G_{{\rm
w}_{\lambda,t}}(z))z^{2}\\}}}{zG_{{\rm
w}_{\lambda,0}}\Big{(}\sqrt{\big{(}1\\!-\\!\frac{t}{z}G_{{\rm
w}_{\lambda,t}}(z)\big{)}\\{(1\\!-\\!\lambda)t\\!+\\!\big{(}1\\!-\\!\frac{t}{z}G_{{\rm
w}_{\lambda,t}}(z)\big{)}z^{2}\\}}\Big{)}},\ \,t\geq 0.\\!$ (3.6)

###### Proof.

$(i)$ Consider the PDE (1.2) for $g(t,z)=G_{{\rm w}_{t}}(z)$. In this case
(3.3) becomes

$\displaystyle\frac{{\rm d}z(t)}{{\rm d}t}=g(t,z(t)),$ (3.7)
$\displaystyle\frac{{\rm d}g(t,z(t))}{{\rm d}t}=0.$ (3.8)

By (3.8), we can conclude that

$\displaystyle g(t,z(t))=g(0,z(0))\qquad\forall t\geq 0.$ (3.9)

Therefore, (3.7) is integrated as

$\displaystyle z(t)=z(0)+tg(0,z(0))=z(0)+tg(t,z(t))\Longleftrightarrow
z(0)=z(t)-tg(t,z(t)),\qquad t\geq 0.$

Inserting this into (3.9), we obtain (3.4) for $g(t,z)=G_{{\rm w}_{t}}(z)$.

$(ii)$ Consider the PDE (1.8) for $g(t,z)=G_{{\rm m}_{\lambda,t}}(z)$. In this
case (3.3) becomes

$\displaystyle\frac{{\rm d}z(t)}{{\rm d}t}=2zg(t,z(t))-(1-\lambda),$ (3.10)
$\displaystyle\frac{{\rm d}g(t,z(t))}{{\rm d}t}=-g(t,z(t))^{2}.$ (3.11)

The solution of (3.11) is given by

$\displaystyle g(t,z(t))=\frac{1}{t+1/g(0,z(0))}.$ (3.12)

Then (3.10) is written as

$\displaystyle\frac{{\rm d}z(t)}{{\rm
d}t}=\frac{2z(t)}{t+1/g(0,z(0))}-(1-\lambda).$

This is integrated as

$\displaystyle
z(t)=\bigg{(}t+\frac{1}{g(0,z(0))}\bigg{)}\bigg{\\{}1-\lambda+C\bigg{(}t+\frac{1}{g(0,z(0))}\bigg{)}\bigg{\\}},$
(3.13)

where $C$ is an integral constant. By setting $t=0$ in this equation, we see
that

$\displaystyle C=g(0,z(0))\\{z(0)g(0,z(0))-(1-\lambda)\\}.$

Using this and (3.12), (3.13) is rewritten as

$\displaystyle z(t)=\frac{1}{g(t,z(t))}+\frac{\lambda
tg(t,z(t))-1}{(1-tg(t,z(t)))g(t,z(t))}+\frac{z(0)}{(1-tg(t,z(t)))^{2}},$

which gives

$\displaystyle z(0)=(1-tg(t,z(t)))\\{(1-\lambda)t+(1-tg(t,z(t)))z(t)\\},\qquad
t\geq 0.$

If we insert this expression of $z(0)$ into (3.12) and replace $z(t)$ by $z$,
$g(t,z(t))$ by $G_{{\rm m}_{\lambda,t}}(z)$, and $g(0,\cdot)$ by $G_{{\rm
m}_{\lambda,0}}(\cdot)$, then we obtain (3.5).

$(iii)$ By Lemma 2.5$(ii)$, (3.5) is transformed into (3.6). ∎

##  4 Proofs

###  4.1 Proof of Theorem 1.1

$(i)$ We put $z=G^{\langle-1\rangle}_{{\rm w}_{t}}(\zeta)$ in (3.4). Then we
have $\zeta=G_{{\rm w}_{0}}\big{(}G^{\langle-1\rangle}_{{\rm
w}_{t}}(\zeta)-t\zeta\big{)}$. Next we apply $G^{\langle-1\rangle}_{{\rm
w}_{0}}$ on both sides and obtain

$\displaystyle G^{\langle-1\rangle}_{{\rm
w}_{0}}(\zeta)=G^{\langle-1\rangle}_{{\rm
w}_{t}}(\zeta)-t\zeta\Longleftrightarrow\zeta G^{\langle-1\rangle}_{{\rm
w}_{0}}(\zeta)-1=\big{(}\zeta G^{\langle-1\rangle}_{{\rm
w}_{t}}(\zeta)-1\big{)}-t\zeta^{2}.$

By the definition (1.14), this implies the following equation between
$R$-transforms

$\displaystyle R_{{\rm w}_{0}}(\zeta)=R_{{\rm w}_{t}}(\zeta)-t\zeta^{2},\qquad
t\geq 0.$

The assertion $(i)$ of Theorem 1.1 is concluded by the well-known result [40],
$R_{{\rm w}^{0}_{t}}(z)=tz^{2}$.

$(ii)$ We put $z=G^{\langle-1\rangle}_{{\rm m}_{\lambda,t}}(\zeta)$ in (3.5).
Then we have

$\displaystyle\frac{1}{\zeta}$ $\displaystyle=t+\frac{1}{G_{{\rm
m}_{\lambda,0}}\big{(}(1-t\zeta)\big{\\{}(1-\lambda)t+(1-t\zeta)G^{\langle-1\rangle}_{{\rm
m}_{\lambda,t}}(\zeta)\big{\\}}\big{)}}$ $\displaystyle\Longleftrightarrow
G_{{\rm
m}_{\lambda,0}}\big{(}(1-t\zeta)\big{\\{}(1-\lambda)t+(1-t\zeta)G^{\langle-1\rangle}_{{\rm
m}_{\lambda,t}}(\zeta)\big{\\}}\big{)}=\frac{\zeta}{1-t\zeta}.$

We apply $G^{\langle-1\rangle}_{{\rm m}_{\lambda,0}}$ on both sides and obtain

$\displaystyle(1-t\zeta)\big{\\{}(1-\lambda)t+(1-t\zeta)G^{\langle-1\rangle}_{{\rm
m}_{\lambda,t}}(\zeta)\big{\\}}=G^{\langle-1\rangle}_{{\rm
m}_{\lambda,0}}\bigg{(}\frac{\zeta}{1-t\zeta}\bigg{)}$
$\displaystyle\qquad{}\Longleftrightarrow-\lambda\zeta
t+(1-t\zeta)\big{[}\zeta G^{\langle-1\rangle}_{{\rm
m}_{\lambda,t}}(\zeta)-1\big{]}=\frac{\zeta}{1-t\zeta}G^{\langle-1\rangle}_{{\rm
m}_{\lambda,0}}\bigg{(}\frac{\zeta}{1-t\zeta}\bigg{)}-1.$

By the definition (1.14), this implies the following equations between
$R$-transforms,

$\displaystyle-\lambda tz+(1-tz)R_{{\rm m}_{\lambda,t}}(z)$
$\displaystyle=R_{{\rm m}_{\lambda,0}}\bigg{(}\frac{z}{1-tz}\bigg{)}$
$\displaystyle\Longleftrightarrow R_{{\rm
m}_{\lambda,t}}(z)=\frac{1}{1-tz}R_{{\rm
m}_{\lambda,0}}\bigg{(}\frac{z}{1-tz}\bigg{)}+\frac{\lambda tz}{1-tz}.$

Since $R_{{\rm m}^{0}_{\lambda,t}}(z)=\lambda tz/(1-tz)$ [16, 40], the
assertion $(ii)$ is proved.

$(iii)$ Applying Lemma 2.5$(iii)$ to (1.6), (1.20) gives (1.21).

Hence the proof of Theorem 1.1 is complete.

###  4.2 Proof of Theorem 1.5

$(i)$ Since ${\rm m}_{\lambda,t}\in{\cal P}^{0}(\mathbb{R}_{+})$ is defined by
(1.6), Lemma 2.5$(iv)$ proves the assertion $(i)$.

$(ii)$ We start from (3.4) in Proposition 3.1$(i)$. Replace $z$ by $1/z$ and
then apply (1.13). We have

$\displaystyle z(\Psi_{{\rm w}_{t}}(z)+1)=\frac{1}{1/z-tz(\Psi_{{\rm
w}_{t}}(z)+1)}\bigg{\\{}\Psi_{{\rm w}_{0}}\bigg{(}\frac{1}{1/z-tz(\Psi_{{\rm
w}_{t}}(z)+1)}\bigg{)}+1\bigg{\\}}.$

Put $z=\Psi^{\langle-1\rangle}_{{\rm w}_{t}}(\zeta)=:\chi_{{\rm
w}_{t}}(\zeta)$. Then

$\displaystyle\chi_{{\rm w}_{t}}(\zeta)(\zeta+1)=\frac{1}{1/\chi_{{\rm
w}_{t}}(\zeta)-t\chi_{{\rm w}_{t}}(\zeta)(\zeta+1)}\bigg{\\{}\Psi_{{\rm
w}_{0}}\bigg{(}\frac{1}{1/\chi_{{\rm w}_{t}}(\zeta)-t\chi_{{\rm
w}_{t}}(\zeta)(\zeta+1)}\bigg{)}+1\bigg{\\}}.$

By (1.16), the above is written as follows,

$\displaystyle\zeta S_{{\rm w}_{t}}(\zeta)$
$\displaystyle=\frac{1}{\frac{\zeta+1}{\zeta S_{{\rm w}_{t}}(\zeta)}-t\zeta
S_{{\rm w}_{t}}(\zeta)}\bigg{\\{}\Psi_{{\rm
w}_{0}}\bigg{(}\frac{1}{\frac{\zeta+1}{\zeta S_{{\rm w}_{t}}(\zeta)}-t\zeta
S_{{\rm w}_{t}}(\zeta)}\bigg{)}+1\bigg{\\}}$
$\displaystyle\Longleftrightarrow\zeta\big{\\{}1-t\zeta S_{{\rm
w}_{t}}(\zeta)^{2}\big{\\}}=\Psi_{{\rm w}_{0}}\bigg{(}\frac{\zeta S_{{\rm
w}_{t}}(\zeta)}{\zeta+1-t\zeta^{2}S_{{\rm w}_{t}}(\zeta)^{2}}\bigg{)}.$

Now we apply $\chi_{{\rm w}_{0}}$ on both sides and obtain

$\displaystyle\chi_{{\rm w}_{0}}\big{(}\zeta\big{\\{}1-t\zeta S_{{\rm
w}_{t}}(\zeta)^{2}\big{\\}}\big{)}=\frac{\zeta S_{{\rm
w}_{t}}(\zeta)}{\zeta+1-t\zeta^{2}S_{{\rm w}_{t}}(\zeta)^{2}}.$

Again we use (1.16) and replace the variable $\zeta$ by $z$. Then we obtain

$\displaystyle\frac{S_{{\rm w}_{t}}(z)}{1-tzS_{{\rm w}_{t}}(z)^{2}}=S_{{\rm
w}_{0}}\big{(}z\big{(}1-tzS_{{\rm w}_{t}}(z)^{2}\big{)}\big{)},\qquad t\geq
0.$

which is written as (1.25) by (1.23).

$(iii)$ We can prove (1.26) similarly to (1.25) as shown above.

$(iv)$ By Lemma 2.5$(iv)$, (1.26) is transformed to (1.27).

Hence the proof of Theorem 1.5 is complete.

###  4.3 Proof of Propisition 1.7

It is easy to verify that

$\displaystyle R_{{\rm w}_{t}^{0}}(z)=tz^{2},$ $\displaystyle R_{{\rm
d}_{a}}(z)=\frac{1}{2}\Big{[}\sqrt{1+4a^{2}z^{2}}-1\Big{]}.$

Then (1.28) is immediately concluded from Theorem 1.1$(i)$. We have already
obtained $S_{{\rm d}_{a}}(z)$ as (1.17) and $S_{{\rm w}_{t}^{0}}(z)$ as
(1.23). Then (1.25) of Theorem 1.5$(ii)$ gives

$\displaystyle\frac{S_{{\rm w}_{t}}(z)}{1-tzS_{{\rm
w}_{t}}(z)^{2}}=\frac{1}{a}\sqrt{\frac{z+1-tz^{2}S_{{\rm
w}_{t}}(z)^{2}}{z-tz^{2}S_{{\rm w}_{t}}(z)^{2}}}.$

This is written as

$\displaystyle t^{2}z^{3}S_{{\rm
w}_{t}}(z)^{4}-z\big{\\{}2tz+t+a^{2}\big{\\}}S_{{\rm w}_{t}}(z)^{2}+(z+1)=0,$

which is solved by (1.29).

###  4.4 Proof of Proposition 1.8

It is easy to verify that

$\displaystyle R_{{\rm m}_{\lambda,t}^{0}}(z)=\frac{\lambda
tz}{1-tz}=\lambda\bigg{(}\frac{1}{1-tz}-1\bigg{)},$ $\displaystyle
R_{\delta_{b}}(z)=bz.$

Then (1.30) is immediately concluded from Theorem 1.1$(ii)$. We can see that
$S_{\delta_{b}}(z)=1/b$. Then (1.26) of Theorem 1.5$(iii)$ gives

$\displaystyle\frac{S_{{\rm m}_{\lambda,t}}(z)}{(1-tzS_{{\rm
m}_{\lambda,t}}(z))\\{1-t(\lambda+z)S_{{\rm
m}_{\lambda,t}}(z)\\}}=\frac{1}{b}.$

This is written as

$\displaystyle
t^{2}z(z+\lambda)S_{\mu_{t}}(z)^{2}-(2tz+t\lambda+b)S_{\mu_{t}}(z)+1=0.$

which is solved by (1.31).

##  5 Concluding remarks

We list out some concluding remarks.

$1.$ In addition to the free Brownian motion [11] and the free Wishart process
[22], Demni introduced the free Jacobi process in [30]. This process has two
parameters $\lambda$ and $\theta$. He derived the following PDE for the Cauchy
transform of the measure-valued process $({\rm k}_{\lambda,\theta,t})_{t\geq
0}$,

$\displaystyle\frac{\partial G_{{\rm k}_{\lambda,\theta,t}}(z)}{\partial t}$
$\displaystyle+[2\lambda\theta z(1-z)G_{{\rm
k}_{\lambda,\theta,t}}(z)+\\{(2\lambda\theta-1)z+\theta(1-\lambda)\\}]\frac{\partial
G_{{\rm k}_{\lambda,\theta,t}}(z)}{\partial z}$
$\displaystyle+\\{\lambda\theta(1-2z)G_{{\rm
k}_{\lambda,\theta,t}}(z)+(2\lambda\theta-1)\\}G_{{\rm
k}_{\lambda,\theta,t}}(z)=0.$ (5.1)

Demni showed that this equation has the stationary measure given by

$\displaystyle{\rm k}_{\lambda,\theta}({\rm
d}x)=\max\bigg{(}0,1-\frac{1}{\lambda}\bigg{)}\delta_{0}({\rm
d}x)+\max\bigg{(}0,1-\frac{1-\theta}{\lambda\theta}\bigg{)}\delta_{1}({\rm
d}x)+g(x)1_{[x_{-},x_{+}]}(x)\,{\rm d}x,$

where

$\displaystyle g(x)=\frac{\sqrt{(x-x_{-})(x_{+}-x)}}{2\lambda\theta\pi
x(1-x)}\qquad\text{with}\quad
x_{\pm}=\big{(}\sqrt{\theta(1-\lambda\theta)}\pm\sqrt{\lambda\theta(1-\theta)}\big{)}^{2}.$

The distribution with the density $g(x)$ is known as the Kesten–McKay law [51,
54]. Is it possible to solve the initial-value problem for (5.1) as we did in
this paper? When $\lambda=1$ and $\theta=1/2$, (5.1) is much simplified and
given by

$\displaystyle\frac{\partial G_{{\rm k}_{1,1/2,t}}(z)}{\partial
t}+\frac{\partial}{\partial z}\bigg{\\{}\frac{1}{2}z(1-z)G_{{\rm
k}_{1,1/2,t}}(z)^{2}\bigg{\\}}=0.$

This equation was solved by Demni, Hamdi, and Hmidi [32] when the initial
probability measure is given by $\delta_{1}$ and by Izumi and Ueda for general
initial probability measure [44]. In these papers, the solutions are related
with the free unitary Brownian motion [11]. See also [31, 39] for further
study.

Another PDE for measure-valued process was reported in [66]

$\displaystyle\frac{\partial G_{{\rm t}_{\alpha,c,t}}(z)}{\partial t}$
$\displaystyle-\big{\\{}2czG_{{\rm
t}_{\alpha,c,t}}(z)+(z+2-\alpha)\big{\\}}\frac{\partial G_{{\rm
t}_{\alpha,c,t}}(z)}{\partial z}-z\frac{\partial^{2}G_{{\rm
t}_{\alpha,c,t}}(z)}{\partial z^{2}}$ $\displaystyle-\big{\\{}cG_{{\rm
t}_{\alpha,c,t}}(z)+1\big{\\}}G_{{\rm t}_{\alpha,c,t}}(z)=0,$

where $\alpha$ and $c$ are positive parameters. This describes the
hydrodynamic limit of the Bru–Wishart (Laguerre) process in a high temperature
regime (see also [58]). Notice that this equation involves a second-order
derivative of $G_{{\rm t}_{\alpha,c,t}}(z)$ and hence it is regarded as a
viscous Burgers-type equation. As proved by [25, 63], the hydrodynamic limit
of Dyson’s Brownian motion model is described by the inviscid Burgers equation
(1.2). In this paper we have studied only such inviscid cases of Burgers-type
equations (see Theorem 2.4 in Section 2.1). As shown by [18, 23, 24], however,
if we consider the system of SDEs which have the same drift terms with Dyson’s
Brownian motion model (2.1) but the martingale terms are replaced as ${\rm
d}B_{t}^{i}\to\sqrt{N}{\rm d}B_{t}^{i}$ (compare [23, equation (5.93)] with
[63, equation (7)]), then we obtain the viscous Burgers equation in the
hydrodynamic limit (see [23, equation (5.98)]). How can we solve such viscous
Burgers-type equations?

$2.$ Forrester and Grela [36] studied the hydrodynamic limits of the circular
ensemble as well as the Jacobi ensemble. In the former case, they considered
the following type of Cauchy transform,

${G^{\circ}_{\mu}(z)=\frac{1}{2}\oint\cot\bigg{(}\frac{z-x}{2}\bigg{)}\mu({\rm
d}x)}.$

This seems to be a trigonometric extension of the usual Cauchy transform
(1.1), since if we introduce a parameter $r>0$, then we see
$(1/2r)\cot((z-x)/2r)\to 1/(z-x)$ as $r\to\infty$. See also [24, 43]. Some
elliptic extensions of Cauchy-type transform have been also considered in a
recent study of elliptic integrable systems [9, 10]. Is it meaningful to
consider trigonometric and elliptic extensions of free probability theory?

$3.$ In the present paper, we have studied the complex Burgers-type equations
which are obtained in the hydrodynamic limits of the stochastic log-gases
studied in random matrix theory [35, 46]. Recently, one of the present authors
and Koshida proposed a new construction of the multiple Schramm–Loewner
evolutions (SLEs) driven by stochastic log-gases using the notion of the
coupling between the multiple SLEs and the Gaussian free fields [47, 48, 49].
Then the infinite-slit limits of the Loewner equations studied by [27, 28, 43]
in the case that the slits are growing simultaneously can be interpreted as
the hydrodynamic limits of the multiple SLEs [42] in the same context of
stochastic processes as explained in Section 2.1. In the study of the
hydrodynamic limits of the multiple SLEs, exact solutions of the complex
Burgers-type equations for interesting initial conditions is very important
[42]. On the other hand, new connections between the free probability theory
and the Loewner chains have been reported [37, 43]. Moreover, an interesting
discussion was given such that the complex Burgers-type equations themselves
can be regarded as Loewner equations for certain subordination processes [43].
The method to solve the initial-value problems for the complex Burgers-type
equations by deriving the functional equations reported in this paper shall be
developed to analyze the hydrodynamic limits of multiple SLEs.

### Acknowledgements

The present authors would like to thank Shinji Koshida and Yoshimichi Ueda for
useful comments on the manuscript. They are grateful to the anonymous referees
for valuable suggestions for future studies on this subject. MK was supported
by the Grant-in-Aid for Scientific Research (C) (No. 19K03674), (B) (No.
18H01124), (S) (No. 16H06338), (A) (No. 21H04432) of Japan Society for the
Promotion of Science. NS was supported by the Grant-in-Aid for Scientific
Research (B) (No. 19H01791) and (C)(No. 19K03515) of Japan Society for the
Promotion of Science.

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