Elliptic Determinantal Processes and Elliptic Dyson Models

Elliptic Determinantal Processes
and Elliptic Dyson Models††This paper is a contribution to the Special Issue
on Elliptic Hypergeometric Functions and Their Applications. The full
collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html

Makoto KATORI

M. Katori

Department of Physics, Faculty of Science and Engineering, Chuo University,
Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan<EMAIL_ADDRESS>
Received April 19, 2017, in final form September 29, 2017; Published online
October 04, 2017

We introduce seven families of stochastic systems of interacting particles in
one-dimension corresponding to the seven families of irreducible reduced
affine root systems. We prove that they are determinantal in the sense that
all spatio-temporal correlation functions are given by determinants controlled
by a single function called the spatio-temporal correlation kernel. For the
four families ${A}_{N-1}$, ${B}_{N}$, ${C}_{N}$ and ${D}_{N}$, we identify the
systems of stochastic differential equations solved by these determinantal
processes, which will be regarded as the elliptic extensions of the Dyson
model. Here we use the notion of martingales in probability theory and the
elliptic determinant evaluations of the Macdonald denominators of irreducible
reduced affine root systems given by Rosengren and Schlosser.

elliptic determinantal processes; elliptic Dyson models; determinantal
martingales; elliptic determinant evaluations; irreducible reduced affine root
systems

60J65; 60G44; 82C22; 60B20; 33E05; 17B22

##  1 Introduction and main results

Stochastic analysis on interacting particle systems is important to provide
useful models describing equilibrium and non-equilibrium phenomena studied in
statistical physics [15]. Determinantal process is a stochastic system of
interacting particles which is integrable in the sense that all spatio-
temporal correlation functions are given by determinants controlled by a
single function called the spatio-temporal correlation kernel [4, 20]. Since
the generating functions of correlation functions are generally given by the
Laplace transforms of probability densities, the stochastic integrability of
determinantal processes is proved by showing that the Laplace transform of any
multi-time joint probability density is expressed by the spatio-temporal
Fredholm determinant associated with the correlation kernel. The purpose of
this paper is to present new kinds of determinantal processes in which the
interactions between particles are described by the logarithmic derivatives of
Jacobi’s theta functions. A classical example of determinantal processes is
Dyson’s Brownian motion model with parameter $\beta=2$, which is a dynamical
version of the eigenvalue statistics of random matrices in the Gaussian
unitary ensemble (GUE), and we call it simply the Dyson model [6, 15, 38]. We
will extend the Dyson model to the elliptic-function-level in this paper. We
use the notion of martingales in probability theory [14, 15] and the elliptic
determinantal evaluations of the Macdonald denominators of seven families of
irreducible reduced affine root systems given by Rosengren and Schlosser [34]
(see also [26, 42]).

Among the seven families of irreducible reduced affine root systems,
${R}={A}_{N-1},{B}_{N},{B}^{\vee}_{N},{C}_{N}$, ${C}^{\vee}_{N},{BC}_{N}$, and
${D}_{N}$, we reported the results only for the system ${R}={A}_{N-1}$ in the
previous papers [16, 17] as follows. Assume $0<t_{\ast}<\infty$, $0<r<\infty$,
$0<{\cal N}<\infty$, and let

$\displaystyle A_{{\cal N}}^{2\pi r}(t_{\ast}-t,x)=\left[\frac{1}{2\pi
r}\frac{\partial}{\partial v}\log\vartheta_{1}(v;\tau)\right]_{v=x/2\pi
r,\,\tau=i{\cal N}(t_{\ast}-t)/2\pi r^{2}},\qquad t\in[0,t_{\ast}),$ (1.1)

where $\theta_{1}(v;\tau)$ denotes one of the Jacobi theta functions. See
Appendix A for the Jacobi theta functions and related functions. For
$N\in\\{2,3,\dots\\}$, we define the Weyl chamber

$\displaystyle\mathbb{W}_{N}=\big{\\{}{\bf
x}=(x_{1},x_{2},\dots,x_{N})\in\mathbb{R}^{N}\colon
x_{1}<x_{2}<\cdots<x_{N}\big{\\}}.$

We introduced a one-parameter ($\beta>0$) family of systems of stochastic
differential equations (SDEs) for $t\in[0,t_{\ast})$ [17]

$\displaystyle({A}_{N-1})\quad$ $\displaystyle
X^{{A}_{N-1}}_{j}(t)=u_{j}+W_{j}(t)+\frac{\beta}{2}\int_{0}^{t}A^{2\pi
r}_{N}\left(t_{\ast}-s,\sum_{\ell=1}^{N}X^{{A}_{N-1}}_{\ell}(s)-\kappa_{N}\right)ds$
$\displaystyle\hphantom{X^{{A}_{N-1}}_{j}(t)=}{}+\frac{\beta}{2}\sum_{\begin{subarray}{c}1\leq
k\leq N,\\\ k\not=j\end{subarray}}\int_{0}^{t}A^{2\pi
r}_{N}\big{(}t_{\ast}-s,X^{{A}_{N-1}}_{j}(s)-X^{{A}_{N-1}}_{k}(s)\big{)}ds,$
$\displaystyle j=1,\dots,N,\ \ \text{in $\mathbb{R}$},$ (1.2)

for ${\bf u}=(u_{1},\dots,u_{N})\in\mathbb{W}_{N}$, where $W_{j}(t)$, $t\geq
0$, $j=1,\dots,N$ are independent one-dimensional standard Brownian motions,
and

$\displaystyle\kappa_{N}=\begin{cases}\pi r(N-1),&\text{if $N$ is even},\\\
\pi r(N-2),&\text{if $N$ is odd}.\end{cases}$

We called this family of $N$-particle systems on $\mathbb{R}$, ${\bf
X}^{{A}_{N-1}}(t)=\big{(}X^{{A}_{N-1}}_{1}(t),\dots,X^{{A}_{N-1}}_{N}(t)\big{)}$,
$t\in[0,t_{\ast})$, the elliptic Dyson model of type A with parameter $\beta$.
By (A.9) in Appendix A.2, (1.2) gives

$\displaystyle dX^{{A}_{N-1}}_{j}(t)\sim
dW_{j}(t)+\frac{\beta}{2}\frac{v^{{A}_{N-1}}_{j}-X^{{A}_{N-1}}_{j}(t)}{t_{\ast}-t}dt,\qquad
j=1,2,\dots,N,\quad\text{in $t\uparrow t_{\ast}$},$

with

$\displaystyle v^{{A}_{N-1}}_{j}=\begin{cases}\displaystyle\frac{\pi
r}{N}(2j-1),&\text{if $N$ is even},\vspace{1mm}\\\ \displaystyle\frac{2\pi
r}{N}(j-1),&\text{if $N$ is odd},\end{cases}\qquad j=1,2,\dots,N,$

which give equidistant-spacing configurations ${\bf
v}^{{A}_{N-1}}=\big{(}v^{{A}_{N-1}}_{1},\dots,v^{{A}_{N-1}}_{N}\big{)}$. This
implies that the elliptic Dyson model of type A is realized as a system of
interacting Brownian bridges (see, for instance, [5, Part I, Section IV.4.22])
pinned at the configuration ${\bf v}^{{A}_{N-1}}$ at time $t_{\ast}$ [17].
When the system of SDEs is temporally homogeneous, it is well known that the
corresponding Kolmogorov (Fokker–Planck) equation for transition probability
density can be transformed into a Calogero–Moser–Sutherland quantum system
(see, for instance, [8, Chapter 11]). The present system of interacting
Brownian bridges is temporally inhomogeneous, however, and the Kolmogorov
equation is mapped into a Schrödinger-type equation with time-dependent
Hamiltonian and the time-dependent ground energy [17]. The obtained quantum
system is elliptic, but different from the elliptic Calogero–Moser–Sutherland
model extensively studied as a quantum integrable system [7, 25, 31, 39, 40].
We found at the same time that the interaction among particles vanishes when
the parameter is chosen to be a special value ($\beta=2$ in our case) as found
in the usual Calogero–Moser–Sutherland models. We applied the determinantal-
martingale-method [14] and proved that the elliptic Dyson model of type A with
$\beta=2$ is an integrable stochastic process in a sense that it is
determinantal for a set of observables [16, 17].

In the present paper, we report the results for other six systems,
${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N}$, ${BC}_{N},{D}_{N}$. Here
we first construct the six families of determinantal processes (Theorem 1.1).
Then for the three families ${B}_{N}$, ${C}_{N}$ and ${D}_{N}$, we clarify the
systems of SDEs (with parameter $\beta=2$) which are solved by our new
determinantal processes (Theorem 1.2).

For $N\in\\{2,3,\dots\\}$, $0<r<\infty$, define

$\displaystyle\mathbb{W}_{N}^{(0,\pi r)}=\big{\\{}{\bf
x}=(x_{1},x_{2},\dots,x_{N})\in\mathbb{R}^{N}\colon
0<x_{1}<x_{2}<\cdots<x_{N}<\pi r\big{\\}}.$

Let ${\bf u}=(u_{1},u_{2},\dots,u_{N})\in\mathbb{W}_{N}^{(0,\pi r)}$, and

$\displaystyle\tau^{{R}}(t)=\frac{i{\cal N}^{{R}}(t_{\ast}-t)}{2\pi r^{2}},$
(1.3)

where the numbers ${\cal N}^{{R}}$ are given by

$\displaystyle{\cal N}^{{R}}=\begin{cases}N,&{R}={A}_{N-1},\\\
2N-1,&{R}={B}_{N},\\\ 2N,&{R}={B}^{\vee}_{N},{C}^{\vee}_{N},\\\
2(N+1),&{R}={C}_{N},\\\ 2N+1,&{R}={BC}_{N},\\\
2(N-1),&{R}={D}_{N}.\end{cases}$ (1.4)

For $t\in[0,t_{\ast})$ put

$\displaystyle D^{{R}}_{{\bf u}}(t,{\bf
x})=\frac{c^{{R}}_{0}\big{(}\tau^{{R}}(t)\big{)}}{c^{{R}}_{0}\big{(}\tau^{{R}}(0)\big{)}}\prod_{\ell=1}^{N}\frac{\vartheta_{1}\big{(}c^{{R}}_{1}x_{\ell}/2\pi
r;c^{{R}}_{2}\tau^{{R}}(t)\big{)}}{\vartheta_{1}\big{(}c^{{R}}_{1}u_{\ell}/2\pi
r;c^{{R}}_{2}\tau^{{R}}(0)\big{)}}$ $\displaystyle\hphantom{D^{{R}}_{{\bf
u}}(t,{\bf x})=}{}\times\prod_{1\leq j<k\leq
N}\frac{\vartheta_{1}\big{(}(x_{k}-x_{j})/2\pi
r;\tau^{{R}}(t)\big{)}}{\vartheta_{1}\big{(}(u_{k}-u_{j})/2\pi
r;\tau^{{R}}(0)\big{)}}\frac{\vartheta_{1}\big{(}(x_{k}+x_{j})/2\pi
r;\tau^{{R}}(t)\big{)}}{\vartheta_{1}\big{(}(u_{k}+u_{j})/2\pi
r;\tau^{{R}}(0)\big{)}}$ $\displaystyle\hphantom{D^{{R}}_{{\bf u}}(t,{\bf
x})=}{}\mbox{for ${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N}$},$ (1.5)
$\displaystyle D^{{BC}_{N}}_{{\bf u}}(t,{\bf
x})=\frac{c^{{BC}_{N}}_{0}\big{(}\tau^{{BC}_{N}}(t)\big{)}}{c^{{BC}_{N}}_{0}\big{(}\tau^{{BC}_{N}}(0)\big{)}}\prod_{\ell=1}^{N}\frac{\vartheta_{1}\big{(}x_{\ell}/2\pi
r;\tau^{{BC}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}u_{\ell}/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}}\frac{\vartheta_{0}\big{(}x_{\ell}/\pi
r;2\tau^{{BC}_{N}}(t)\big{)}}{\vartheta_{0}\big{(}u_{\ell}/\pi
r;2\tau^{{BC}_{N}}(0)\big{)}}$ $\displaystyle\hphantom{D^{{BC}_{N}}_{{\bf
u}}(t,{\bf x})=}{}\times\prod_{1\leq j<k\leq
N}\frac{\vartheta_{1}\big{(}(x_{k}-x_{j})/2\pi
r;\tau^{{BC}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(u_{k}-u_{j})/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}}\frac{\vartheta_{1}\big{(}(x_{k}+x_{j})/2\pi
r;\tau^{{BC}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(u_{k}+u_{j})/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}},$ (1.6) $\displaystyle D^{{D}_{N}}_{{\bf
u}}(t,{\bf
x})=\frac{c^{{D}_{N}}_{0}\big{(}\tau^{{D}_{N}}(t)\big{)}}{c^{{D}_{N}}_{0}\big{(}\tau^{{D}_{N}}(0)\big{)}}$
$\displaystyle\hphantom{D^{{D}_{N}}_{{\bf u}}(t,{\bf x})=}{}\times\prod_{1\leq
j<k\leq N}\frac{\vartheta_{1}\big{(}(x_{k}-x_{j})/2\pi
r;\tau^{{D}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(u_{k}-u_{j})/2\pi
r;\tau^{{D}_{N}}(0)\big{)}}\frac{\vartheta_{1}\big{(}(x_{k}+x_{j})/2\pi
r;\tau^{{D}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(u_{k}+u_{j})/2\pi
r;\tau^{{D}_{N}}(0)\big{)}},$ (1.7)

where

$\displaystyle c^{{R}}_{0}(\tau)=\eta(\tau)^{-N(N-1)}\qquad\text{for
${R}={B}_{N},{C}_{N}$},$ $\displaystyle
c^{{B}^{\vee}_{N}}_{0}(\tau)=\eta(\tau)^{-(N-1)^{2}}\eta(2\tau)^{-(N-1)},$
$\displaystyle
c^{{C}^{\vee}_{N}}_{0}(\tau)=\eta(\tau)^{-(N-1)^{2}}\eta(\tau/2)^{-(N-1)},$
$\displaystyle c^{{BC}_{N}}_{0}(\tau)=\eta(\tau)^{-N(N-1)}\eta(2\tau)^{-N},$
$\displaystyle c^{{D}_{N}}_{0}(\tau)=\eta(\tau)^{-N(N-2)},$ (1.8)

and

$\displaystyle c^{{B}_{N}}_{1}=c^{{C}^{\vee}_{N}}_{1}=1,\qquad
c^{{B}^{\vee}_{N}}_{1}=c^{{C}_{N}}_{1}=2,$ $\displaystyle
c^{{B}_{N}}_{2}=c^{{C}_{N}}_{2}=1,\qquad c^{{B}^{\vee}_{N}}_{2}=2,\qquad
c^{{C}^{\vee}_{N}}_{2}=1/2.$ (1.9)

Here $\eta(\tau)$ denotes the Dedekind modular function (see, for instance,
[32, Section 23.15])

$\displaystyle\eta(\tau)=e^{\tau\pi
i/12}\prod_{n=1}^{\infty}\big{(}1-e^{2n\tau\pi i}\big{)},\qquad\Im\tau>0.$
(1.10)

We call (1.5)–(1.7) the determinantal martingale-functions [14] (see Sections
2.2 and 2.5).

In the interval $[0,\pi r]$, we consider the $N$-particle system of one-
dimensional standard Brownian motions ${\bf B}(t)=(B_{1}(t),\dots,B_{N}(t))$,
$t\geq 0$ started at ${\bf u}=(u_{1},\dots,u_{N})\in\mathbb{W}_{N}^{(0,\pi
r)}$ with either an absorbing or reflecting boundary condition at the
endpoints of the interval, $0$ and $\pi r$. The transition probability density
of each particle is generally denoted by $p^{[0,\pi r]}$. The boundary
conditions at 0 and $\pi r$ are indicated by $b$ and $b^{\prime}$,
respectively, and the transition probability density with the specified
boundary conditions $b$, $b^{\prime}$ is written as
$p^{[0,\pi]}_{b,b^{\prime}}$. In this paper the absorbing (resp. reflecting)
boundary condition is abbreviated as ‘a’ (resp. ‘r’). By the reflection
principle of Brownian motion (see, for instance, [5, Appendixes 1.5 and 1.6]),
if both boundaries are absorbing, the transition probability density is given
by

$\displaystyle p^{[0,\pi r]}_{\rm
aa}(t,y|x)=\sum_{k=-\infty}^{\infty}\big{\\{}p_{\rm BM}(t,y+2\pi rk|x)-p_{\rm
BM}(t,y+2\pi rk|-x)\big{\\}},$ (1.11)

and if both are reflecting, it is given by

$\displaystyle p^{[0,\pi r]}_{\rm
rr}(t,y|x)=\sum_{k=-\infty}^{\infty}\big{\\{}p_{\rm BM}(t,y+2\pi rk|x)+p_{\rm
BM}(t,y+2\pi rk|-x)\big{\\}},$ (1.12)

for $x,y\in[0,\pi r]$, $t\geq 0$, where $p_{\rm BM}(t,y|x)$ denotes the
transition probability density of the one-dimensional standard Brownian motion

$\displaystyle p_{\rm
BM}(t,y|x)=\begin{cases}\displaystyle\frac{e^{-(x-y)^{2}/2t}}{\sqrt{2\pi
t}},&t>0,\\\ \delta(x-y),&t=0.\end{cases}$ (1.13)

We write the probability law of such a system of boundary-conditioned Brownian
motions in $[0,\pi r]$ as ${\rm P}^{[0,\pi r]}_{{\bf u}}$. In ${\rm P}^{[0,\pi
r]}_{{\bf u}}$, put

$\displaystyle T_{\rm collision}=\inf\big{\\{}t>0\colon B_{j}(t)=B_{k}(t)\
\text{for any $j\not=k$}\big{\\}},$ (1.14)

i.e., the first collision-time of the $N$-particle system of Brownian motions
in the interval $[0,\pi r]$. Let ${\bf 1}(\omega)$ be the indicator function
of a condition $\omega$; ${\bf 1}(\omega)=1$ if $\omega$ is satisfied, and
${\bf 1}(\omega)=0$ otherwise. Then we define

$\displaystyle\mathbb{P}_{{\bf u}}^{{R}}\big{|}_{{\cal F}_{t}}={\bf 1}(T_{\rm
collision}>t)D^{{R}}_{{\bf u}}(t,{\bf B}(t)){\rm P}_{{\bf u}}^{[0,\pi
r]}\big{|}_{{\cal F}_{t}},\qquad t\in[0,t_{\ast}),$ (1.15)

where $D^{{R}}_{{\bf u}}$ are given by (1.5)–(1.7) and ${\cal F}_{t}$ denotes
the filtration associated with the Brownian motion (see Section 2.1). That is,
the Radon–Nikodym derivative of $\mathbb{P}_{{\bf u}}^{{R}}$ with respect to
the Wiener measure ${\rm P}_{{\bf u}}^{[0,\pi r]}$ of Brownian motion is given
by ${\bf 1}(T_{\rm collision}>t)D^{{R}}_{{\bf u}}(t,{\bf B}(t))$ at each time
$t\in[0,t_{\ast})$. Therefore, a stochastic process with $N$ particles
governed by $\mathbb{P}_{{\bf u}}^{{R}}$ is well-defined as a realization of
non-intersecting paths which are absolutely continuous to the $N$-particle
paths of independent Brownian motions in $[0,\pi r]$.

For $y\in\mathbb{R}$, $\delta_{y}(\cdot)$ denotes the delta measure such that
$\delta_{y}(\\{x\\})=1$ if $x=y$ and $\delta_{y}(\\{x\\})=0$ otherwise. The
first theorem of this paper is the following.

######  Theorem 1.1.

Assume that $0<t_{\ast}<\infty$, $0<r<\infty$. For each $N\in\\{2,3,\dots\\}$,
${R}={B}_{N}$, ${B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N},{BC}_{N},{D}_{N}$, ${\bf
u}=(u_{1},\dots,u_{N})\in\mathbb{W}_{N}^{(0,\pi r)}$, $\mathbb{P}_{{\bf
u}}^{{R}}$ defined by (1.15) gives a probability measure and defines a
measure-valued stochastic process

$\displaystyle\Xi^{{R}}(t,\cdot)=\sum_{j=1}^{N}\delta_{X^{{R}}_{j}(t)}(\cdot),\qquad
t\in[0,t_{\ast}).$ (1.16)

The process
$\big{(}\big{(}\Xi^{{R}}(t)\big{)}_{t\in[0,t_{\ast})},\mathbb{P}_{{\bf
u}}^{{R}}\big{)}$ is determinantal with the spatio-temporal correlation kernel

$\displaystyle\mathbb{K}^{{R}}_{{\bf u}}(s,x;t,y)=\sum_{j=1}^{N}p^{[0,\pi
r]}(s,x|u_{j})M^{{R}}_{{\bf u},u_{j}}(t,y)-{\bf 1}(s>t)p^{[0,\pi
r]}(s-t,x|y),$ (1.17)

$(s,x),(t,y)\in[0,t_{\ast})\times[0,\pi r]$, where

$\displaystyle M^{{R}}_{{\bf
u},u_{j}}(t,x)=\int_{-\infty}^{\infty}\Phi^{{R}}_{{\bf
u},u_{j}}(x+i\widetilde{x})p_{\rm BM}(t,\widetilde{x}|0)d\widetilde{x}$ (1.18)

with the sets of entire functions $($the elliptic Lagrange interpolation
functions$)$

$\displaystyle\Phi^{{R}}_{{\bf
u},u_{j}}(z)=\frac{\vartheta_{1}\big{(}c^{{R}}_{1}z/2\pi
r;c^{{R}}_{2}\tau^{{R}}(0)\big{)}}{\vartheta_{1}\big{(}c^{{R}}_{1}u_{j}/2\pi
r;c^{{R}}_{2}\tau^{{R}}(0)\big{)}}$ $\displaystyle\hphantom{\Phi^{{R}}_{{\bf
u},u_{j}}(z)=}{}\times\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{\vartheta_{1}\big{(}(z-u_{\ell})/2\pi
r;\tau^{{R}}(0)\big{)}}{\vartheta_{1}\big{(}(u_{j}-u_{\ell})/2\pi
r;\tau^{{R}}(0)\big{)}}\frac{\vartheta_{1}\big{(}(z+u_{\ell})/2\pi
r;\tau^{{R}}(0)\big{)}}{\vartheta_{1}\big{(}(u_{j}+u_{\ell})/2\pi
r;\tau^{{R}}(0)\big{)}}$ $\displaystyle\hphantom{\Phi^{{R}}_{{\bf
u},u_{j}}(z)=}{}\text{for
${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N}$},$ (1.19)
$\displaystyle\Phi^{{BC}_{N}}_{{\bf
u},u_{j}}(z)=\frac{\vartheta_{1}\big{(}z/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}}{\vartheta_{1}\big{(}u_{j}/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}}\frac{\vartheta_{0}\big{(}z/\pi
r;2\tau^{{BC}_{N}}(0)\big{)}}{\vartheta_{0}\big{(}u_{j}/\pi
r;2\tau^{{BC}_{N}}(0)\big{)}}$ $\displaystyle\hphantom{\Phi^{{BC}_{N}}_{{\bf
u},u_{j}}(z)=}{}\times\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{\vartheta_{1}\big{(}(z-u_{\ell})/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}}{\vartheta_{1}\big{(}(u_{j}-u_{\ell})/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}}\frac{\vartheta_{1}\big{(}(z+u_{\ell})/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}}{\vartheta_{1}\big{(}(u_{j}+u_{\ell})/2\pi
r;\tau^{{BC}_{N}}(0)\big{)}},$ (1.20) $\displaystyle\Phi^{{D}_{N}}_{{\bf
u},u_{j}}(z)=\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{\vartheta_{1}\big{(}(z-u_{\ell})/2\pi
r;\tau^{{D}_{N}}(0)\big{)}}{\vartheta_{1}\big{(}(u_{j}-u_{\ell})/2\pi
r;\tau^{{D}_{N}}(0)\big{)}}\frac{\vartheta_{1}\big{(}(z+u_{\ell})/2\pi
r;\tau^{{D}_{N}}(0)\big{)}}{\vartheta_{1}\big{(}(u_{j}+u_{\ell})/2\pi
r;\tau^{{D}_{N}}(0)\big{)}},$ (1.21)

$j=1,2,\dots,N$. Here $\tau^{{R}}(t)$ are given by (1.3) with (1.4), and
$c^{{R}}_{1}$, $c^{{R}}_{2}$ for
${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N}$ are given by (1.9).

The second theorem of this paper is the following.

######  Theorem 1.2.

Let $N\in\\{2,3,\dots\\}$, $0<t_{\ast}<\infty$, $0<r<\infty$. Assume ${\bf
u}=(u_{1},\dots,u_{N})\in\mathbb{W}_{N}^{(0,\pi r)}$. In the interval $[0,\pi
r]$, for
$\big{(}\big{(}\Xi^{{B}_{N}}(t)\big{)}_{t\in[0,t_{\ast})},\mathbb{P}^{{B}_{N}}_{{\bf
u}}\big{)}$ we put an absorbing boundary condition at $0$ and a reflecting
boundary condition at $\pi r$, for
$\big{(}\big{(}\Xi^{{C}_{N}}(t)\big{)}_{t\in[0,t_{\ast})},\mathbb{P}^{{C}_{N}}_{{\bf
u}}\big{)}$ we put an absorbing boundary condition both at $0$ and $\pi r$,
and for
$\big{(}\big{(}\Xi^{{D}_{N}}(t)\big{)}_{t\in[0,t_{\ast})},\mathbb{P}^{{D}_{N}}_{{\bf
u}}\big{)}$ we put a reflecting boundary condition both at $0$ and $\pi r$,
respectively. If we set
$\Xi^{{R}}(t)=\sum\limits_{j=1}^{N}\delta_{X^{{R}}_{j}(t)}$, then ${\bf
X}^{{R}}(t)=\big{(}X^{{R}}_{1}(t),\dots,X^{{R}}_{N}(t)\big{)}$,
${R}={B}_{N},{C}_{N},{D}_{N}$, solve the following systems of SDEs

$\displaystyle({B}_{N})\quad
X^{{B}_{N}}_{j}(t)=u_{j}+W_{j}(t)+\int_{0}^{t}A^{2\pi
r}_{2N-1}\big{(}t_{\ast}-s,X^{{B}_{N}}_{j}(s)\big{)}ds$
$\displaystyle\hphantom{({B}_{N})\quad
X^{{B}_{N}}_{j}(t)=}{}+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\big{\\{}A^{2\pi
r}_{2N-1}\big{(}t_{\ast}-s,X^{{B}_{N}}_{j}(s)-X^{{B}_{N}}_{k}(s)\big{)}$
$\displaystyle\hphantom{({B}_{N})\quad X^{{B}_{N}}_{j}(t)=}{}+A^{2\pi
r}_{2N-1}\big{(}t_{\ast}-s,X^{{B}_{N}}_{j}(s)+X^{{B}_{N}}_{k}(s)\big{)}\big{\\}}ds,\qquad
j=1,2,\dots,N,$ $\displaystyle\hphantom{({B}_{N})\quad
X^{{B}_{N}}_{j}(t)=}{}\text{in $[0,\pi r]$ with a reflecting boundary
condition at $\pi r$},$ (1.22) $\displaystyle({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=u_{j}+W_{j}(t)+2\int_{0}^{t}A^{2\pi
r}_{2(N+1)}\big{(}t_{\ast}-s,2X^{{C}_{N}}_{j}(s)\big{)}ds$
$\displaystyle\hphantom{({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=}{}+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\big{\\{}A^{2\pi
r}_{2(N+1)}\big{(}t_{\ast}-s,X^{{C}_{N}}_{j}(s)-X^{{C}_{N}}_{k}(s)\big{)}$
$\displaystyle\hphantom{({C}_{N})\quad X^{{C}_{N}}_{j}(t)=}{}+A^{2\pi
r}_{2(N+1)}\big{(}t_{\ast}-s,X^{{C}_{N}}_{j}(s)+X^{{C}_{N}}_{k}(s)\big{)}\big{\\}}ds,\qquad
j=1,2,\dots,N,$ $\displaystyle\hphantom{({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=}{}\text{in $[0,\pi r]$},$ (1.23)
$\displaystyle({D}_{N})\quad
X^{{D}_{N}}_{j}(t)=u_{j}+W_{j}(t)+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\big{\\{}A^{2\pi
r}_{2(N-1)}\big{(}t_{\ast}-s,X^{{D}_{N}}_{j}(s)-X^{{D}_{N}}_{k}(s)\big{)}$
$\displaystyle\hphantom{({D}_{N})\quad X^{{D}_{N}}_{j}(t)=}{}+A^{2\pi
r}_{2(N-1)}\big{(}t_{\ast}-s,X^{{D}_{N}}_{j}(s)+X^{{D}_{N}}_{k}(s)\big{)}\big{\\}}ds,\qquad
j=1,2,\dots,N,$ $\displaystyle\hphantom{({D}_{N})\quad
X^{{D}_{N}}_{j}(t)=}{}\text{in $[0,\pi r]$ with a reflecting boundary
condition both at $0$ and $\pi r$},$ (1.24)

where $W_{j}(t)$, $t\geq 0$, $j=1,2,\dots,N$ are independent one-dimensional
standard Brownian motions.

We call the systems (1.22)–(1.24) the elliptic Dyson models of types B, C, D,
respectively. By (A.9) in Appendix A.2, we see that in $t\uparrow t_{\ast}$

$\displaystyle dX^{{R}}_{j}(t)\sim
dW_{j}(t)+\frac{v^{{R}}_{j}-X^{{R}}_{j}(t)}{t_{\ast}-t}dt,\qquad
j=1,2,\dots,N,\qquad{R}={B}_{N},{C}_{N},{D}_{N}$

with

$\displaystyle v^{{B}_{N}}_{j}=\frac{2j-1}{2N-1}\pi r,\qquad
v^{{C}_{N}}_{j}=\frac{j}{N+1}\pi r,\qquad v^{{D}_{N}}_{j}=\frac{j-1}{N-1}\pi
r,\qquad j=1,2,\dots,N.$

The above implies that the elliptic Dyson models of types B, C, and D are
realized as the systems of interacting Brownian bridges pinned at the
configurations ${\bf v}^{{R}}=\big{(}v^{{R}}_{1},\dots,v^{{R}}_{N}\big{)}$,
${R}={B}_{N},{C}_{N},{D}_{N}$, at time $t_{\ast}$. Note that
$v^{{B}_{N}}_{1}>0$, $v^{{B}_{N}}_{N}=\pi r$; $v^{{C}_{N}}_{1}>0$,
$v^{{C}_{N}}_{N}<\pi r$; $v^{{D}_{N}}_{1}=0$, $v^{{D}_{N}}_{N}=\pi r$,
corresponding to the situation such that the absorbing boundary condition is
imposed at 0 for ${B}_{N}$, and at 0 and $\pi r$ for ${C}_{N}$, while the
reflecting boundary condition is imposed at other endpoints of the interval
$[0,\pi r]$.

We see

$\displaystyle\lim_{t_{\ast}\to\infty}A_{{\cal N}}^{2\pi
r}(t_{\ast}-t,x)=\frac{1}{2r}\cot\left(\frac{x}{2r}\right),$

by (1.1) and (A.4). Hence in the limit $t_{\ast}\to\infty$, the systems of
SDEs (1.2) with $\beta=2$, and of (1.22)–(1.24) become the following
temporally homogeneous systems of SDEs for $t\in[0,\infty)$:

$\displaystyle({A}_{N-1})\quad
X^{{A}_{N-1}}_{j}(t)=u_{j}+W_{j}(t)-\frac{1}{2r}\int_{0}^{t}\tan\left(\frac{1}{2r}\sum_{\ell=1}^{N}X^{{A}_{N-1}}_{\ell}(s)\right)ds$
$\displaystyle\hphantom{({A}_{N-1})\quad
X^{{A}_{N-1}}_{j}(t)=}{}+\frac{1}{2r}\sum_{\begin{subarray}{c}1\leq k\leq
N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\cot\left(\frac{X^{{A}_{N-1}}_{j}(s)-X^{{A}_{N-1}}_{k}(s)}{2r}\right)ds,$
$\displaystyle\hphantom{({A}_{N-1})\quad
X^{{A}_{N-1}}_{j}(t)=}{}j=1,2,\dots,N,\quad\text{in $\mathbb{R}$},$ (1.25)
$\displaystyle({B}_{N})\quad
X^{{B}_{N}}_{j}(t)=u_{j}+W_{j}(t)+\frac{1}{2r}\int_{0}^{t}\cot\left(\frac{X^{{B}_{N}}_{j}(s)}{2r}\right)ds$
$\displaystyle\hphantom{({B}_{N})\quad
X^{{B}_{N}}_{j}(t)=}{}+\frac{1}{2r}\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\left\\{\cot\left(\frac{X^{{B}_{N}}_{j}(s)-X^{{B}_{N}}_{k}(s)}{2r}\right)\right.$
$\displaystyle\left.\hphantom{({B}_{N})\quad
X^{{B}_{N}}_{j}(t)=}{}+\cot\left(\frac{X^{{B}_{N}}_{j}(s)+X^{{B}_{N}}_{k}(s)}{2r}\right)\right\\}ds,\quad
j=1,2,\dots,N,$ $\displaystyle\hphantom{({B}_{N})\quad
X^{{B}_{N}}_{j}(t)=}{}\text{in $[0,\pi r]$ with a ref\/lecting boundary
condition at $\pi r$},$ (1.26) $\displaystyle({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=u_{j}+W_{j}(t)+\frac{1}{r}\int_{0}^{t}\cot\left(\frac{X^{{C}_{N}}_{j}(s)}{r}\right)ds$
$\displaystyle\hphantom{({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=}{}+\frac{1}{2r}\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\left\\{\cot\left(\frac{X^{{C}_{N}}_{j}(s)-X^{{C}_{N}}_{k}(s)}{2r}\right)\right.$
$\displaystyle\left.\hphantom{({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=}{}+\cot\left(\frac{X^{{C}_{N}}_{j}(s)+X^{{C}_{N}}_{k}(s)}{2r}\right)\right\\}ds,\quad
j=1,2,\dots,N,\quad\text{in $[0,\pi r]$},$ (1.27) $\displaystyle({D}_{N})\quad
X^{{D}_{N}}_{j}(t)=u_{j}+W_{j}(t)+\frac{1}{2r}\sum_{\begin{subarray}{c}1\leq
k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\left\\{\cot\left(\frac{X^{{D}_{N}}_{j}(s)-X^{{D}_{N}}_{k}(s)}{2r}\right)\right.$
$\displaystyle\left.\hphantom{({D}_{N})\quad
X^{{D}_{N}}_{j}(t)=}{}+\cot\left(\frac{X^{{D}_{N}}_{j}(s)+X^{{D}_{N}}_{k}(s)}{2r}\right)\right\\}ds,\quad
j=1,2,\dots,N,$ $\displaystyle\hphantom{({D}_{N})\quad
X^{{D}_{N}}_{j}(t)=}{}\text{in $[0,\pi r]$ with a ref\/lecting boundary
condition both at $0$ and $\pi r$},$ (1.28)

respectively. We will call the above systems (1.25)–(1.28) the trigonometric
Dyson models of types A, B, C and D, respectively.

Moreover, for

$\displaystyle\lim_{r\to\infty}\frac{1}{2r}\cot\left(\frac{x}{2r}\right)=\frac{1}{x},\qquad\lim_{r\to\infty}\frac{1}{2r}\tan\left(\frac{x}{2r}\right)=0,$

the $r\to\infty$ limit of the systems (1.25)–(1.28) are given as follows, for
$t\in[0,\infty)$

$\displaystyle({A}_{N-1})\quad
X^{{A}_{N-1}}_{j}(t)=u_{j}+W_{j}(t)+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\frac{1}{X^{{A}_{N-1}}_{j}(s)-X^{{A}_{N-1}}_{k}(s)}ds,$
$\displaystyle\hphantom{({A}_{N-1})\quad
X^{{A}_{N-1}}_{j}(t)=}{}j=1,2,\dots,N,\quad\text{in $\mathbb{R}$},$ (1.29)
$\displaystyle({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=u_{j}+W_{j}(t)+\int_{0}^{t}\frac{1}{X^{{C}_{N}}_{j}(s)}ds$
$\displaystyle\hphantom{({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=}{}+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\left\\{\frac{1}{X^{{C}_{N}}_{j}(s)-X^{{C}_{N}}_{k}(s)}+\frac{1}{X^{{C}_{N}}_{j}(s)+X^{{C}_{N}}_{k}(s)}\right\\}ds,$
$\displaystyle\hphantom{({C}_{N})\quad
X^{{C}_{N}}_{j}(t)=}{}j=1,2,\dots,N,\quad\text{in $[0,\infty)$},$ (1.30)
$\displaystyle({D}_{N})\quad X^{{D}_{N}}_{j}(t)=u_{j}+W_{j}(t)$
$\displaystyle\hphantom{({D}_{N})\quad
X^{{D}_{N}}_{j}(t)=}{}+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\int_{0}^{t}\left\\{\frac{1}{X^{{D}_{N}}_{j}(s)-X^{{D}_{N}}_{k}(s)}+\frac{1}{X^{{D}_{N}}_{j}(s)+X^{{D}_{N}}_{k}(s)}\right\\}ds,$
(1.31) $\displaystyle\hphantom{({D}_{N})\quad
X^{{D}_{N}}_{j}(t)=}{}j=1,2,\dots,N,\quad\text{in $[0,\infty)$ with a
ref\/lecting boundary condition at 0}.$

That is, the system (1.25) is reduced to (1.29), (1.26) and (1.27) are
degenerated to (1.30), and (1.28) to (1.31). The system (1.29) is the original
Dyson model (Dyson’s Brownian motion model with parameter $\beta=2$) [6, 15].
The SDEs (1.30) and (1.31) are known as the system of noncolliding absorbing
Brownian motions (or the Dyson model of type C) and the system of noncolliding
reflecting Brownian motions (or the Dyson model of type D), respectively. See
[2, 18, 19, 23, 24, 28, 41].

The correlation kernel for a determinantal process is in general a function of
two points on the spatio-temporal plane, say, $(s,x)$ and $(t,y)$. In the
present formula (1.17) in Theorem 1.1, the dependence of the correlation
kernel $\mathbb{K}^{{R}}_{{\bf u}}$ on one spatio-temporal point $(s,x)$ is
explicitly given by the transition probability density $p^{[0,\pi r]}$ of a
single Brownian motion in an interval $[0,\pi r]$ with given boundary
conditions. The dependence of $\mathbb{K}^{{R}}_{{\bf u}}$ on another spatio-
temporal point $(t,y)$ is described by the following two procedures. The
information of interaction among particles and initial configuration is
expressed by a set of entire functions $\big{\\{}\Phi^{{R}}_{{\bf
u},u_{j}}(z)\big{\\}}_{j=1}^{N}$, (1.19)–(1.21), which are static functions
without time-variable $t$. Then evolution in time $t$ is given by the integral
transformation (1.18) specified by the transition probability density $p_{\rm
BM}$ with time duration $t$ of a single Brownian motion. One of the benefits
of such separation of static information and dynamics in the formula is that
we can trace a relaxation process to equilibrium, which is a typical non-
equilibrium phenomenon, for any initial configuration in
$\mathbb{W}_{N}^{(0,\pi r)}$ [14]. In order to demonstrate this fact, in
Section 5 we will take the temporally homogeneous limit $t_{\ast}\to\infty$ to
make the systems have equilibrium processes and study the relaxation to
equilibria for the trigonometric Dysom models of types C and D. Another
possible benefit of the present formula for spatio-temporal correlation
kernels is that the infinite particle systems will be studied if we can
control the set of entire functions $\big{\\{}\Phi^{{R}}_{{\bf
u},u_{j}}(z)\big{\\}}_{j=1}^{N}$ in the infinite-particle limit $N\to\infty$.
For the original Dyson model [22] and other related systems [21, 23], we have
applied the Hadamard theorem on the Weierstrass canonical product-formulas of
entire functions [27] to analyze the infinite particle systems. In the present
case with (1.19)–(1.21), we have to treat the rations of infinite products of
the Jacobi theta functions in this limit $N\to\infty$. Asymptotic analysis
with $N\to\infty$ will be an interesting future problem.

The paper is organized as follows. In Section 2 we explain the relationship
between determinantal martingales and determinantal processes used in this
paper, and determinantal equalities (Lemma 2.4) obtained from the elliptic
determinant evaluations of the Macdonald denominators given by Rosengren and
Schlosser [34]. There we explain how to derive the determinantal martingale-
functions (1.5)–(1.7), which define the interacting systems of Brownian
motions by (1.15). Proofs of Theorems 1.1 and 1.2 are given in Sections 3 and
4, respectively. We study the temporally homogeneous limit $t_{\ast}\to\infty$
in Section 5, and relaxation processes to equilibria are clarified for the
trigonometric Dyson models of types C and D. Concluding remarks and open
problems are given in Section 6. Notations and formulas of the Jacobi theta
functions and related functions are listed in Appendix A.

##  2 Determinantal martingales and determinantal equalities

###  2.1 Notion of martingale

Martingales are the stochastic processes preserving their mean values and thus
they represent fluctuations. A typical example of martingale is the one-
dimensional standard Brownian motion as explained below. Let $B(t)$, $t\geq 0$
denote the position of the standard Brownian motion in $\mathbb{R}$ starting
from the origin 0 at time $t=0$. The transition probability density from the
position $x\in\mathbb{R}$ to $y\in\mathbb{R}$ in time duration $t\geq 0$ is
given by (1.13). Let ${\cal B}(\mathbb{R})$ be the Borel set on $\mathbb{R}$.
Then for an arbitrary time sequence $0\equiv t_{0}<t_{1}<\cdots<t_{M}<\infty$,
$M\in\mathbb{N}\equiv\\{1,2,\dots\\}$, and for any $A_{m}\in{\cal
B}(\mathbb{R})$, $m=1,2,\dots,M$,

$\displaystyle{\rm P}\big{[}B(t_{m})\in A_{m},\,m=1,2,\dots,M\big{]}$
$\displaystyle\qquad{}=\int_{A_{1}}dx^{(1)}\cdots\int_{A_{M}}dx^{(M)}\prod_{m=1}^{M}p_{\rm
BM}\big{(}t_{m}-t_{m-1},x^{(m)}|x^{(m-1)}\big{)}$

with $x^{(0)}\equiv 0$. The collection of all paths is denoted by $\Omega$ and
there is a subset $\widetilde{\Omega}\subset\Omega$ such that ${\rm
P}[\widetilde{\Omega}]=1$ and for any realization of path
$\omega\in\widetilde{\Omega}$, $B(t)=B(t,\omega)$, $t\geq 0$ is a real
continuous function of $t$. In other words, $B(t)$, $t\geq 0$ has a continuous
path almost surely (a.s. for short). For each $t\in[0,\infty)$, we write the
smallest $\sigma$-field (completely additive class of events) generated by the
Brownian motion up to time $t$ as ${\cal F}_{t}=\sigma(B(s)\colon 0\leq s\leq
t)$. We have a nondecreasing family $\\{{\cal F}_{t}\colon t\geq 0\\}$ such
that ${\cal F}_{s}\subset{\cal F}_{t}$ for $0\leq s<t<\infty$, which we call a
filtration, and put ${\cal F}=\bigcup_{t\geq 0}{\cal F}_{t}$. The triplet
$(\Omega,{\cal F},{\rm P})$ is called the probability space for the one-
dimensional standard Brownian motion, and ${\rm P}$ is especially called the
Wiener measure. The expectation with respect to the probability law ${\rm P}$
is written as ${\rm E}$. When we see $p_{\rm BM}(t,y|x)$ as a function of $y$,
it is nothing but the probability density of the normal distribution with mean
$x$ and variance $t$, and hence it is easy to verify that

$\displaystyle{\rm E}[B(t)|{\cal F}_{s}]=\int_{-\infty}^{\infty}xp_{\rm
BM}(t-s,x|B(s))dx=B(s)\qquad\text{a.s.}\quad 0\leq s<t<\infty,$

which means that $B(t)$, $t\geq 0$ is a martingale. We see, however,
$B(t)^{n}$, $t\geq 0$, $n\in\\{2,3,\dots\\}$ are not martingales, since the
generating function of ${\rm E}[B(t)^{n}|{\cal F}_{s}]$,
$n\in\mathbb{N}_{0}\equiv\\{0,1,2,\dots\\}$, $0<s\leq t<\infty$, with
parameter $\alpha\in\mathbb{C}$ is calculated as

$\displaystyle\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}{\rm E}[B(t)^{n}|{\cal
F}_{s}]={\rm E}[e^{\alpha B(t)}|{\cal F}_{s}]=\int_{-\infty}^{\infty}e^{\alpha
x}\frac{e^{-(x-B(s))^{2}/2(t-s)}}{\sqrt{2\pi(t-s)}}dx$
$\displaystyle\hphantom{\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}{\rm
E}[B(t)^{n}|{\cal F}_{s}]={\rm E}[e^{\alpha B(t)}|{\cal F}_{s}]}{}=e^{\alpha
B(s)+(t-s)\alpha^{2}/2}\not=e^{\alpha B(s)}\quad\text{a.s.}$

Now we assume that $\widetilde{B}(t)$, $t\geq 0$ is a one-dimensional standard
Brownian motion which is independent of $B(t)$, $t\geq 0$, and its probability
space is denoted by $\big{(}\widetilde{\Omega},\widetilde{{\cal
F}},\widetilde{{\rm P}}\big{)}$. Then we introduce a complex-valued martingale
called the complex Brownian motion

$\displaystyle Z(t)=B(t)+i\widetilde{B}(t),\qquad t\geq 0,$

with $i=\sqrt{-1}$. The probability space of $Z(t)$, $t\geq 0$ is given by the
product space $(\Omega,{\cal F},{\rm
P})\otimes(\widetilde{\Omega},\widetilde{{\cal F}},\widetilde{{\rm P}})$ and
we write the expectation as ${\bf E}={\rm E}\otimes\widetilde{{\rm E}}$. For
the complex Brownian motion, by the independence of its real and imaginary
parts, we see that

$\displaystyle\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}{\bf
E}\big{[}Z(t)^{n}|{\cal F}_{s}\otimes\widetilde{{\cal F}}_{s}\big{]}={\bf
E}\big{[}e^{\alpha Z(t)}|{\cal F}_{s}\otimes\widetilde{{\cal
F}}_{s}\big{]}={\rm E}\big{[}e^{\alpha B(t)}|{\cal
F}_{s}\big{]}\times\widetilde{{\rm
E}}\big{[}e^{i\alpha\widetilde{B}(t)}|\widetilde{{\cal F}}_{s}\big{]}$
$\displaystyle\hphantom{\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}{\bf
E}\big{[}Z(t)^{n}|{\cal F}_{s}\otimes\widetilde{{\cal
F}}_{s}\big{]}}{}=e^{\alpha B(s)+(t-s)\alpha^{2}/2}\times
e^{i\alpha\widetilde{B}(s)-(t-s)\alpha^{2}/2}=e^{\alpha
Z(s)}=\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}Z(s)^{n}$
$\displaystyle\hphantom{\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}{\bf
E}\big{[}Z(t)^{n}|{\cal F}_{s}\otimes\widetilde{{\cal
F}}_{s}\big{]}=}{}\text{a.s.}\quad 0\leq s<t<\infty,\quad\alpha\in\mathbb{C},$

which implies that for any $n\in\mathbb{N}_{0}$, $Z(t)^{n}$, $t\geq 0$ is a
martingale. This observation will be generalized as the following stronger
statement; if $F$ is an entire and non-constant function, then $F(Z(t))$,
$t\geq 0$ is a time change of a complex Brownian motion (see, for instance,
[33, Theorem 2.5 in Section V.2]). Hence $F(Z(t))$, $t\geq 0$ is a martingale:

$\displaystyle{\bf E}[F(Z(t))|{\cal F}_{s}\otimes\widetilde{{\cal
F}}_{s}]=F(Z(s))\qquad\mbox{a.s.}\quad 0\leq s<t<\infty.$

If we take the expectation $\widetilde{{\rm E}}$ with respect to $\Im
Z(\cdot)=\widetilde{B}(\cdot)$ of the both sides of the above equality, we
have

$\displaystyle{\rm E}\big{[}\widetilde{{\rm E}}[F(Z(t))]\big{|}{\cal
F}_{s}\big{]}=\widetilde{{\rm E}}[F(Z(s))]\qquad\text{a.s.}\quad 0\leq
s<t<\infty.$

In this way we can obtain a martingale
$\widehat{F}(t,B(t))\equiv\widetilde{{\rm E}}[F(Z(t))]$, $t\geq 0$ with
respect to the filtration ${\cal F}_{t}$ of the one-dimensional Brownian
motion. The present argument implies that if we have proper entire functions,
then we will obtain useful martingales describing intrinsic fluctuations
involved in interacting particle systems.

###  2.2 Basic equalities and determinantal martingales

Let $f_{j}$, $j\in\mathbb{I}$, be an infinite series of linearly independent
entire functions, where the index set $\mathbb{I}$ is
$\mathbb{Z}=\\{\dots,-1,0,1,2,\dots\\}$ or $\mathbb{N}$. For
$N\in\\{2,3\dots\\}$, we assume

$\displaystyle{\bf u}=(u_{1},u_{2},\dots,u_{N})\in\mathbb{W}_{N}.$ (2.1)

We then define a set of $N$ distinct entire and non-constant functions of
$z\in\mathbb{C}$ by

$\displaystyle\Phi_{{\bf u},u_{j}}(z)=\frac{{\det\limits_{1\leq\ell,m\leq
N}[f_{\ell}(u_{m}+(z-u_{m})\delta_{mj})]}}{{\det\limits_{1\leq\ell,m\leq
N}[f_{\ell}(u_{m})]}},\qquad j=1,2,\dots,N.$ (2.2)

By definition, it is easy to verify that

$\displaystyle\Phi_{{\bf u},u_{j}}(u_{k})=\delta_{jk},\qquad
j,k\in\\{1,2,\dots,N\\}.$ (2.3)

It should be noted that given (2.1), $\Phi_{{\bf u},u_{j}}(z)$,
$j\in\\{1,2,\dots,N\\}$ satisfying (2.3) can be regarded as the Lagrange
interpolation functions [10, 11, 12]. We can prove the following lemmas.

######  Lemma 2.1.

Let $N\in\\{2,3,\dots\\}$. The functions (2.2) have the following expansions

$\displaystyle\Phi_{{\bf u},u_{j}}(z)=\sum_{k=1}^{N}\phi_{{\bf
u},u_{j}}(k)f_{k}(z),\qquad j\in\\{1,2,\dots,N\\},$ (2.4)

and the coefficients $\phi_{{\bf u},u_{j}}(k)$, $j,k\in\\{1,2,\dots,N\\}$
satisfy the relations

$\displaystyle\sum_{\ell=1}^{N}f_{j}(u_{\ell})\phi_{{\bf
u},u_{\ell}}(k)=\frac{{\det\limits_{1\leq m,n\leq
N}[f_{m+(j-m)\delta_{mk}}(u_{n})]}}{{\det\limits_{1\leq m,n\leq
N}[f_{m}(u_{n})]}},\qquad j\in\mathbb{Z},\qquad k\in\\{1,2,\dots,N\\}.$ (2.5)

In particular, if $j$ is an element of $\\{1,2,\dots,N\\}$,

$\displaystyle\sum_{\ell=1}^{N}f_{j}(u_{\ell})\phi_{{\bf
u},u_{\ell}}(k)=\delta_{jk},\qquad j,k\in\\{1,2,\dots,N\\}.$ (2.6)

###### Proof.

Consider an $N\times N$ matrix

$\displaystyle{\sf f}_{{\bf u}}=(f_{\ell}(u_{m}))_{1\leq\ell,m\leq N},$

and write its determinant as $|{\sf f}_{{\bf u}}|\not=0$. The minor
determinant $|{\sf f}_{{\bf u}}(k,j)|$ is defined as the determinant of the
$(N-1)\times(N-1)$ matrix, ${\sf f}_{{\bf u}}(k,j)$, which is obtained from
${\sf f}_{{\bf u}}$ by deleting the $k$-th row and the $j$-th column,
$k,j\in\\{1,2,\dots,N\\}$. Then the determinant in the numerator of (2.2) is
expanded along the $j$-th column and we obtain (2.4) with

$\displaystyle\phi_{{\bf u},u_{j}}(k)=(-1)^{j+k}\frac{|{\sf f}_{{\bf
u}}(k,j)|}{|{\sf f}_{{\bf u}}|}.$ (2.7)

Hence

$\displaystyle\sum_{\ell=1}^{N}f_{j}(u_{\ell})\phi_{{\bf
u},u_{\ell}}(k)=\frac{1}{|{\sf f}_{{\bf
u}}|}\sum_{\ell=1}^{N}(-1)^{\ell+k}|{\sf f}_{{\bf
u}}(k,\ell)|f_{j}(u_{\ell}),$

which proves (2.5), for the summation in the r.h.s. is the expansion of
$\det\limits_{1\leq m,n\leq N}[f_{m+(j-m)\delta_{mk}}(u_{n})]$ along the
$k$-th row. It is immediate to conclude (2.6) from (2.5), if both of $j$ and
$k$ are elements of $\\{1,2,\dots,N\\}$. ∎

Then we obtain the following equalities.

######  Lemma 2.2.

  1. $(i)$

For $z\in\mathbb{C}$,

$\displaystyle\sum_{\ell=1}^{N}f_{j}(u_{\ell})\Phi_{{\bf
u},u_{\ell}}(z)=f_{j}(z),\qquad j\in\\{1,2,\dots,N\\}.$ (2.8)

  2. $(ii)$

For ${\bf z}=(z_{1},z_{2},\dots,z_{N})\in\mathbb{C}^{N}$, the equality

$\displaystyle\det_{1\leq j,k\leq N}[\Phi_{{\bf
u},u_{j}}(z_{k})]=\frac{{\det\limits_{1\leq j,k\leq
N}[f_{j}(z_{k})]}}{{\det\limits_{1\leq j,k\leq N}[f_{j}(u_{k})]}},\qquad{\bf
z}=(z_{1},z_{2},\dots,z_{N})\in\mathbb{C}^{N}$ (2.9)

holds.

###### Proof.

(i) Multiply the both sides of (2.6) by $f_{k}(z)$ and take summation over
$k\in\\{1,2,\dots,N\\}$. Then use (2.4) to obtain (2.8). (ii) Consider $N$
equations by putting $z=z_{j}$, $j\in\\{1,2,\dots,N\\}$ in (2.8). Then
calculate the determinant $|{\sf f}_{{\bf z}}|$ to prove the determinantal
equality (2.9). ∎

We consider $N$ pairs of independent copies $(B_{k}(t),\widetilde{B}_{k}(t))$,
$k=1,2,\dots,N$, of $(B(t),\widetilde{B}(t))$, $t\geq 0$, and define $N$
independent complex Brownian motions

$\displaystyle Z_{k}(t)=u_{k}+B_{k}(t)+i\widetilde{B}_{k}(t),\qquad t\geq
0,\qquad k=1,2,\dots,N,$ (2.10)

each of which starts from $u_{k}\in\mathbb{R}$. The probability law and
expectation of them are given by ${\rm P}_{{\bf u}}\otimes\widetilde{{\rm P}}$
and ${\rm E}_{{\bf u}}\otimes\widetilde{{\rm E}}$, respectively. Then for each
complex Brownian motion $Z_{k}(t)$, $t\geq 0$, $k=1,2,\dots,N$, we have $N$
distinct time-changes of complex Brownian motions, $\Phi_{{\bf
u},u_{j}}(Z_{k}(t))$, $t\geq 0$, $j=1,2,\dots,N$, started from the real
values, 0 or 1; $\Phi_{{\bf u},u_{j}}(Z_{k}(0))=\Phi_{{\bf
u},u_{j}}(u_{k})=\delta_{jk}$, $j,k=1,2,\dots,N$. Therefore, we can conclude
that, if we take expectation $\widetilde{{\rm E}}$, we will obtain $N$
distinct martingales

$\displaystyle M_{{\bf u},u_{j}}(t,B_{k}(t))\equiv\widetilde{{\rm
E}}\big{[}\Phi_{{\bf u},u_{j}}(B_{k}(t)+i\widetilde{B}_{k}(t))\big{]}$
$\displaystyle\hphantom{M_{{\bf
u},u_{j}}(t,B_{k}(t))}{}=\frac{{\det\limits_{1\leq\ell,m\leq
N}\big{[}\widehat{f}_{\ell}(t\delta_{mj},u_{m}+(B_{k}(t)-u_{m})\delta_{mj})\big{]}}}{{\det\limits_{1\leq\ell,m\leq
N}\big{[}\widehat{f}_{\ell}(0,u_{m})\big{]}}},\qquad t\geq 0,\quad
j=1,2,\dots,N,$

for each one-dimensional Brownian motion $B_{k}(t)$, $t\geq 0$,
$k=1,2,\dots,N$, where

$\displaystyle\widehat{f}_{\ell}(t,x)\equiv\widetilde{{\rm
E}}\big{[}f_{\ell}(x+i\widetilde{B}(t))\big{]}=\int_{-\infty}^{\infty}f_{\ell}(x+i\widetilde{x})p_{\rm
BM}(t,\widetilde{x}|0)d\widetilde{x},\qquad\ell\in\mathbb{I}.$

By this definition, $\widehat{f}_{\ell}(0,x)=f_{\ell}(x)$,
$\ell\in\mathbb{I}$. Then we define a multivariate function of $t\geq 0$ and
${\bf x}=(x_{1},x_{2},\dots,x_{N})\in\mathbb{R}^{N}$ by

$\displaystyle D_{{\bf u}}(t,{\bf x})=\det_{1\leq j,k\leq N}[M_{{\bf
u},u_{j}}(t,x_{k})],$ (2.11)

which gives a martingale as a functional of $t\geq 0$ and ${\bf
B}(t)=(B_{1}(t),B_{2}(t),\dots,B_{N}(t))$, $t\geq 0$,

$\displaystyle{\rm E}_{{\bf u}}[D_{{\bf u}}(t,{\bf B}(t))|{\cal
F}_{s}]=D_{{\bf u}}(s,{\bf B}(s)),\qquad 0\leq s<t<\infty.$

By the multi-linearity of determinant and (2.9), we see

$\displaystyle D_{{\bf u}}(t,{\bf x})=\widetilde{{\rm E}}\Big{[}\det_{1\leq
j,k\leq N}\big{[}\Phi_{{\bf
u},u_{j}}(x_{k}+i\widetilde{B}_{k}(t))\big{]}\Big{]}=\frac{{\det\limits_{1\leq
j,k\leq N}\big{[}\widehat{f}_{j}(t,x_{k})\big{]}}}{{\det\limits_{1\leq j,k\leq
N}\big{[}\widehat{f}_{j}(0,u_{k})\big{]}}}.$ (2.12)

We call $D_{{\bf u}}(t,{\bf B}(t))$, $t\geq 0$, the determinantal martingale
[14].

###  2.3 Auxiliary measure and spatio-temporal Fredholm determinant

Now we introduce an auxiliary measure $\widehat{\mathbb{P}}_{{\bf u}}$, which
is complex-valued in general, but is absolutely continuous to the Wiener
measure ${\rm P}_{{\bf u}}$ as

$\displaystyle\widehat{\mathbb{P}}_{{\bf u}}\big{|}_{{\cal F}_{t}}=D_{{\bf
u}}(t,{\bf B}(t)){\rm P}_{{\bf u}}\big{|}_{{\cal F}_{t}},\qquad t\geq 0.$
(2.13)

(Note that this is generally different from $\mathbb{P}^{{R}}_{{\bf u}}$ given
by (1.15), since the condition ${\bf 1}(T_{\rm collision}>t)$ is omitted, and
hence it is not the measure representing any noncolliding particles.) Here we
consider the corresponding auxiliary system of $N$ particles on $\mathbb{R}$
governed by $\widehat{\mathbb{P}}_{{\bf u}}$

$\displaystyle\widehat{{\bf
X}}(t)=\big{(}\widehat{X}_{1}(t),\dots,\widehat{X}_{N}(t)\big{)},\qquad t\geq
0,$

starting from (2.1) and each particle of which has a continuous path a.s. We
consider the unlabeled configuration of $\widehat{{\bf X}}(t)$ as

$\displaystyle\widehat{\Xi}(t,\cdot)=\sum_{j=1}^{N}\delta_{\widehat{X}_{j}(t)}(\cdot),\qquad
t\geq 0.$ (2.14)

Consider an arbitrary number $M\in\mathbb{N}$ and an arbitrary set of strictly
increasing times ${\bf t}=\\{t_{1},t_{2},\dots,t_{M}\\}$, $0\equiv
t_{0}<t_{1}<\cdots<t_{M}<\infty$. Let ${\rm C}_{{\rm c}}(\mathbb{R})$ be the
set of all continuous real-valued functions with compact supports on
$\mathbb{R}$. For ${\bf g}=(g_{t_{1}},g_{t_{1}},\dots,g_{t_{M}})\in{\rm
C}_{{\rm c}}(\mathbb{R})^{M}$, we consider the following functional of ${\bf
g}$

$\displaystyle\widehat{\mathbb{L}}_{{\bf u}}[{\bf t},{\bf
g}]=\widehat{\mathbb{E}}_{{\bf
u}}\left[\exp\left\\{\sum_{m=1}^{M}\int_{\mathbb{R}}g_{t_{m}}(x)\widehat{\Xi}(t_{m},dx)\right\\}\right],$
(2.15)

which is the Laplace transform of the multi-time distribution function
$\widehat{\mathbb{P}}_{{\bf u}}$ on a set of times ${\bf t}$ with the
functions ${\bf g}$. If we put

$\displaystyle\chi_{t}=e^{g_{t}}-1,$

then (2.15) can be written as

$\displaystyle\widehat{\mathbb{L}}_{{\bf u}}[{\bf t},{\bf
g}]=\widehat{\mathbb{E}}_{{\bf
u}}\left[\prod_{m=1}^{M}\prod_{j=1}^{N}\big{\\{}1+\chi_{t_{m}}\big{(}\widehat{X}_{j}(t_{m})\big{)}\big{\\}}\right].$
(2.16)

Explicit expression of (2.16) is given by the following multiple integrals

$\displaystyle\widehat{\mathbb{L}}_{{\bf u}}[{\bf t},{\bf
g}]=\int_{\mathbb{R}^{N}}d{\bf x}^{(1)}\cdots\int_{\mathbb{R}^{N}}d{\bf
x}^{(M)}D_{{\bf u}}\big{(}t_{M},{\bf x}^{(M)}\big{)}$
$\displaystyle\hphantom{\widehat{\mathbb{L}}_{{\bf u}}[{\bf t},{\bf
g}]=}{}\times\prod_{m=1}^{M}\prod_{j=1}^{N}\big{[}p_{\rm
BM}\big{(}t_{m}-t_{m-1},x^{(m)}_{j}|x^{(m-1)}_{j}\big{)}\big{\\{}1+\chi_{t_{m}}\big{(}x^{(m)}_{j}\big{)}\big{\\}}\big{]},$

where $x^{(0)}_{j}=u_{j}$, $j=1,2,\dots,N$, ${\bf
x}^{(m)}=\big{(}x^{(m)}_{1},\dots,x^{(m)}_{N}\big{)}$, and $d{\bf
x}^{(m)}=\prod\limits_{j=1}^{N}dx^{(m)}_{j}$, $m=1,2,\dots,M$.

The following was proved as Theorem 1.3 in [14].

######  Proposition 2.3.

Put

$\displaystyle\widehat{\mathbb{K}}_{{\bf u}}(s,x;t,y)=\sum_{j=1}^{N}p_{\rm
BM}(s,x|u_{j})M_{{\bf u},u_{j}}(t,y)-{\bf 1}(s>t)p_{\rm BM}(s-t,x|y),$

$s,t>0$, $x,y\in\mathbb{R}$. Then the following equality holds for an
arbitrary number $M\in\mathbb{N}$, an arbitrary set of strictly increasing
times ${\bf t}=\\{t_{1},t_{2},\dots,t_{M}\\}$, $0\equiv
t_{0}<t_{1}<\cdots<t_{M}<\infty$, and ${\bf
g}=(g_{t_{1}},g_{t_{1}},\dots,g_{t_{M}})\in{\rm C}_{{\rm c}}(\mathbb{R})^{M}$,

$\displaystyle\widehat{\mathbb{L}}_{{\bf u}}[{\bf t},{\bf g}]=\mathop{{\rm
Det}}_{\begin{subarray}{c}s,t\in{\bf t},\\\
x,y\in\mathbb{R}\end{subarray}}\big{[}\delta_{st}\delta(x-y)+\widehat{\mathbb{K}}_{{\bf
u}}(s,x;t,y)\chi_{t}(y)\big{]},$

where the r.h.s. denotes the spatio-temporal Fredholm determinant with the
kernel $\widehat{\mathbb{K}}_{{\bf u}}$ defined by

$\displaystyle\mathop{{\rm Det}}_{\begin{subarray}{c}s,t\in{\bf t},\\\
x,y\in\mathbb{R}\end{subarray}}\big{[}\delta_{st}\delta(x-y)+\widehat{\mathbb{K}}_{{\bf
u}}(s,x;t,y)\chi_{t}(y)\big{]}$ (2.17)
$\displaystyle\equiv\sum_{\begin{subarray}{c}0\leq N_{m}\leq N,\\\ 1\leq m\leq
M\end{subarray}}\int_{\prod\limits_{m=1}^{M}\mathbb{W}_{N_{m}}}\prod_{m=1}^{M}d{\bf
x}_{N_{m}}^{(m)}\prod_{j=1}^{N_{m}}\chi_{t_{m}}\big{(}x_{j}^{(m)}\big{)}\det_{\begin{subarray}{c}1\leq
j\leq N_{m},\,1\leq k\leq N_{n},\\\ 1\leq m,n\leq
M\end{subarray}}\big{[}\widehat{\mathbb{K}}_{{\bf
u}}\big{(}t_{m},x_{j}^{(m)};t_{n},x_{k}^{(n)}\big{)}\big{]},$

where $d{\bf x}^{(m)}_{N_{m}}=\prod\limits_{j=1}^{N_{m}}dx^{(m)}_{j}$,
$m=1,2,\dots,M$, and the term with $N_{m}=0$, $1\leq\forall\,m\leq M$ in the
r.h.s. should be interpreted as $1$.

This proposition is general and it proves that the auxiliary system (2.14) is
determinantal. The measure (2.13) which governs this particle system is,
however, complex-valued in general, and hence the system is unphysical. The
problem is to clarify the proper conditions which should be added to (2.13) to
construct a non-negative-definite real measure, i.e., the probability measure,
which defines a physical system of interacting particles. As a matter of
course, this problem depends on the choice of an infinite set of linearly
independent entire functions $f_{j}$, $j\in\mathbb{I}$.

###  2.4 Elliptic determinant evaluations of the Macdonald denominators

Here we report the results when we choose the entire functions $f_{j}$,
$j\in\mathbb{Z}$ as follows

$\displaystyle
f^{{A}_{N-1}}_{j}(z;\tau)=e^{iJ^{{A}_{N-1}}(j)z/r}\vartheta_{1}\left(J^{{A}_{N-1}}(j)\tau+\frac{{\cal
N}^{{A}_{N-1}}z}{2\pi r}+\frac{1-(-1)^{N}}{4};{\cal
N}^{{A}_{N-1}}\tau\right),$ $\displaystyle
f^{{R}}_{j}(z;\tau)=e^{iJ^{{R}}(j)z/r}\vartheta_{1}\left(J^{{R}}(j)\tau+\frac{{\cal
N}^{{R}}z}{2\pi r};{\cal N}^{{R}}\tau\right)$
$\displaystyle\hphantom{f^{{R}}_{j}(z;\tau)=}{}-e^{-iJ^{{R}}(j)z/r}\vartheta_{1}\left(J^{{R}}(j)\tau-\frac{{\cal
N}^{{R}}z}{2\pi r};{\cal N}^{{R}}\tau\right)\qquad\text{for
${R}={B}_{N},{B}^{\vee}_{N}$},$ $\displaystyle
f^{{R}}_{j}(z;\tau)=e^{iJ^{{R}}(j)z/r}\vartheta_{1}\left(J^{{R}}(j)\tau+\frac{{\cal
N}^{{R}}z}{2\pi r}+\frac{1}{2};{\cal N}^{{R}}\tau\right)$
$\displaystyle\hphantom{f^{{R}}_{j}(z;\tau)=}{}-e^{-iJ^{{R}}(j)z/r}\vartheta_{1}\left(J^{{R}}(j)\tau-\frac{{\cal
N}^{{R}}z}{2\pi r}+\frac{1}{2};{\cal N}^{{R}}\tau\right)\qquad\text{for
${R}={C}_{N},{C}^{\vee}_{N},{BC}_{N}$},$ $\displaystyle
f^{{D}_{N}}_{j}(z;\tau)=e^{iJ^{{D}_{N}}(j)z/r}\vartheta_{1}\left(J^{{D}_{N}}(j)\tau+\frac{{\cal
N}^{{D}_{N}}z}{2\pi r}+\frac{1}{2};{\cal N}^{{D}_{N}}\tau\right)$
$\displaystyle\hphantom{f^{{D}_{N}}_{j}(z;\tau)=}{}+e^{-iJ^{{D}_{N}}(j)z/r}\vartheta_{1}\left(J^{{D}_{N}}(j)\tau-\frac{{\cal
N}^{{D}_{N}}z}{2\pi r}+\frac{1}{2};{\cal N}^{{D}_{N}}\tau\right),$ (2.18)

$z\in\mathbb{C}$, $j\in\mathbb{Z}$, with $N\in\mathbb{N}$, $0<r<\infty$, and
$\tau\in\mathbb{C}$ with $0<\Im\tau<\infty$, where

$\displaystyle
J^{{R}}(j)=\begin{cases}j-1,&{R}={A}_{N-1},{B}_{N},{B}^{\vee}_{N},{D}_{N},\\\
j,&{R}={C}_{N},{BC}_{N},\\\ j-1/2,&{R}={C}^{\vee}_{N},\end{cases}$ (2.19)

and ${\cal N}^{{R}}$ are given by (1.4). These functions (2.18) were used to
express the determinant evaluations by Rosengren and Schlosser [34] for the
Macdonald denominators $W_{{R}}(x)$ for seven families of irreducible reduced
affine root systems
${R}={A}_{N-1},{B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N},{BC}_{N},{D}_{N}$.
Note that, if $x\in\mathbb{R}$, $\tau\in i\mathbb{R}$, and $0<\Im\tau<\infty$,
then $f^{{R}}(x;\tau)\in\mathbb{R}$ for ${R}={B}_{N},{B}^{\vee}_{N},{D}_{N}$
and $f^{{R}}(x;\tau)\in i\mathbb{R}$ for
${R}={C}_{N},{C}^{\vee}_{N},{BC}_{N}$.

Assume that, for ${\bf u}=(u_{1},u_{2},\dots,u_{N})\in\mathbb{W}_{N}$,
$\det\limits_{1\leq j,k\leq N}[f_{j}(u_{k})]$ is factorized in the form

$\displaystyle\det_{1\leq j,k\leq N}[f_{j}(u_{k})]=k_{0}k_{\rm sym}({\bf
u})\prod_{\ell=1}^{N}k_{1}(u_{\ell})\prod_{1\leq j<k\leq
N}k_{2}(u_{k},u_{j}),$ (2.20)

where $k_{0}$ is a constant, $k_{1}$ is a single variable function, $k_{2}$ is
an antisymmetric function of two variables, $k_{2}(u,v)=-k_{2}(v,u)$, and
$k_{\rm sym}({\bf u})$ is a symmetric function of ${\bf u}$ which cannot be
factorized as $\prod\limits_{\ell=1}^{N}k_{1}(u_{\ell})$. For ${\bf
u}=(u_{1},\dots,u_{j-1},u_{j},u_{j+1},\dots,u_{N})\in\mathbb{W}_{N}$, we
replace the $j$-th component $u_{j}$ by a variable $z$ and write the obtained
vector as ${\bf u}^{(j)}(z)=(u_{1},\dots,u_{j-1},z,u_{j+1},\dots,u_{N})$.
Under the assumption (2.20), $\Phi_{{\bf u},u_{j}}(z)$ defined by (2.2) is
also factorized as

$\displaystyle\Phi_{{\bf u},u_{j}}(z)=\frac{k_{\rm sym}({\bf
u}^{(j)}(z))}{k_{\rm sym}({\bf
u})}\frac{k_{1}(z)}{k_{1}(u_{j})}\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{k_{2}(z,u_{\ell})}{k_{2}(u_{j},u_{\ell})},\qquad
j=1,2,\dots,N.$ (2.21)

Then the determinantal equality (2.9) becomes

$\displaystyle\det_{1\leq j,k\leq N}\left[\frac{k_{\rm sym}({\bf
u}^{(j)}(z_{k}))}{k_{\rm sym}({\bf
u})}\frac{k_{1}(z_{k})}{k_{1}(u_{j})}\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{k_{2}(z_{k},u_{\ell})}{k_{2}(u_{j},u_{\ell})}\right]$
$\displaystyle\qquad{}=\frac{k_{\rm sym}({\bf z})}{k_{\rm sym}({\bf
u})}\prod_{\ell=1}^{N}\frac{k_{1}(z_{\ell})}{k_{1}(u_{\ell})}\prod_{1\leq
j<k\leq N}\frac{k_{2}(z_{k},z_{j})}{k_{2}(u_{k},u_{j})},\qquad{\bf
z}\in\mathbb{C}^{N},\qquad{\bf u}\in\mathbb{W}_{N}.$ (2.22)

In [34] Rosengren and Schlosser gave the elliptic determinant evaluations of
the Macdonald denominators of seven families of irreducible reduced affine
root systems, ${A}_{N-1}$, ${B}_{N}$, ${B}^{\vee}_{N}$, ${C}_{N}$,
${C}^{\vee}_{N}$, ${BC}_{N}$ and ${D}_{N}$. From their determinantal
equalities [34, Proposition 6.1], it is easy to verify that our seven choices
of $f_{j}$, $j\in\mathbb{Z}$ given by (2.18) allow the factorization (2.20)
for their determinants. As functions of $\tau$, we write

$\displaystyle q(\tau)=e^{\tau\pi i},\qquad
q_{0}(\tau)=\prod_{n=1}^{\infty}\big{(}1-q(\tau)^{2n}\big{)}.$ (2.23)

######  Lemma 2.4.

For the seven choices of $f^{{R}}_{j}(\cdot;\tau)$, $j\in\mathbb{Z}$,
$\Im\tau>0$, given by (2.18), the equality

$\displaystyle\det_{1\leq j,k\leq
N}\big{[}f^{{R}}_{j}(u_{k};\tau)\big{]}=k^{{R}}_{0}(\tau)k^{{R}}_{\rm
sym}({\bf u};\tau)\prod_{\ell=1}^{N}k^{{R}}_{1}(u_{\ell};\tau)\prod_{1\leq
j<k\leq N}k^{{R}}_{2}(u_{k},u_{j};\tau),$ (2.24)

holds with the following factors,

$\displaystyle
k_{0}^{{A}_{N-1}}(\tau)=i^{-(N-1)(3N+1-(-1)^{N})/2}q(\tau)^{-(N-1)(3N-2)/8}q_{0}(\tau)^{-(N-1)(N-2)/2},$
$\displaystyle
k_{0}^{{B}_{N}}(\tau)=2q(\tau)^{-N(N-1)/4}q_{0}(\tau)^{-N(N-1)},$
$\displaystyle
k_{0}^{{B}^{\vee}_{N}}(\tau)=2q(\tau)^{-N(N-1)/4}q_{0}(\tau)^{-(N-1)^{2}}q_{0}(2\tau)^{-(N-1)},$
$\displaystyle
k_{0}^{{C}_{N}}(\tau)=i^{-N}q(\tau)^{-N^{2}/4}q_{0}(\tau)^{-N(N-1)},$
$\displaystyle
k_{0}^{{C}^{\vee}_{N}}(\tau)=i^{-N}q(\tau)^{-N(2N-1)/8}q_{0}(\tau)^{-(N-1)^{2}}q_{0}(\tau/2)^{-(N-1)},$
$\displaystyle
k_{0}^{{BC}_{N}}(\tau)=i^{-N}q(\tau)^{-N(N+1)/4}q_{0}(\tau)^{-N(N-1)}q_{0}(2\tau)^{-N},$
$\displaystyle
k_{0}^{{D}_{N}}(\tau)=4q(\tau)^{-N(N-1)/4}q_{0}(\tau)^{-N(N-2)},$ (2.25)
$\displaystyle k_{\rm sym}^{{A}_{N-1}}({\bf
u};\tau)=\vartheta_{1}\left(\frac{\sum\limits_{j=1}^{N}u_{j}-\kappa_{N}}{2\pi
r};\tau\right),$ $\displaystyle k_{\rm sym}^{{R}}({\bf
u};\tau)=1\quad\mbox{for
${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N},{BC}_{N},{D}_{N}$},$
$\displaystyle k_{1}^{{A}_{N-1}}(u;\tau)=k_{1}^{{D}_{N}}(u;\tau)=1,$
$\displaystyle k_{1}^{{B}_{N}}(u;\tau)=\vartheta_{1}\left(\frac{u}{2\pi
r};\tau\right),\qquad
k_{1}^{{B}^{\vee}_{N}}(u;\tau)=\vartheta_{1}\left(\frac{u}{\pi
r};2\tau\right),$ $\displaystyle
k_{1}^{{C}_{N}}(u;\tau)=\vartheta_{1}\left(\frac{u}{\pi r};\tau\right),\qquad
k_{1}^{{C}^{\vee}_{N}}(u;\tau)=\vartheta_{1}\left(\frac{u}{2\pi
r};\frac{\tau}{2}\right),$ $\displaystyle
k_{1}^{{BC}_{N}}(u;\tau)=\vartheta_{1}\left(\frac{u}{2\pi
r};\tau\right)\vartheta_{0}\left(\frac{u}{\pi r};2\tau\right),$ $\displaystyle
k_{2}^{{A}_{N-1}}(u,v;\tau)=\vartheta_{1}\left(\frac{u-v}{2\pi
r};\tau\right),$ $\displaystyle
k_{2}^{{R}}(u,v;\tau)=\vartheta_{1}\left(\frac{u-v}{2\pi
r};\tau\right)\vartheta_{1}\left(\frac{u+v}{2\pi r};\tau\right)\quad\text{for
${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N},{BC}_{N},{D}_{N}$}.$ (2.26)

###  2.5 Determinantal martingale-functions

Since $f_{j}^{{R}}(z;\tau)$, $j\in\mathbb{Z}$ given by (2.18) are entire and
non-constant functions of $z\in\mathbb{C}$,

$\displaystyle\widehat{f}^{{R}}_{j}(t,x;\tau)=\widetilde{{\rm
E}}[f^{{R}}_{j}(x+i\widetilde{B}(t);\tau)],\qquad t\in[0,\infty),\qquad
j\in\mathbb{Z},$ (2.27)

give the single-variable martingale-functions such that, if
$\widehat{f}^{{R}}_{j}(t,B(t);\tau)$, $t\geq 0$, $j\in\mathbb{Z}$ are finite,
then they give martingales with respect to the filtration ${\cal F}_{t}$ of
one-dimensional Brownian motion $B(t)$, $t\geq 0$,

$\displaystyle{\rm E}[\widehat{f}^{{R}}_{j}(t,B(t);\tau)|{\cal
F}_{s}]=\widehat{f}^{{R}}_{j}(s,B(s);\tau),\qquad 0\leq
s<t<\infty\qquad\mbox{a.s.}\quad j\in\mathbb{Z}.$

We have found that $\widehat{f}^{{R}}_{j}(t,x;\tau)$’s are expressed using
$f^{{R}}_{j}(t,x;\cdot)$ by shifting

$\displaystyle\tau\to\tau-\frac{i{\cal N}^{{R}}t}{2\pi r^{2}}$

with (1.4) and multiplying time-dependent factors. With $0<t_{\ast}<\infty$,
we put (1.3).

######  Lemma 2.5.

For $f^{{R}}_{j}$, $j\in\mathbb{Z}$ given by (2.18), we obtain the following
infinite series of linearly independent martingale-functions

$\displaystyle\widehat{f}^{{R}}_{j}\big{(}t,x;\tau^{{R}}(0)\big{)}=e^{J^{{R}}(j)^{2}t/2r^{2}}f^{{R}}_{j}\big{(}x;\tau^{{R}}(t)\big{)},\qquad
t\in[0,t_{\ast}),\qquad j\in\mathbb{Z},$ (2.28)

where $J^{{R}}(j)$, $j\in\mathbb{Z}$ are given by (2.19).

###### Proof.

By (2.18), it is enough to calculate

$\displaystyle\widetilde{{\rm E}}\left[e^{\pm
iJ^{{R}}(j)(x+i\widetilde{B}(t))/r}\vartheta_{1}\left(J^{{R}}(j)\tau\pm\frac{{\cal
N}^{{R}}(x+i\widetilde{B}(t))}{2\pi r}+\alpha;{\cal N}^{{R}}t\right)\right]$
(2.29)

with a constant $\alpha$. By the definition (A.1) of $\vartheta_{1}$, this is
equal to

$\displaystyle e^{\pm iJ^{{R}}(j)x}i\sum_{n\in\mathbb{Z}}(-1)^{n}e^{{\cal
N}^{{R}}\tau\pi i(n-1/2)^{2}+(J^{{R}}(j)\tau\pm{\cal N}^{{R}}x/2\pi
r+\alpha)\pi i(2n-1)}$ $\displaystyle\qquad{}\times\widetilde{{\rm
E}}\big{[}e^{\mp\\{J^{{R}}(j)+{\cal
N}^{{R}}(2n-1)/2\\}\widetilde{B}(t)/r}\big{]}.$

Since

$\displaystyle\widetilde{{\rm E}}\big{[}e^{\mp\\{J^{{R}}(j)+{\cal
N}^{{R}}(2n-1)/2\\}\widetilde{B}(t)/r}\big{]}=e^{\\{J^{{R}}(j)+{\cal
N}^{{R}}(2n-1)/2\\}^{2}t/2r^{2}}$
$\displaystyle\qquad{}=e^{J^{{R}}(j)^{2}t/2r^{2}}e^{{\cal N}^{{R}}(-i{\cal
N}^{{R}}t/2\pi r^{2})\pi i(n-1/2)^{2}+J^{{R}}(j)(-i{\cal N}^{{R}}t/2\pi
r^{2})\pi i(2n-1)},$

we can see that (2.29) is equal to

$\displaystyle e^{J^{{R}}(j)^{2}t/2r^{2}}e^{\pm
iJ^{{R}}(j)x/r}\vartheta_{1}\left(J^{{R}}(j)\left(\tau-\frac{i{\cal
N}^{{R}}t}{2\pi r^{2}}\right)\pm\frac{{\cal N}^{{R}}x}{2\pi r}+\alpha;{\cal
N}^{{R}}\left(\tau-\frac{i{\cal N}^{{R}}t}{2\pi r^{2}}\right)\right).$

Therefore, if we set $\tau=i{\cal N}^{{R}}t_{\ast}/2\pi r^{2}$, (2.28) are
obtained. ∎

We call the multivariate function $D_{{\bf u}}(t,{\bf x})$ of ${\bf
x}\in\mathbb{R}^{N}$ defined by (2.11) the determinantal martingale-function,
since if it is finite, it gives the determinantal martingale when we put the
$N$-dimensional Brownian motion ${\bf B}(t)$, $t\geq 0$ into ${\bf x}$. The
following is proved.

######  Proposition 2.6.

For $f^{{R}}_{j}$, $j\in\mathbb{Z}$ given by (2.18), the determinantal
martingale-functions are factorized as follows

$\displaystyle D^{{R}}_{{\bf u}}(t,{\bf
x})=\frac{c^{{R}}_{0}\big{(}\tau^{{R}}(t)\big{)}}{c^{{R}}_{0}\big{(}\tau^{{R}}(0)\big{)}}\frac{k_{\rm
sym}^{{R}}\big{(}{\bf x};\tau^{{R}}(t)\big{)}}{k^{{R}}_{\rm sym}\big{(}{\bf
u};\tau^{{R}}(0)\big{)}}\prod_{\ell=1}^{N}\frac{k^{{R}}_{1}\big{(}x_{\ell};\tau^{{R}}(t)\big{)}}{k^{{R}}_{1}\big{(}u_{\ell};\tau^{{R}}(0)\big{)}}\prod_{1\leq
j<k\leq
N}\frac{k^{{R}}_{2}\big{(}x_{k},x_{j};\tau^{{R}}(t)\big{)}}{k^{{R}}_{2}\big{(}u_{k},u_{j};\tau^{{R}}(0)\big{)}},\\!\\!\\!$
(2.30)

where

$\displaystyle c^{{A}_{N-1}}_{0}(\tau)=\eta(\tau)^{-(N-1)(N-2)/2}$ (2.31)

and for ${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N},{BC}_{N},{D}_{N}$,
$c^{{R}}_{0}(\tau)$ are given by (1.8). For
${R}={B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N}$, ${BC}_{N},{D}_{N}$,
(2.30) are written as (1.5)–(1.7) with (1.8) and (1.9).

###### Proof.

By the second equality of (2.12) and Lemma 2.5,

$\displaystyle D^{{R}}_{{\bf u}}(t,{\bf x})=\frac{{\det\limits_{1\leq j,k\leq
N}\big{[}e^{J^{{R}}(j)^{2}t/2r^{2}}f^{{R}}_{j}(x_{k};\tau^{{R}}(t))\big{]}}}{{\det\limits_{1\leq
j,k\leq
N}\big{[}f^{{R}}_{j}(u_{k};\tau^{{R}}(0))\big{]}}}=e^{(t/2r^{2})\sum\limits_{j=1}^{N}J^{{R}}(j)^{2}}\frac{{\det\limits_{1\leq
j,k\leq N}\big{[}f^{{R}}_{j}(x_{k};\tau^{{R}}(t))\big{]}}}{{\det\limits_{1\leq
j,k\leq N}\big{[}f^{{R}}_{j}(u_{k};\tau^{{R}}(0))\big{]}}},$

where the multi-linearity of determinant was used in the second equality. Then
applying Lemma 2.4, we obtain (2.30) with

$\displaystyle\frac{c^{{R}}_{0}(\tau^{{R}}(t))}{c^{{R}}_{0}(\tau^{{R}}(0))}=e^{(t/2r^{2})\sum\limits_{j=1}^{N}J^{{R}}(j)^{2}}\frac{k^{{R}}_{0}(\tau^{{R}}(t))}{k^{{R}}_{0}(\tau^{{R}}(0))}.$

By (2.25) and the summation formulas

$\displaystyle\sum\limits_{j=1}^{N}j=N(N+1)/2,\qquad\sum\limits_{j=1}^{N}j^{2}=N(N+1)(2N+1)/6,$

we obtain the expressions (1.8) using the Dedekind modular function (1.10),
since $\eta(\tau)=e^{\tau\pi i/12}q_{0}(\tau)$ with (2.23). Thus we can derive
readily the expressions (1.5)–(1.7) with (1.8) and (1.9) from (2.30). ∎

##  3 Proof of Theorem 1.1

Assume $t\in[0,t_{\ast})$. By (A.3), we see that if $0\leq x\leq 2\pi r$,
$0\leq k_{1}^{{R}}\big{(}x;\tau^{{R}}(t)\big{)}<\infty$ for
${R}={B}_{N},{C}^{\vee}_{N},{BC}_{N}$; if $0\leq x\leq\pi r$, $0\leq
k_{1}^{{R}}\big{(}x;\tau^{{R}}(t)\big{)}<\infty$ for
${R}={B}^{\vee}_{N},{C}_{N}$; and $k_{1}^{{D}_{N}}\equiv 1$. We also see that
if

$\displaystyle|x_{1}|<x_{2}<\cdots<x_{N-1}<x_{N}\wedge(2\pi r-x_{N}),$ (3.1)

then $0<\prod\limits_{1\leq j<k\leq
N}k_{2}^{{R}}\big{(}x_{k},x_{j};\tau^{{R}}(t)\big{)}<\infty$. The inequality
$|x_{1}|<x_{2}$ means that $x_{2}$ is greater than both of $x_{1}$ and its
reflection at $0$, and $x_{N-1}<x_{N}\wedge(2\pi r-x_{N})$ means that
$x_{N-1}$ is smaller than both of $x_{N}$ and its reflection at $\pi r$. The
above observation implies that if ${\bf u}\in\mathbb{W}_{N}^{(0,\pi r)}$, then

$\displaystyle\begin{split}&0\leq\prod_{\ell=1}^{N}k^{{R}}_{1}\big{(}B_{\ell}(t);\tau^{{R}}(t)\big{)}\prod_{1\leq
j<k\leq N}k^{{R}}_{2}\big{(}B_{k}(t),B_{j}(t);\tau^{{R}}(t)\big{)}<\infty,\\\
&\text{for}\quad 0\leq t\leq T_{\rm collision},\qquad\text{a.s.\ in ${\rm
P}^{[0,\pi r]}_{{\bf u}}$}.\end{split}$

Due to the Karlin–McGregor–Lindström–Gessel–Viennot (KMLGV) formula [9, 13,
29] for non-intersecting paths, the transition probability density of the $N$
particle system in $[0,\pi r]$ governed by the probability law
$\mathbb{P}^{{R}}_{{\bf u}}$ defined by (1.15) is given by

$\displaystyle p^{{R}}(t,{\bf y}|s,{\bf
x})=\frac{c^{{R}}_{0}\big{(}\tau^{{R}}(t)\big{)}}{c^{{R}}_{0}\big{(}\tau^{{R}}(s)\big{)}}\prod_{\ell=1}^{N}\frac{k^{{R}}_{1}\big{(}y_{\ell};\tau^{{R}}(t)\big{)}}{k^{{R}}_{1}\big{(}x_{\ell};\tau^{{R}}(s)\big{)}}\prod_{1\leq
j<k\leq
N}\frac{k^{{R}}_{2}\big{(}y_{k},y_{j};\tau^{{R}}(t)\big{)}}{k^{{R}}_{2}\big{(}x_{k},x_{j};\tau^{{R}}(s)\big{)}}$
$\displaystyle\hphantom{p^{{R}}(t,{\bf y}|s,{\bf x})=}{}\times\det_{1\leq
n,n^{\prime}\leq N}\big{[}p^{[0,\pi
r]}(t-s,y_{n}|x_{n^{\prime}})\big{]},\qquad 0\leq s<t<t_{\ast},\quad{\bf
x},{\bf y}\in\mathbb{W}_{N}^{(0,\pi r)}.\\!\\!\\!$ (3.2)

Since we consider the measure-valued stochastic process (1.16), the
configuration is unlabeled and hence all the observables at each time should
be symmetric functions of particle positions. For an arbitrary number
$M\in\mathbb{N}$ and an arbitrary strictly increasing series of times $0\equiv
t_{0}<t_{1}<\cdots<t_{M}<t_{\ast}$, let $g_{t_{m}}({\bf x})$, $m=1,2,\dots,M$,
be symmetric functions of ${\bf x}\in\mathbb{W}_{N}^{(0,\pi r)}$. Then (1.15)
and (3.2) give

$\displaystyle\mathbb{E}^{{R}}_{{\bf u}}\left[\prod_{m=1}^{M}g_{t_{m}}({\bf
X}(t_{m}))\right]={\rm E}_{{\bf u}}^{[0,\pi
r]}\left[\prod_{m=1}^{M}g_{t_{m}}({\bf B}(t_{m})){\bf 1}(T_{\rm
collision}>t_{M})D^{{R}}_{{\bf u}}(t,{\bf B}(t_{M}))\right]$
$\displaystyle{}=\int_{\mathbb{W}_{N}^{(0,\pi r)}}d{\bf
x}^{(1)}\cdots\int_{\mathbb{W}_{N}^{(0,\pi r)}}d{\bf
x}^{(M)}\prod_{m=1}^{M}\det_{1\leq n,{n^{\prime}}\leq N}\big{[}p^{[0,\pi
r]}\big{(}t_{m}-t_{m-1},x^{(m)}_{n}|x^{(m-1)}_{n^{\prime}}\big{)}\big{]}g_{t_{m}}\big{(}{\bf
x}^{(m)}\big{)}$
$\displaystyle\quad{}\times\frac{c^{{R}}_{0}\big{(}\tau^{{R}}(t_{M})\big{)}}{c^{{R}}_{0}\big{(}\tau^{{R}}(0)\big{)}}\prod_{\ell=1}^{N}\frac{k^{{R}}_{1}\big{(}x^{(M)}_{\ell};\tau^{{R}}(t_{M})\big{)}}{k^{{R}}_{1}\big{(}u_{\ell};\tau^{{R}}(0)\big{)}}\prod_{1\leq
j<k\leq
N}\frac{k^{{R}}_{2}\big{(}x^{(M)}_{k},x^{(M)}_{j};\tau^{{R}}(t_{M})\big{)}}{k^{{R}}_{2}\big{(}u_{k},u_{j};\tau^{{R}}(0)\big{)}}.$
(3.3)

By definition of determinant, the above is equal to

$\displaystyle\sum_{\sigma_{1}\in{\cal S}_{N}}\cdots\sum_{\sigma_{M}\in{\cal
S}_{N}}\int_{\mathbb{W}_{N}^{(0,\pi r)}}d{\bf
x}^{(1)}\cdots\int_{\mathbb{W}_{N}^{(0,\pi r)}}d{\bf x}^{(M)}$
$\displaystyle\qquad{}\times\prod_{m=1}^{M}\left\\{{\rm
sgn}(\sigma_{m})\prod_{n=1}^{N}p^{[0,\pi
r]}\big{(}t_{m}-t_{m-1},x^{(m)}_{\sigma^{(m)}(n)}|x^{(m-1)}_{\sigma^{(m-1)}(n)}\big{)}g_{t_{m}}\big{(}{\bf
x}^{(m)}\big{)}\right.$
$\displaystyle\qquad\qquad{}\times\frac{c^{{R}}_{0}\big{(}\tau^{{R}}(t_{m})\big{)}}{c^{{R}}_{0}\big{(}\tau^{{R}}(t_{m-1})\big{)}}\prod_{\ell=1}^{N}\frac{k^{{R}}_{1}\big{(}x^{(m)}_{\ell};\tau^{{R}}(t_{m})\big{)}}{k^{{R}}_{1}\big{(}x^{(m-1)}_{\ell};\tau^{{R}}(t_{m-1})\big{)}}$
$\displaystyle\left.\qquad\qquad{}\times\left|\prod_{1\leq j<k\leq
N}\frac{k^{{R}}_{2}\big{(}x^{(m)}_{\sigma^{(m)}(k)},x^{(m)}_{\sigma^{(m)}(j)};\tau^{{R}}(t_{m})\big{)}}{k^{{R}}_{2}\big{(}x^{(m-1)}_{\sigma^{(m-1)}(k)},x^{(m-1)}_{\sigma^{(m-1)}(j)};\tau^{{R}}(t_{m-1})\big{)}}\right|\right\\},$

where ${\cal S}_{N}$ denotes the collection of all permutations of $N$ indices
$\\{1,2,\dots,N\\}$,
$\sigma^{(m)}\equiv\sigma_{1}\circ\sigma_{2}\circ\cdots\circ\sigma_{m}$,
$m\geq 1$, $\sigma^{(0)}={\rm id}$, and $x^{(0)}_{j}=u_{j}$, $j=1,\dots,N$.
Here we used the fact that $g_{t_{m}}({\bf x})$’s and
$\prod\limits_{\ell=1}^{N}k^{{R}}_{1}(x_{\ell};\cdot)$ are symmetric functions
of ${\bf x}$. By the definition of $\mathbb{W}_{N}^{(0,\pi r)}$ and the fact
that

$\displaystyle\prod_{1\leq j<k\leq
N}k^{{R}}_{2}(x_{\sigma(k)},x_{\sigma(j)};\cdot)={\rm sgn}(\sigma)\prod_{1\leq
j<k\leq N}k^{{R}}_{2}(x_{k},x_{j};\cdot),\qquad\sigma\in{\cal S}_{N},$

this is equal to

$\displaystyle\int_{[0,\pi r]^{N}}d{\bf x}^{(1)}\cdots\int_{[0,\pi
r]^{N}}d{\bf x}^{(M)}\prod_{m=1}^{M}\left\\{\prod_{n=1}^{N}p^{[0,\pi
r]}\big{(}t_{m}-t_{m-1},x^{(m)}_{n}|x^{(m-1)}_{n}\big{)}g_{t_{m}}\big{(}{\bf
x}^{(m)}\big{)}\right\\}$
$\displaystyle\qquad{}\times\frac{c^{{R}}_{0}\big{(}\tau^{{R}}(t_{M})\big{)}}{c^{{R}}_{0}\big{(}\tau^{{R}}(0)\big{)}}\prod_{\ell=1}^{N}\frac{k^{{R}}_{1}\big{(}x^{(M)}_{\ell};\tau^{{R}}(t_{M})\big{)}}{k^{{R}}_{1}\big{(}u_{\ell};\tau^{{R}}(0)\big{)}}\prod_{1\leq
j<k\leq
N}\frac{k^{{R}}_{2}\big{(}x^{(M)}_{k},x^{(M)}_{j};\tau^{{R}}(t_{M})\big{)}}{k^{{R}}_{2}\big{(}u_{k},u_{j};\tau^{{R}}(0)\big{)}}.$

Therefore, we obtain the equality

$\displaystyle\mathbb{E}^{{R}}_{{\bf u}}\left[\prod_{m=1}^{M}g_{t_{m}}({\bf
X}(t_{m}))\right]={\rm E}^{[0,\pi r]}_{{\bf
u}}\left[\prod_{m=1}^{M}g_{t_{m}}({\bf B}(t_{m}))D^{{R}}_{{\bf u}}(t_{M},{\bf
B}(t_{M}))\right],$ (3.4)

that is, the noncolliding condition, $T_{\rm collision}>t_{M}$, can be omitted
in (3.3). The expression (3.4) is interpreted as the expectation of the
product of symmetric functions $\prod\limits_{m=1}^{M}g_{t_{m}}(\cdot)$ with
respect to the signed measure

$\displaystyle\widehat{\mathbb{P}}^{{R}}_{{\bf u}}\big{|}_{{\cal
F}_{t}}=D^{{R}}_{{\bf u}}(t,{\bf B}(t)){\rm P}^{[0,\pi r]}_{{\bf
u}}\big{|}_{{\cal F}_{t}},\qquad t\in[0,t_{\ast}),$ (3.5)

which is a modification of (2.13) obtained by replacing ${\rm P}_{{\bf u}}$ by
${\rm P}_{{\bf u}}^{[0,\pi r]}$ and $D_{{\bf u}}$ by $D^{{R}}_{{\bf u}}$. Then
we can apply Proposition 2.3, where we replace $p_{\rm BM}$ by $p^{[0,\pi r]}$
and use the proper martingale functions given by (1.18) with (1.19)–(1.21).
Thus the proof is completed.

##  4 Proof of Theorem 1.2

Theorem 1.2 is concluded from the following key lemma.

######  Lemma 4.1.

The transition probability densities $p^{{B}_{N}}$, $p^{{C}_{N}}$ and
$p^{{D}_{N}}$ given by (3.2) for ${R}={B}_{N},{C}_{N}$, and ${D}_{N}$ solve
the following backward Kolmogorov equations

$\displaystyle-\frac{\partial p^{{B}_{N}}(t,{\bf y}|s,{\bf x})}{\partial
s}=\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}p^{{B}_{N}}(t,{\bf y}|s,{\bf
x})}{\partial x_{j}^{2}}+\sum_{j=1}^{N}A^{2\pi
r}_{2N-1}(t_{\ast}-s,x_{j})\frac{\partial p^{{B}_{N}}(t,{\bf y}|s,{\bf
x})}{\partial x_{j}}$ $\displaystyle\qquad{}+\sum_{\begin{subarray}{c}1\leq
j,k\leq N,\cr j\not=k\end{subarray}}\big{(}A^{2\pi
r}_{2N-1}(t_{\ast}-s,x_{j}-x_{k})+A^{2\pi
r}_{2N-1}(t_{\ast}-s,x_{j}+x_{k})\big{)}\frac{\partial p^{{B}_{N}}(t,{\bf
y}|s,{\bf x})}{\partial x_{j}},$ (4.1) $\displaystyle-\frac{\partial
p^{{C}_{N}}(t,{\bf y}|s,{\bf x})}{\partial
s}=\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}p^{{C}_{N}}(t,{\bf y}|s,{\bf
x})}{\partial x_{j}^{2}}+2\sum_{j=1}^{N}A^{2\pi
r}_{2(N+1)}(t_{\ast}-s,2x_{j})\frac{\partial p^{{C}_{N}}(t,{\bf y}|s,{\bf
x})}{\partial x_{j}}$ $\displaystyle\qquad{}+\sum_{\begin{subarray}{c}1\leq
j,k\leq N,\cr j\not=k\end{subarray}}\big{(}A^{2\pi
r}_{2(N+1)}(t_{\ast}-s,x_{j}-x_{k})+A^{2\pi
r}_{2(N+1)}(t_{\ast}-s,x_{j}+x_{k})\big{)}\frac{\partial p^{{C}_{N}}(t,{\bf
y}|s,{\bf x})}{\partial x_{j}},\\!\\!\\!$ (4.2) $\displaystyle-\frac{\partial
p^{{D}_{N}}(t,{\bf y}|s,{\bf x})}{\partial
s}=\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}p^{{D}_{N}}(t,{\bf y}|s,{\bf
x})}{\partial x_{j}^{2}}$
$\displaystyle\qquad{}+\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\big{(}A^{2\pi
r}_{2(N-1)}(t_{\ast}-s,x_{j}-x_{k})+A^{2\pi
r}_{2(N-1)}(t_{\ast}-s,x_{j}+x_{k})\big{)}\frac{\partial p^{{D}_{N}}(t,{\bf
y}|s,{\bf x})}{\partial x_{j}},\\!\\!\\!\\!$ (4.3)

for ${\bf x},{\bf y}\in\mathbb{W}_{N}^{(0,\pi r)}$, $0\leq s<t<t_{\ast}$,
under the conditions

$\displaystyle\lim_{s\uparrow t}p^{{B}_{N}}(t,{\bf y}|s,{\bf
x})=\lim_{s\uparrow t}p^{{C}_{N}}(t,{\bf y}|s,{\bf x})=\lim_{s\uparrow
t}p^{{D}_{N}}(t,{\bf y}|s,{\bf x})=\prod_{j=1}^{N}\delta(x_{j}-y_{j}).$ (4.4)

###### Proof.

For given $t\in(0,t_{\ast})$ and ${\bf y}\in\mathbb{W}_{N}^{(0,\pi r)}$, put

$\displaystyle u(s,{\bf x})=w(s,{\bf x})q^{[0,\pi r]}(t-s,{\bf y}|{\bf
x}),\qquad 0<s<t,\qquad{\bf x}\in\mathbb{W}_{N}^{(0,\pi r)}$ (4.5)

with the KMLGV determinant

$\displaystyle q^{[0,\pi r]}(t-s,{\bf y}|{\bf x})=\det_{1\leq j,k\leq
N}\big{[}p^{[0,\pi r]}(t-s,y_{j}|x_{k})\big{]},$ (4.6)

where $w(s,{\bf x})$ is a ${\rm C}^{1,2}$-function which will be specified
later, and $p^{[0,\pi r]}=p^{[0,\pi r]}_{\rm aa}$ given by (1.11) for
${R}={C}_{N}$, $p^{[0,\pi r]}=p^{[0,\pi r]}_{\rm rr}$ given by (1.12) for
${R}={D}_{N}$. Since $p^{[0,\pi r]}$ is the transition probability density of
a boundary-conditioned Brownian motion, it solves the diffusion equation

$\displaystyle-\frac{\partial p^{[0,\pi r]}(t-s,y|x)}{\partial
s}=\frac{1}{2}\frac{\partial^{2}p^{[0,\pi r]}(t-s,y|x)}{\partial x^{2}}$

with $\lim\limits_{s\uparrow t}p^{[0,\pi r]}(t-s,y|x)=\delta(x-y)$. The KMLGV
determinant (4.6) is the summation of products of $p^{[0,\pi r]}$’s, and hence

$\displaystyle-\frac{\partial q(t-s,{\bf y}|{\bf x})}{\partial
s}=\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}q(t-s,{\bf y}|{\bf x})}{\partial
x_{j}^{2}}$

is satisfied. It is easy to verify that

$\displaystyle\lim_{s\uparrow t}q(t-s,{\bf y}|{\bf
x})=\prod_{j=1}^{N}\delta(x_{j}-y_{j}).$ (4.7)

Therefore, the following equation holds

$\displaystyle\left\\{\frac{\partial}{\partial
s}+\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}}{\partial
x_{j}^{2}}\right\\}u(s,{\bf x})$ $\displaystyle\qquad{}=q^{[0,\pi r]}(t-s,{\bf
y}|{\bf x})\left\\{\frac{\partial}{\partial
s}+\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}}{\partial
x_{j}^{2}}\right\\}w(s,{\bf x})+\sum_{j=1}^{N}\frac{\partial w(s,{\bf
x})}{\partial x_{j}}\frac{\partial q^{[0,\pi r]}(t-s,{\bf y}|{\bf
x})}{\partial x_{j}}.$

Since

$\displaystyle\sum_{j=1}^{N}\frac{\partial w(s,{\bf x})}{\partial
x_{j}}\frac{\partial u(s,{\bf x})}{\partial x_{j}}$
$\displaystyle\qquad{}=q^{[0,\pi r]}(t-s,{\bf y}|{\bf
x})\sum_{j=1}^{N}\left(\frac{\partial w(s,{\bf x})}{\partial
x_{j}}\right)^{2}+w(s,{\bf x})\sum_{j=1}^{N}\frac{\partial w(s,{\bf
x})}{\partial x_{j}}\frac{\partial q^{[0,\pi r]}(t-s,{\bf y}|{\bf
x})}{\partial x_{j}},$

the above equation is written as

$\displaystyle\left\\{\frac{\partial}{\partial
s}+\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}}{\partial
x_{j}^{2}}-\frac{1}{w(s,{\bf x})}\sum_{j=1}^{N}\frac{\partial w(s,{\bf
x})}{\partial x_{j}}\frac{\partial}{\partial x_{j}}\right\\}u(s,{\bf x})$
$\displaystyle\qquad{}=q^{[0,\pi r]}(t-s,{\bf y}|{\bf
x})\left[\left\\{\frac{\partial}{\partial
s}+\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}}{\partial
x_{j}^{2}}\right\\}w(s,{\bf x})-\frac{1}{w(s,{\bf
x})}\sum_{j=1}^{N}\left(\frac{\partial w(s,{\bf x})}{\partial
x_{j}}\right)^{2}\right].$ (4.8)

Now we put

$\displaystyle w(s,{\bf x})=w^{{R}}(s,{\bf x})$ (4.9)
$\displaystyle\equiv\left\\{g^{{R}}(s)\prod_{\ell=1}^{N}\vartheta_{1}\left(\frac{c_{1}^{{R}}x_{\ell}}{2\pi
r};\tau^{{R}}(s)\right)\prod_{1\leq j<k\leq
N}\vartheta_{1}\left(\frac{x_{k}-x_{j}}{2\pi
r};\tau^{{R}}(s)\right)\vartheta_{1}\left(\frac{x_{k}+x_{j}}{2\pi
r};\tau^{{R}}(s)\right)\right\\}^{-1}$

for ${R}={B}_{N},{C}_{N}$, and

$\displaystyle w(s,{\bf x})=w^{{D}_{N}}(s,{\bf x})$
$\displaystyle\hphantom{w(s,{\bf
x})}{}\equiv\left\\{g^{{D}_{N}}(s)\prod_{1\leq j<k\leq
N}\vartheta_{1}\left(\frac{x_{k}-x_{j}}{2\pi
r};\tau^{{D}_{N}}(s)\right)\vartheta_{1}\left(\frac{x_{k}+x_{j}}{2\pi
r};\tau^{{D}_{N}}(s)\right)\right\\}^{-1}$ (4.10)

for ${R}={D}_{N}$, where $g^{{R}}(s)$, ${R}={B}_{N},{C}_{N},{D}_{N}$, are
${\rm C}^{1}$-functions of $s$, which will be determined later, and
$c_{1}^{{B}_{N}}=1$, $c_{1}^{{C}_{N}}=2$. We define $\alpha^{{R}}=1$ for
${R}={B}_{N}$ and ${C}_{N}$, and $\alpha^{{R}}=0$ for ${R}={D}_{N}$. We put
$c_{1}^{{D}_{N}}=1$ for convention. By (1.1) with (A.7), we see that

$\displaystyle-\frac{1}{w^{{R}}(s,{\bf x})}\frac{\partial w^{{R}}(s,{\bf
x})}{\partial x_{j}}=\alpha^{{R}}c_{1}^{{R}}A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x_{j}\big{)}$
$\displaystyle\hphantom{-\frac{1}{w^{{R}}(s,{\bf x})}\frac{\partial
w^{{R}}(s,{\bf x})}{\partial x_{j}}=}{}+\sum_{\begin{subarray}{c}1\leq k\leq
N,\cr k\not=j\end{subarray}}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)},$ (4.11)

$j=1,2,\dots,N$, and if we multiply (4.11) by $\partial p^{{R}}(t,{\bf
y}|s,{\bf x})/\partial x_{j}$ and take summation over $j=1,2,\dots,N$ for
${R}={B}_{N},{C}_{N},{D}_{N}$, they give the drift terms in (4.1)–(4.3).
Therefore, if we can choose $g^{{R}}(s)$ so that the equation

$\displaystyle\left\\{\frac{\partial}{\partial
s}+\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}}{\partial
x_{j}^{2}}\right\\}w^{{R}}(s,{\bf x})-\frac{1}{w^{{R}}(s,{\bf
x})}\sum_{j=1}^{N}\left(\frac{\partial w^{{R}}(s,{\bf x})}{\partial
x_{j}}\right)^{2}=0$ (4.12)

holds, then the r.h.s. of (4.8) vanishes and we can conclude that $u(s,{\bf
x})$ given in the form (4.5) with (4.9) and (4.10) solve the Kolmogorov
equations (4.1)–(4.3).

From (4.9) and (4.10) with (1.3), we find that

$\displaystyle\frac{1}{w^{{R}}(s,{\bf x})}\frac{\partial w^{{R}}(s,{\bf
x})}{\partial
s}=-\frac{1}{g^{{R}}(s)}\frac{dg^{{R}}(s)}{ds}+\alpha^{{R}}\frac{i{\cal
N}^{{R}}}{2\pi
r^{2}}\sum_{j=1}^{N}\frac{\dot{\vartheta_{1}}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}$ $\displaystyle\qquad{}+\frac{1}{2}\frac{i{\cal
N}^{{R}}}{2\pi r^{2}}\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\left\\{\frac{\dot{\vartheta_{1}}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}+\frac{\dot{\vartheta_{1}}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}\right\\},$

where
$\dot{\vartheta}_{1}(v;\tau)=\partial\vartheta_{1}(v;\tau)/\partial\tau$. If
we use the equation (A.2), then the above is written as

$\displaystyle\frac{1}{w^{{R}}(s,{\bf x})}\frac{\partial w^{{R}}(s,{\bf
x})}{\partial
s}=-\frac{1}{g^{{R}}(s)}\frac{dg^{{R}}(s)}{ds}+\alpha^{{R}}\frac{{\cal
N}^{{R}}}{8\pi^{2}r^{2}}\sum_{j=1}^{N}\frac{\vartheta_{1}^{\prime\prime}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}$ $\displaystyle\qquad{}+\frac{1}{2}\frac{{\cal
N}^{{R}}}{8\pi^{2}r^{2}}\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\left\\{\frac{\vartheta_{1}^{\prime\prime}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}+\frac{\vartheta_{1}^{\prime\prime}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}\right\\},$

where
$\vartheta_{1}^{\prime\prime}(v;\tau)=\partial^{2}\vartheta_{1}(v;\tau)/\partial
v^{2}$. From (4.11), we find that

$\displaystyle\frac{\partial^{2}w^{{R}}(s,{\bf x})}{\partial
x_{j}^{2}}=-w^{{R}}(s,{\bf x})\frac{\partial}{\partial
x_{j}}\Bigg{\\{}\alpha^{{R}}c_{1}^{{R}}A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x_{j}\big{)}$
$\displaystyle{}+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)}\Bigg{\\}}+\frac{1}{w^{{R}}(s,{\bf
x})}\left(\frac{\partial w^{{R}}(s,{\bf x})}{\partial x_{j}}\right)^{2},$

and hence

$\displaystyle\frac{1}{2}\sum_{j=1}^{N}\frac{\partial^{2}w^{{R}}(s,{\bf
x})}{\partial x_{j}^{2}}-\frac{1}{w^{{R}}(s,{\bf
x})}\sum_{j=1}^{N}\left(\frac{\partial w^{{R}}(s,{\bf x})}{\partial
x_{j}}\right)^{2}=-\frac{1}{2}w^{{R}}(s,{\bf x})S$

with

$\displaystyle S=\sum_{j=1}^{N}\Bigg{[}\frac{\partial}{\partial
x_{j}}\bigg{\\{}\alpha^{{R}}c_{1}^{{R}}A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x_{j}\big{)}$
$\displaystyle{}+\sum_{\begin{subarray}{c}1\leq k\leq N,\cr
k\not=j\end{subarray}}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)}\bigg{\\}}$
$\displaystyle{}+\bigg{\\{}\alpha^{{R}}c_{1}^{{R}}A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x_{j}\big{)}+\sum_{\begin{subarray}{c}1\leq
k\leq N,\cr k\not=j\end{subarray}}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)}\bigg{\\}}^{2}\Bigg{]}.$

By definition (1.1),

$\displaystyle\frac{\partial A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}}{\partial x}=\frac{c_{1}^{{R}}}{(2\pi
r)^{2}}\frac{\vartheta_{1}^{\prime\prime}\big{(}c_{1}^{{R}}x/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}c_{1}^{{R}}x/2\pi
r;\tau^{{R}}(s)\big{)}}-c_{1}^{{R}}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}\big{)}^{2}.$ (4.13)

Hence

$\displaystyle S=\alpha^{{R}}\frac{\big{(}c_{1}^{{R}}\big{)}^{2}}{(2\pi
r)^{2}}\sum_{j=1}^{N}\frac{\vartheta_{1}^{\prime\prime}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}$ $\displaystyle\hphantom{S=}{}+\frac{1}{(2\pi
r)^{2}}\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\left\\{\frac{\vartheta_{1}^{\prime\prime}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}+\frac{\vartheta_{1}^{\prime\prime}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}\right\\}$
$\displaystyle\hphantom{S=}{}+2\alpha^{{R}}c_{1}^{{R}}\sum_{\begin{subarray}{c}1\leq
j,k\leq N,\cr j\not=k\end{subarray}}A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x_{j}\big{)}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)}$
$\displaystyle\hphantom{S=}{}+\sum_{\begin{subarray}{c}1\leq j,k,\ell\leq
N,\cr j\not=k\not=\ell\not=j\end{subarray}}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)}$
$\displaystyle\hphantom{S=}{}\times\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{\ell})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{\ell})\big{)},$

where we have used the fact

$\displaystyle\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})=0$

concluded from (A.7). Then (4.12) holds, if

$\displaystyle-\frac{1}{g^{{R}}(s)}\frac{dg^{{R}}(s)}{ds}+\alpha^{{R}}\frac{{\cal
N}^{{R}}-\big{(}c_{1}^{{R}}\big{)}^{2}}{8\pi^{2}r^{2}}\sum_{j=1}^{N}\frac{\vartheta_{1}^{\prime\prime}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}c_{1}^{{R}}x_{j}/2\pi
r;\tau^{{R}}(s)\big{)}}$ $\displaystyle\qquad{}+\frac{{\cal
N}^{{R}}-2}{16\pi^{2}r^{2}}\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\left\\{\frac{\vartheta_{1}^{\prime\prime}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}-x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}+\frac{\vartheta_{1}^{\prime\prime}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}(x_{j}+x_{k})/2\pi
r;\tau^{{R}}(s)\big{)}}\right\\}$
$\displaystyle\qquad{}-\alpha^{{R}}c_{1}^{{R}}\sum_{\begin{subarray}{c}1\leq
j,k\leq N,\cr j\not=k\end{subarray}}A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x_{j}\big{)}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)}$
$\displaystyle\qquad{}-\frac{1}{2}\sum_{\begin{subarray}{c}1\leq j,k,\ell\leq
N,\cr j\not=k\not=\ell\not=j\end{subarray}}\big{(}A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}-x_{k})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{k})\big{)}$ $\displaystyle\qquad{}\times\big{(}A_{{\cal
N}^{{R}}}^{2\pi r}(t_{\ast}-s,x_{j}-x_{\ell})+A_{{\cal N}^{{R}}}^{2\pi
r}(t_{\ast}-s,x_{j}+x_{\ell})\big{)}=0.$ (4.14)

Now we use the following expressions for $A_{{\cal N}^{{R}}}^{2\pi r}$ and its
spatial derivative [16, 17]

$\displaystyle A_{{\cal N}^{{R}}}^{2\pi
r}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}=\zeta_{{\cal
N}^{{R}}}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}-c_{1}^{{R}}\frac{\eta_{{\cal
N}^{{R}}}^{1}(t_{\ast}-s)x}{\pi r},$ $\displaystyle\frac{\partial A_{{\cal
N}^{{R}}}^{2\pi r}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}}{\partial
x}=-c_{1}^{{R}}\wp_{{\cal
N}^{{R}}}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}-c_{1}^{{R}}\frac{\eta_{{\cal
N}^{{R}}}^{1}(t_{\ast}-s)}{\pi r}.$ (4.15)

Here $\eta_{{\cal N}^{{R}}}^{1}(t_{\ast}-s)$ is given by (A.10) and we put

$\displaystyle\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x)=\zeta(x|2\omega_{1},2\omega_{3})\big{|}_{\omega_{1}=\pi
r,\omega_{3}=i{\cal N}^{{R}}(t_{\ast}-s)/2r},$ $\displaystyle\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,x)=\wp(x|2\omega_{1},2\omega_{3})\big{|}_{\omega_{1}=\pi
r,\omega_{3}=i{\cal N}^{{R}}(t_{\ast}-s)/2r},$ (4.16)

where the Weierstrass $\wp$ function and zeta function $\zeta$ are defined by
(A.14) in Appendix A.5. Applying (4.15) to (4.13) gives

$\displaystyle\frac{1}{(2\pi
r)^{2}}\frac{\vartheta_{1}^{\prime\prime}\big{(}c_{1}^{{R}}x/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}c_{1}^{{R}}x/2\pi
r;\tau^{{R}}(s)\big{)}}=\left(\zeta_{{\cal
N}^{{R}}}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}-c_{1}^{{R}}\frac{\eta_{{\cal
N}^{{R}}}^{1}(t_{\ast}-s)x}{\pi r}\right)^{2}$
$\displaystyle\hphantom{\frac{1}{(2\pi
r)^{2}}\frac{\vartheta_{1}^{\prime\prime}\big{(}c_{1}^{{R}}x/2\pi
r;\tau^{{R}}(s)\big{)}}{\vartheta_{1}\big{(}c_{1}^{{R}}x/2\pi
r;\tau^{{R}}(s)\big{)}}=}{}-\wp_{{\cal
N}^{{R}}}\big{(}t_{\ast}-s,c_{1}^{{R}}x\big{)}-\frac{\eta_{{\cal
N}^{{R}}}^{1}(t_{\ast}-s)}{\pi r}.$ (4.17)

Using (4.15) and (4.17), the l.h.s. of (4.14) can be written using
$\zeta_{{\cal N}^{{R}}},\wp_{{\cal N}^{{R}}}$ and $\eta_{{\cal N}^{{R}}}^{1}$.
Moreover, from the functional equation (A.15) given in Appendix A.5, we can
derive the following equalities

$\displaystyle\sum_{\begin{subarray}{c}1\leq j,k,\ell\leq N,\cr
j\not=k\not=\ell\not=j\end{subarray}}(\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k}))$
$\displaystyle\qquad\quad{}\times(\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{\ell})+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{\ell}))$
$\displaystyle\qquad{}=(N-2)\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\big{(}\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})^{2}+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k})^{2}\big{)}$
$\displaystyle\qquad\quad{}-(N-2)\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}(\wp_{{\cal N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})+\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k})),$ (4.18)
$\displaystyle\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\zeta_{{\cal N}^{{R}}}(t_{\ast}-s,x_{j})(\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k}))$
$\displaystyle\qquad{}=\frac{1}{4}\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\big{(}\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})^{2}+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k})^{2}\big{)}$
$\displaystyle\qquad\quad{}-\frac{1}{4}\sum_{\begin{subarray}{c}1\leq j,k\leq
N,\cr j\not=k\end{subarray}}(\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})+\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k}))$
$\displaystyle\qquad\quad{}+(N-1)\sum_{j=1}^{N}\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j})^{2}-(N-1)\sum_{j=1}^{N}\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}),$ (4.19)

and

$\displaystyle\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\zeta_{{\cal N}^{{R}}}(t_{\ast}-s,2x_{j})(\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k}))$
$\displaystyle\qquad{}=\frac{1}{2}\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\big{(}\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})^{2}+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k})^{2}\big{)}$
$\displaystyle\qquad\quad{}-\frac{1}{2}\sum_{\begin{subarray}{c}1\leq j,k\leq
N,\cr j\not=k\end{subarray}}(\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})+\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k}))$
$\displaystyle\qquad\quad{}+\frac{N-1}{2}\sum_{j=1}^{N}\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,2x_{j})^{2}-\frac{N-1}{2}\sum_{j=1}^{N}\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,2x_{j}).$ (4.20)

Since $\eta_{{\cal N}^{{R}}}(t_{\ast}-s,-x)=-\eta_{{\cal
N}^{{R}}}(t_{\ast}-s,x)$, we have

$\displaystyle\sum_{\begin{subarray}{c}1\leq j,k,\ell\leq N,\cr
j\not=k\not=\ell\not=j\end{subarray}}(\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k}))\\{(x_{j}-x_{k})+(x_{j}+x_{\ell})\\}$
$\displaystyle{}=(N-2)\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\\{\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}-x_{k})(x_{j}-x_{k})+\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,x_{j}+x_{k})(x_{j}+x_{k})\\},\\!\\!\\!\\!$ (4.21)

and

$\displaystyle\sum_{\begin{subarray}{c}1\leq j,k,\ell\leq N,\cr
j\not=k\not=\ell\not=j\end{subarray}}\\{(x_{j}-x_{k})(x_{j}-x_{\ell})+(x_{j}+x_{k})(x_{j}+x_{\ell})\\}$
$\displaystyle\qquad{}=(N-2)\sum_{\begin{subarray}{c}1\leq j,k\leq N,\cr
j\not=k\end{subarray}}\big{\\{}(x_{j}-x_{k})^{2}+(x_{j}+x_{k})^{2}\big{\\}}=4(N-1)(N-2)\sum_{j=1}^{N}x_{j}^{2}.$
(4.22)

Using the above formulas and the values of ${\cal N}^{{R}}$ given by (1.4), we
can show that (4.14) is reduced to the following simple equations

$\displaystyle\frac{d\log
g^{{B}_{N}}(s)}{ds}=-N(N-1)(2N-1)\frac{\eta_{2N-1}^{1}(t_{\ast}-s)}{2\pi r},$
$\displaystyle\frac{d\log
g^{{C}_{N}}(s)}{ds}=-N(N^{2}-1)\frac{\eta_{2(N+1)}^{1}(t_{\ast}-s)}{\pi r},$
$\displaystyle\frac{d\log
g^{{D}_{N}}(s)}{ds}=-N(N-1)(N-2)\frac{\eta_{2(N-1)}^{1}(t_{\ast}-s)}{\pi r}.$
(4.23)

They give the conditions for $g^{{R}}(s)$ so that (4.12) holds. Since (A.11)
gives for ${R}={B}_{N},{C}_{N},{D}_{N}$

$\displaystyle\frac{d\log\eta\big{(}\tau^{{B}_{N}}(s)\big{)}}{ds}=(2N-1)\frac{\eta_{2N-1}^{1}(t_{\ast}-s)}{2\pi
r},$
$\displaystyle\frac{d\log\eta\big{(}\tau^{{C}_{N}}(s)\big{)}}{ds}=(N+1)\frac{\eta_{2(N+1)}^{1}(t_{\ast}-s)}{\pi
r},$
$\displaystyle\frac{d\log\eta\big{(}\tau^{{D}_{N}}(s)\big{)}}{ds}=(N-1)\frac{\eta_{2(N-1)}^{1}(t_{\ast}-s)}{\pi
r},$

we find that

$\displaystyle
g^{{B}_{N}}(s)=c^{{B}_{N}}\eta\big{(}\tau^{{B}_{N}}(s)\big{)}^{-N(N-1)},$
$\displaystyle
g^{{C}_{N}}(s)=c^{{C}_{N}}\eta\big{(}\tau^{{C}_{N}}(s)\big{)}^{-N(N-1)},$
$\displaystyle
g^{{D}_{N}}(s)=c^{{D}_{N}}\eta\big{(}\tau^{{D}_{N}}(s)\big{)}^{-N(N-2)},$

with constants $c^{{R}}$, ${R}={B}_{N},{C}_{N},{D}_{N}$. By (4.7), the
conditions (4.4) determine the constants.

In conclusion, under the condition (4.4),

$\displaystyle p^{{B}_{N}}(t,{\bf y}|s,{\bf
x})=\left(\frac{\eta\big{(}\tau^{{B}_{N}}(t)\big{)}}{\eta\big{(}\tau^{{B}_{N}}(s)\big{)}}\right)^{-N(N-1)}\prod_{\ell=1}^{N}\frac{\vartheta_{1}\big{(}y_{\ell}/2\pi
r;\tau^{{B}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}x_{\ell}/2\pi
r;\tau^{{B}_{N}}(s)\big{)}}$ $\displaystyle\qquad{}\times\prod_{1\leq j<k\leq
N}\frac{\vartheta_{1}\big{(}(y_{k}-y_{j})/2\pi
r;\tau^{{B}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(x_{k}-x_{j})/2\pi
r;\tau^{{B}_{N}}(s)\big{)}}\frac{\vartheta_{1}\big{(}(y_{k}+y_{j})/2\pi
r;\tau^{{B}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(x_{k}+x_{j})/2\pi
r;\tau^{{B}_{N}}(s)\big{)}}q^{[0,\pi r]}_{\rm ar}(t-s,{\bf y}|{\bf x})$

solves (4.1),

$\displaystyle p^{{C}_{N}}(t,{\bf y}|s,{\bf
x})=\left(\frac{\eta\big{(}\tau^{{C}_{N}}(t)\big{)}}{\eta\big{(}\tau^{{C}_{N}}(s)\big{)}}\right)^{-N(N-1)}\prod_{\ell=1}^{N}\frac{\vartheta_{1}\big{(}y_{\ell}/\pi
r;\tau^{{C}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}x_{\ell}/\pi
r;\tau^{{C}_{N}}(s)\big{)}}$ $\displaystyle\qquad{}\times\prod_{1\leq j<k\leq
N}\frac{\vartheta_{1}\big{(}(y_{k}-y_{j})/2\pi
r;\tau^{{C}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(x_{k}-x_{j})/2\pi
r;\tau^{{C}_{N}}(s)\big{)}}\frac{\vartheta_{1}\big{(}(y_{k}+y_{j})/2\pi
r;\tau^{{C}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(x_{k}+x_{j})/2\pi
r;\tau^{{C}_{N}}(s)\big{)}}q^{[0,\pi r]}_{\rm aa}(t-s,{\bf y}|{\bf x})$

solves (4.2), and

$\displaystyle p^{{D}_{N}}(t,{\bf y}|s,{\bf
x})=\left(\frac{\eta\big{(}\tau^{{D}_{N}}(t)\big{)}}{\eta\big{(}\tau^{{D}_{N}}(s)\big{)}}\right)^{-N(N-2)}$
$\displaystyle\qquad{}\times\prod_{1\leq j<k\leq
N}\frac{\vartheta_{1}\big{(}(y_{k}-y_{j})/2\pi
r;\tau^{{D}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(x_{k}-x_{j})/2\pi
r;\tau^{{D}_{N}}(s)\big{)}}\frac{\vartheta_{1}\big{(}(y_{k}+y_{j})/2\pi
r;\tau^{{D}_{N}}(t)\big{)}}{\vartheta_{1}\big{(}(x_{k}+x_{j})/2\pi
r;\tau^{{D}_{N}}(s)\big{)}}q^{[0,\pi r]}_{\rm rr}(t-s,{\bf y}|{\bf x})$

solves (4.3). The proof is thus completed. ∎

Now we prove Theorem 1.2.

###### Proof.

It is obvious that the backward Kolmogorov equations (4.1)–(4.3) correspond to
the systems of SDEs (1.22)–(1.24), respectively [15, 33]. Due to the behavior
(A.8) of $A^{2\pi r}_{{\cal N}}$, we can show that [15, 33] particles in
$[0,\pi r]$ following (1.22) do not arrive at 0, and those following (1.23) do
not arrive at 0 nor $\pi r$ with probability one, and thus we do not need to
impose any boundary condition at these endpoints of $[0,\pi r]$ for these
systems of SDEs. Hence the proof is completed. ∎

##  5 Relaxation to equilibrium processes
in trigonometric Dyson models of types C and D

As a corollary of Theorems 1.1 and 1.2, we obtain the following trigonometric
determinantal processes by taking the temporally homogeneous limit
$t_{\ast}\to\infty$.

######  Corollary 5.1.

Assume that ${\bf u}\in\mathbb{W}^{(0,\pi r)}$. Then the trigonometric Dyson
models of types C and D given by (1.27) and (1.28), respectively, are
determinantal with the spatio-temporal correlation kernels

$\displaystyle\check{{\bf K}}^{{C}_{N}}(s,x;t,y)=\sum_{j=1}^{N}p^{[0,\pi
r]}_{\rm aa}(s,x|u_{j})\check{M}^{{C}_{N}}_{{\bf u},u_{j}}(t,y)-{\bf
1}(s>t)p^{[0,\pi r]}_{\rm aa}(s-t,x|y),$ $\displaystyle\check{{\bf
K}}^{{D}_{N}}(s,x;t,y)=\sum_{j=1}^{N}p^{[0,\pi r]}_{\rm
rr}(s,x|u_{j})\check{M}^{{D}_{N}}_{{\bf u},u_{j}}(t,y)-{\bf 1}(s>t)p^{[0,\pi
r]}_{\rm rr}(s-t,x|y),$ (5.1)

$(s,x),(t,y)\in[0,\infty)\times[0,\pi r]$, where

$\displaystyle\check{M}^{{R}}_{{\bf u},u_{j}}(t,x)=\widetilde{{\rm
E}}\big{[}\check{\Phi}^{{R}}_{{\bf
u},u_{j}}\big{(}x+i\widetilde{B}(t)\big{)}\big{]},\qquad{R}={C}_{N},{D}_{N}$

with

$\displaystyle\check{\Phi}^{{C}_{N}}_{{\bf
u},u_{j}}(z)=\frac{\sin(z/r)}{\sin(u_{j}/r)}\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{\sin((z-u_{\ell})/2r)}{\sin((u_{j}-u_{\ell})/2r)}\frac{\sin((z+u_{\ell})/2r)}{\sin((u_{j}+u_{\ell})/2r)},$
$\displaystyle\check{\Phi}^{{D}_{N}}_{{\bf
u},u_{j}}(z)=\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{\sin((z-u_{\ell})/2r)}{\sin((u_{j}-u_{\ell})/2r)}\frac{\sin((z+u_{\ell})/2r)}{\sin((u_{j}+u_{\ell})/2r)},\qquad
j=1,2,\dots,N.$ (5.2)

We find that in the limit $t_{\ast}\to\infty$, the equality (2.24) with
${R}={C}_{N}$ and ${D}_{N}$ gives

$\displaystyle\det_{1\leq j,k\leq
N}\big{[}\check{f}^{{C}_{N}}_{j}(u_{k})\big{]}=(-1)^{N(N-1)/2}2^{N(N-1)}\prod_{\ell=1}^{N}\sin\left(\frac{u_{\ell}}{r}\right)$
$\displaystyle\hphantom{\det_{1\leq j,k\leq
N}\big{[}\check{f}^{{C}_{N}}_{j}(u_{k})\big{]}=}{}\times\prod_{1\leq j<k\leq
N}\sin\left(\frac{u_{k}-u_{j}}{2r}\right)\sin\left(\frac{u_{k}+u_{j}}{2r}\right),$
$\displaystyle\det_{1\leq j,k\leq
N}\big{[}\check{f}^{{D}_{N}}_{j}(u_{k})\big{]}=(-1)^{N(N-1)/2}2^{(N-1)^{2}}\prod_{1\leq
j<k\leq
N}\sin\left(\frac{u_{k}-u_{j}}{2r}\right)\sin\left(\frac{u_{k}+u_{j}}{2r}\right)$
(5.3)

with

$\displaystyle\check{f}_{j}^{{C}_{N}}(z)=\sin\left(\frac{jz}{r}\right),\qquad\check{f}_{j}^{{D}_{N}}(z)=\cos\left(\frac{(j-1)z}{r}\right),\qquad
j\in\mathbb{Z}.$ (5.4)

By Lemma 2.1, we see that (5.2) can be expanded as

$\displaystyle\check{\Phi}^{{R}}_{{\bf
u},u_{j}}(z)=\sum_{k=1}^{N}\check{\phi}^{{R}}_{{\bf
u},u_{j}}(k)\check{f}^{{R}}_{k}(z),\qquad{R}={C}_{N},{D}_{N},$

and

$\displaystyle\sum_{\ell=1}^{N}\check{f}^{{R}}_{j}(u_{\ell})\check{\phi}^{{R}}_{{\bf
u},u_{\ell}}(k)=\delta_{jk},\qquad j,k\in\\{1,2,\dots,N\\}.$ (5.5)

We note that the transition probability densities (1.11) and (1.12) are
written as (A.12) and (A.13), respectively. Hence we have

$\displaystyle p^{[0,\pi r]}_{\rm aa}(t,y|x)=\frac{1}{\pi
r}\sum_{n\in\mathbb{Z}}e^{-n^{2}t/2r^{2}}\check{f}^{{C}_{N}}_{n}(y)\check{f}^{{C}_{N}}_{n}(x)=\frac{2}{\pi
r}\sum_{\ell=1}^{\infty}e^{-\ell^{2}t/2r^{2}}\check{f}^{{C}_{N}}_{\ell}(y)\check{f}^{{C}_{N}}_{\ell}(x),$
$\displaystyle p^{[0,\pi r]}_{\rm rr}(t,y|x)=\frac{1}{\pi
r}\left(\check{f}^{{D}_{N}}_{1}(y)\check{f}^{{D}_{N}}_{1}(x)+2\sum_{\ell=2}^{\infty}e^{-(\ell-1)^{2}t/2r^{2}}\check{f}^{{D}_{N}}_{\ell}(y)\check{f}^{{D}_{N}}_{\ell}(x)\right).$
(5.6)

Now we show relaxation processes to equilibria, which are typical non-
equilibrium phenomena.

######  Proposition 5.2.

For any initial configuration ${\bf u}\in\mathbb{W}^{(0,\pi r)}$, the
trigonometric Dyson models of types C and D exhibit relaxation to the
equilibrium processes. The equilibrium processes are determinantal with the
correlation kernels

$\displaystyle\check{{\bf K}}^{{C}_{N}}_{\rm eq}(t-s,x,y)$
$\displaystyle\qquad{}=\begin{cases}\displaystyle\frac{1}{\pi
r}\sum_{n\colon|n|\leq
N}e^{n^{2}(t-s)/2r^{2}}\sin\left(\frac{nx}{r}\right)\sin\left(\frac{ny}{r}\right),&\text{if
$t>s$},\vspace{1mm}\\\ \displaystyle\frac{1}{2\pi
r}\left[\frac{\sin\\{(2N+1)(y-x)/2r\\}}{\sin\\{(y-x)/2r\\}}-\frac{\sin\\{(2N+1)(y+x)/2r\\}}{\sin\\{(y+x)/2r\\}}\right],&\text{if
$t=s$},\vspace{1mm}\\\ \displaystyle-\frac{1}{\pi r}\sum_{n\colon|n|\geq
N+1}e^{n^{2}(t-s)/2r^{2}}\sin\left(\frac{nx}{r}\right)\sin\left(\frac{ny}{r}\right),&\text{if
$t<s$},\end{cases}$ (5.7)

and

$\displaystyle\check{{\bf K}}^{{D}_{N}}_{\rm eq}(t-s,x,y)$
$\displaystyle\qquad{}=\begin{cases}\displaystyle\frac{1}{\pi
r}\sum_{n\colon|n|\leq
N-1}e^{n^{2}(t-s)/2r^{2}}\cos\left(\frac{nx}{r}\right)\cos\left(\frac{ny}{r}\right),&\text{if
$t>s$},\vspace{1mm}\\\ \displaystyle\frac{1}{2\pi
r}\left[\frac{\sin\\{(2N-1)(y-x)/2r\\}}{\sin\\{(y-x)/2r\\}}+\frac{\sin\\{(2N-1)(y+x)/2r\\}}{\sin\\{(y+x)/2r\\}}\right],&\text{if
$t=s$},\vspace{1mm}\\\ \displaystyle-\frac{1}{\pi r}\sum_{n\colon|n|\geq
N}e^{n^{2}(t-s)/2r^{2}}\cos\left(\frac{nx}{r}\right)\cos\left(\frac{ny}{r}\right),&\text{if
$t<s$},\end{cases}$ (5.8)

respectively.

###### Proof.

Here we prove the convergence of correlation kernels. It implies the
convergence of the generating functions of spatio-temporal correlation
functions (i.e., the Laplace transformations of multi-time joint probability
densities). Hence the convergence of process is concluded in the sense of
finite-dimensional distributions [22]. Assume ${\bf u}\in\mathbb{W}^{(0,\pi
r)}$ and let

$\displaystyle{\bf g}^{{C}_{N}}_{{\bf u}}(s,x;t,y)=\sum_{j=1}^{N}p^{[0,\pi
r]}_{\rm aa}(s,x|u_{j})\check{M}^{{C}_{N}}_{{\bf u},u_{j}}(t,y),$
$\displaystyle{\bf g}^{{D}_{N}}_{{\bf u}}(s,x;t,y)=\sum_{j=1}^{N}p^{[0,\pi
r]}_{\rm rr}(s,x|u_{j})\check{M}^{{D}_{N}}_{{\bf
u},u_{j}}(t,y),\qquad(s,x),(t,y)\in[0,\infty)\times[0,\pi r].$

It is easy to verify that

$\displaystyle\widetilde{{\rm
E}}\big{[}\check{f}^{{C}_{N}}_{k}\big{(}x+i\widetilde{B}(t)\big{)}\big{]}=e^{k^{2}t/2r^{2}}\check{f}^{{C}_{N}}_{k}(x),$
$\displaystyle\widetilde{{\rm
E}}\big{[}\check{f}^{{D}_{N}}_{k}\big{(}x+i\widetilde{B}(t)\big{)}\big{]}=e^{(k-1)^{2}t/2r^{2}}\check{f}^{{D}_{N}}_{k}(x),\qquad
k\in\mathbb{Z},$

and thus

$\displaystyle\check{M}^{{C}_{N}}_{{\bf
u},u_{j}}(t,y)=\sum_{k=1}^{N}\check{\phi}^{{C}_{N}}_{{\bf
u},u_{j}}(k)e^{k^{2}t/2r^{2}}\check{f}^{{C}_{N}}_{k}(y),$
$\displaystyle\check{M}^{{D}_{N}}_{{\bf
u},u_{j}}(t,y)=\sum_{k=1}^{N}\check{\phi}^{{D}_{N}}_{{\bf
u},u_{j}}(k)e^{(k-1)^{2}t/2r^{2}}\check{f}^{{D}_{N}}_{k}(y),\qquad
j\in\\{1,2,\dots,N\\}.$

Then we have

$\displaystyle{\bf g}^{{R}}_{{\bf u}}(s,x;t,y)={\bf g}^{{R}}_{\rm
eq}(t-s,x,y)+{\bf r}^{{R}}_{{\bf u}}(s,x;t,y),\qquad{R}={C}_{N},{D}_{N}$

with

$\displaystyle{\bf g}^{{C}_{N}}_{\rm eq}(t-s,x,y)=\frac{2}{\pi
r}\sum_{\ell=1}^{N}e^{\ell^{2}(t-s)/2r^{2}}\check{f}^{{C}_{N}}_{\ell}(x)\check{f}^{{C}_{N}}_{\ell}(y)$
$\displaystyle\hphantom{{\bf g}^{{C}_{N}}_{\rm eq}(t-s,x,y)}{}=\frac{1}{\pi
r}\sum_{n\colon|n|\leq
N}e^{n^{2}(t-s)/2r^{2}}\sin\left(\frac{nx}{r}\right)\sin\left(\frac{ny}{r}\right),$
$\displaystyle{\bf r}^{{C}_{N}}_{{\bf u}}(s,x;t,y)=\frac{2}{\pi
r}\sum_{k=1}^{N}e^{k^{2}(t-s)/2r^{2}}\check{f}^{{C}_{N}}_{k}(y)\sum_{\ell=N+1}^{\infty}e^{-(\ell^{2}-k^{2})s/2r^{2}}\check{f}^{{C}_{N}}_{\ell}(x)$
$\displaystyle\hphantom{{\bf r}^{{C}_{N}}_{{\bf
u}}(s,x;t,y)=}{}\times\sum_{j=1}^{N}\check{f}^{{C}_{N}}_{\ell}(u_{j})\check{\phi}^{{C}_{N}}_{{\bf
u},u_{j}}(k),$ (5.9)

and

$\displaystyle{\bf g}^{{D}_{N}}_{\rm eq}(t-s,x,y)=\frac{1}{\pi
r}\left(\check{f}^{{D}_{N}}_{1}(x)\check{f}^{{D}_{N}}_{1}(y)+2\sum_{\ell=2}^{N}e^{(\ell-1)^{2}(t-s)/2r^{2}}\check{f}^{{D}_{N}}_{\ell}(x)\check{f}^{{D}_{N}}_{\ell}(y)\right)$
$\displaystyle\hphantom{{\bf g}^{{D}_{N}}_{\rm eq}(t-s,x,y)}{}=\frac{1}{\pi
r}\sum_{n:|n|\leq
N-1}e^{n^{2}(t-s)/2r^{2}}\cos\left(\frac{nx}{r}\right)\cos\left(\frac{ny}{r}\right),$
$\displaystyle{\bf r}^{{D}_{N}}_{{\bf u}}(s,x;t,y)=\frac{2}{\pi
r}\sum_{k=1}^{N}e^{(k-1)^{2}(t-s)/2r^{2}}\check{f}^{{D}_{N}}_{k}(y)$
$\displaystyle\hphantom{{\bf r}^{{D}_{N}}_{{\bf
u}}(s,x;t,y)=}{}\times\sum_{\ell=N+1}^{\infty}e^{-\\{(\ell-1)^{2}-(k-1)^{2}\\}s/2r^{2}}\check{f}^{{D}_{N}}_{\ell}(x)\sum_{j=1}^{N}\check{f}^{{D}_{N}}_{\ell}(u_{j})\check{\phi}^{{D}_{N}}_{{\bf
u},u_{j}}(k),$ (5.10)

where (5.5) and (5.6) were used. For any fixed $s,t\in[0,\infty)$, it is
obvious that

$\displaystyle\lim_{T\to\infty}{\bf r}^{{R}}_{{\bf
u}}(s+T,x;t+T,y)=0,\qquad{R}={C}_{N},{D}_{N},$

uniformly on any subset of $(x,y)\in(0,\pi r)^{2}$, since the summations of
$k$ are taken for $1\leq k\leq N$, while those of $\ell$ are taken for
$\ell\geq N+1$ in (5.9) and (5.10). Hence

$\displaystyle\lim_{T\to\infty}\check{{\bf K}}^{{C}_{N}}_{{\bf
u}}(s+T,x;t+T,y)={\bf g}^{{C}_{N}}_{\rm eq}(t-s,x,y)-{\bf 1}(s>t)p^{[0,\pi
r]}_{\rm aa}(s-t,x|y),$ $\displaystyle\lim_{T\to\infty}\check{{\bf
K}}^{{D}_{N}}_{{\bf u}}(s+T,x;t+T,y)={\bf g}^{{D}_{N}}_{\rm eq}(t-s,x,y)-{\bf
1}(s>t)p^{[0,\pi r]}_{\rm rr}(s-t,x|y),$

in the same sense. It is easy to confirm that the limit kernels are
represented by (5.7) and (5.8), if we use (5.6). The limit kernels (5.7) and
(5.8) depend on time difference $t-s$, which implies that the determinantal
processes defined by them are temporally homogeneous. The determinantal
processes with the spatio-temporal correlation kernels (5.7) and (5.8) are
equilibrium processes, which are reversible with respect to the determinantal
point processes with the spatial correlation kernels

$\displaystyle\check{\rm K}_{\rm eq}^{{C}_{N}}(x,y)=\frac{1}{2\pi
r}\left[\frac{\sin\\{(2N+1)(y-x)/2r\\}}{\sin\\{(y-x)/2r\\}}-\frac{\sin\\{(2N+1)(y+x)/2r\\}}{\sin\\{(y+x)/2r\\}}\right],$
$\displaystyle\check{\rm K}_{\rm eq}^{{D}_{N}}(x,y)=\frac{1}{2\pi
r}\left[\frac{\sin\\{(2N-1)(y-x)/2r\\}}{\sin\\{(y-x)/2r\\}}+\frac{\sin\\{(2N-1)(y+x)/2r\\}}{\sin\\{(y+x)/2r\\}}\right],$
(5.11)

$(x,y)\in[0,\pi r]^{2}$, respectively. The convergence of processes is
irreversible. Thus all statements of the present proposition have been proved.
∎

We note that

$\displaystyle\check{\rho}^{{C}_{N}}_{\rm eq}(x)=\lim_{y\to x}\check{\rm
K}^{{C}_{N}}_{\rm eq}(x,y)=\frac{2}{\pi
r}\sum_{n=1}^{N}\sin^{2}\left(\frac{nx}{r}\right),$
$\displaystyle\check{\rho}^{{D}_{N}}_{\rm eq}(x)=\lim_{y\to x}\check{\rm
K}^{{D}_{N}}_{\rm eq}(x,y)=\frac{1}{\pi
r}\left\\{1+2\sum_{n=1}^{N-1}\cos^{2}\left(\frac{nx}{r}\right)\right\\},$

and hence

$\displaystyle\int_{0}^{\pi r}\check{\rho}^{{C}_{N}}_{\rm
eq}(x)dx=\int_{0}^{\pi r}\check{\rho}^{{D}_{N}}_{\rm eq}(x)dx=N,$

as required.

##  6 Concluding remarks and open problems

In the previous paper [16, 17] and in this paper (Theorem 1.1), we have
introduced seven families of interacting particle systems
$\Xi^{{R}}(t)=\sum\limits_{j=1}^{N}\delta_{X^{{R}}_{j}(t)}$,
$t\in[0,t_{\ast})$ governed by the probability laws $\mathbb{P}^{{R}}_{{\bf
u}}$ associated with the irreducible reduced affine root systems denoted by
${R}={A}_{N-1},{B}_{N},{B}^{\vee}_{N},{C}_{N}$,
${C}^{\vee}_{N},{BC}_{N},{D}_{N}$. When we proved that they are determinantal
processes, we showed that without change of expectations of symmetric
functions of $\big{\\{}X^{{R}}_{j}(\cdot)\big{\\}}_{j=1}^{N}$,
$\mathbb{P}^{{R}}_{{\bf u}}$ can be replaced by the signed measures
$\widehat{\mathbb{P}}^{{R}}_{{\bf u}}$. Define $\widehat{{\cal
F}}^{{R}}_{t}=\sigma\big{(}\widehat{\Xi}^{{R}}(s)\colon 0\leq s\leq t\big{)}$,
$t\in[0,t_{\ast})$. As the simplest corollary of this fact, we can conclude
that at any time $0<t<t_{\ast}$,

$\displaystyle\widehat{\mathbb{E}}^{{R}}_{{\bf u}}\big{[}{\bf
1}\big{(}\widehat{\Xi}^{{R}}(t)\in\widehat{{\cal
F}}^{{R}}_{t}\big{)}\big{]}=1\qquad\text{for}\quad{R}={A}_{N-1},{B}_{N},{B}^{\vee}_{N},{C}_{N},{C}^{\vee}_{N},{BC}_{N},{D}_{N}.$

This is nothing but a rather trivial statement such that the processes
$\big{(}\widehat{\Xi}^{{R}}(t)\big{)}_{t\in[0,t_{\ast})}$ are well-normalized,
but it provides nontrivial multiple-integral equalities including parameters
$t\in[0,t_{\ast})$ and ${\bf u}=(u_{1},\dots,u_{N})$. For example, for
${R}={D}_{N}$, we will have

$\displaystyle\int_{[0,\pi r]^{N}}d{\bf x}\prod_{\ell=1}^{N}\frac{1}{2\pi
r}\left\\{\vartheta_{3}\left(\frac{x_{\ell}-u_{\ell}}{2\pi r};\frac{it}{2\pi
r^{2}}\right)+\vartheta_{3}\left(\frac{x_{\ell}+u_{\ell}}{2\pi
r};\frac{it}{2\pi r^{2}}\right)\right\\}$ (6.1)
$\displaystyle{}\times\prod_{1\leq j<k\leq
N}\\!\\!\frac{\vartheta_{1}((x_{k}-x_{j})/2\pi
r;\tau^{{D}_{N}}(t))}{\vartheta_{1}((u_{k}-u_{j})/2\pi
r;\tau^{{D}_{N}}(0))}\frac{\vartheta_{1}((x_{k}+x_{j})/2\pi
r;\tau^{{D}_{N}}(t))}{\vartheta_{1}((u_{k}+u_{j})/2\pi
r;\tau^{{D}_{N}}(0))}=\left(\frac{\eta(\tau^{{D}_{N}}(t))}{\eta(\tau^{{D}_{N}}(0))}\right)^{N(N-2)}\\!\\!,$

where $\eta(\tau)$ is the Dedekind modular function (1.10) and
$\tau^{{D}_{N}}(t)=i(N-1)(t_{\ast}-t)/\pi r^{2}$.

In the previous papers [16, 17] and in Theorem 1.2 we have identified the
systems of SDEs which are solved by the four families of determinantal
processes,
$\big{(}\big{(}\Xi^{{A}_{N-1}}(t)\big{)}_{t\in[0,t_{\ast})},\mathbb{P}^{{A}_{N-1}}_{{\bf
u}}\big{)}$, ${\bf u}\in\mathbb{W}_{N}^{(0,2\pi r)}$, and
$\big{(}\big{(}\Xi^{{R}}(t)\big{)}_{t\in[0,t_{\ast})},\mathbb{P}^{{R}}_{{\bf
u}}\big{)}$, ${\bf u}\in\mathbb{W}_{N}^{(0,\pi r)}$ with ${R}={B}_{N},{C}_{N}$
and ${D}_{N}$. The systems of SDEs for other cases
${R}={B}^{\vee}_{N},{C}^{\vee}_{N},{BC}_{N}$ are not yet clarified. The
exceptional cases of reduced irreducible affine root systems and the non-
reduced irreducible affine root systems [30] will be studied from the view
point of the present stochastic analysis.

The classical Dyson models of type A in $\mathbb{R}$ given by (1.29) and of
types C and D in $[0,\infty)$ given by (1.30) and (1.31) are realized as the
eigenvalue processes of Hermitian-matrix-valued Brownian motions with
specified symmetry [6, 18]. It will be an interesting problem to construct the
matrix-valued Brownian motions such that the eigenvalue processes of them
provide the elliptic Dyson models; (1.2) with $\beta=2$ and (1.22)–(1.24).

As shown in Section 4, the factors represented by the Dedekind modular
function (1.10) in the determinantal martingale-functions (1.5)–(1.7) are
essential in the proof of Theorem 1.2. We have noted that ${\cal N}^{{R}}$
given by (1.4) is identified with the quantity g given for the reduced
irreducible affine root systems in [30, Appendix 1]. Interpretation of the
formulas (1.8) from the view point of representation theory is desired.

As mentioned after Theorem 1.2 in Section 1, if we take the temporally
homogeneous limit $t_{\ast}\to\infty$, the elliptic Dyson models of types A,
B, C, and D become the corresponding trigonometric Dyson models, and in the
further limit $r\to\infty$, they are reduced to the Dyson models in
$\mathbb{R}$, in $[0,\infty)$ with an absorbing boundary condition at the
origin, and in $[0,\infty)$ with a reflecting boundary condition at the
origin, respectively. In the limit $r\to\infty$, $p^{[0,\pi r]}_{\rm aa}$ and
$p^{[0,\pi r]}_{\rm rr}$ given by (1.11) and (1.12) become

$\displaystyle p^{[0,\infty)}_{\rm aa}(t,y|x)=p_{\rm BM}(t,y|x)-p_{\rm
BM}(t,y|-x),$ $\displaystyle p^{[0,\infty)}_{\rm rr}(t,y|x)=p_{\rm
BM}(t,y|x)+p_{\rm BM}(t,y|-x),$

and in the double limit $t_{\ast}\to\infty$, $r\to\infty$,
$\Phi^{{C}_{N}}_{{\bf u},u_{j}}(z)$ and $\Phi^{{D}_{N}}_{{\bf u},u_{j}}(z)$
given by (1.19)–(1.21) become

$\displaystyle\Phi^{{C}_{N}}_{{\bf
u},u_{j}}(z)\to\frac{z_{j}}{u_{j}}\Phi^{[0,\infty)}_{{\bf
u},u_{j}}(z),\qquad\Phi^{{D}_{N}}_{{\bf u},u_{j}}(z)\to\Phi^{[0,\infty)}_{{\bf
u},u_{j}}(z)$

with

$\displaystyle\Phi^{[0,\infty)}_{{\bf
u},u_{j}}(z)=\prod_{\begin{subarray}{c}1\leq\ell\leq
N,\cr\ell\not=j\end{subarray}}\frac{z^{2}-u_{\ell}^{2}}{u_{j}^{2}-u_{\ell}^{2}}.$

Then the spatio-temporal correlation kernels (1.17) are reduced to the
following

$\displaystyle{\bf K}^{{C}_{N}}_{{\bf
u}}(s,x;t,y)=\sum_{j=1}^{N}p^{[0,\infty)}_{\rm aa}(s,x|u_{j})\widetilde{{\rm
E}}\left[\frac{y+i\widetilde{B}(t)}{u_{j}}\Phi^{[0,\infty)}_{{\bf
u},u_{j}}\big{(}y+i\widetilde{B}(t)\big{)}\right]$
$\displaystyle\hphantom{{\bf K}^{{C}_{N}}_{{\bf u}}(s,x;t,y)=}{}-{\bf
1}(s>t)p^{[0,\infty)}_{\rm aa}(s-t,x|y),$ $\displaystyle{\bf
K}^{{D}_{N}}_{{\bf u}}(s,x;t,y)=\sum_{j=1}^{N}p^{[0,\infty)}_{\rm
rr}(s,x|u_{j})\widetilde{{\rm E}}\big{[}\Phi^{[0,\infty)}_{{\bf
u},u_{j}}\big{(}y+i\widetilde{B}(t)\big{)}\big{]}-{\bf
1}(s>t)p^{[0,\infty)}_{\rm rr}(s-t,x|y),$

$(s,x),(t,y)\in[0,\infty)\times[0,\infty)$. Consider the $D$-dimensional
Bessel processes BES(D) in $[0,\infty)$, whose transition probability
densities $p^{{\rm BES}^{(D)}}$ are expressed using the modified Bessel
function $I_{\nu}(z)$ with $\nu=(D-2)/2$ [15, 33]. In particular, we find that

$\displaystyle p^{{\rm BES}^{(3)}}(t,y|x)=\frac{y}{x}p^{[0,\infty)}_{\rm
aa}(t,y|x),\qquad p^{{\rm BES}^{(1)}}(t,y|x)=p^{[0,\infty)}_{\rm rr}(t,y|x),$

$x,y\in(0,\infty)$, $t\geq 0$. If we put

$\displaystyle{\bf K}^{{\rm BES}^{(3)}}_{{\bf
u}}(s,x;t,y)=\sum_{j=1}^{N}p^{{\rm BES}^{(3)}}(s,x|u_{j})\widetilde{{\rm
E}}\left[\frac{y+i\widetilde{B}(t)}{y}\Phi^{[0,\infty)}_{{\bf
u},u_{j}}\big{(}y+i\widetilde{B}(t)\big{)}\right]$
$\displaystyle\hphantom{{\bf K}^{{\rm BES}^{(3)}}_{{\bf u}}(s,x;t,y)=}{}-{\bf
1}(s>t)p^{{\rm BES}^{(3)}}(s-t,x|y),$ (6.2) $\displaystyle{\bf K}^{{\rm
BES}^{(1)}}_{{\bf u}}(s,x;t,y)=\sum_{j=1}^{N}p^{{\rm
BES}^{(1)}}(s,x|u_{j})\widetilde{{\rm E}}\big{[}\Phi^{[0,\infty)}_{{\bf
u},u_{j}}\big{(}y+i\widetilde{B}(t)\big{)}\big{]}$
$\displaystyle\hphantom{{\bf K}^{{\rm BES}^{(1)}}_{{\bf u}}(s,x;t,y)=}{}-{\bf
1}(s>t)p^{{\rm BES}^{(1)}}(s-t,x|y),$ (6.3)

the following equalities are established

$\displaystyle{\bf K}^{{C}_{N}}_{{\bf u}}(s,x;t,y)=\frac{y}{x}{\bf K}^{{\rm
BES}^{(3)}}_{{\bf u}}(s,x;t,y),\qquad{\bf K}^{{D}_{N}}_{{\bf u}}(s,x;t,y)={\bf
K}^{{\rm BES}^{(1)}}_{{\bf u}}(s,x;t,y),$ (6.4)

$(s,x),(t,y)\in[0,\infty)\times[0,\infty)$. The functions (6.2) and (6.3) are
the spatio-temporal correlation kernels of the $N$-particle systems of BES(3)
and BES(1) with noncolliding condition (see [14, Section 7]). It is easy to
verify that the spatio-temporal Fredholm determinant (2.17) is invariant under
the transformation of kernel

$\displaystyle\widehat{\mathbb{K}}_{{\bf
u}}(s,x;t,y)\to\frac{a(t,y)}{a(s,x)}\widehat{\mathbb{K}}_{{\bf u}}(s,x;t,y)$

with an arbitrary continuous function $a$. Then (6.4) implies that the
elliptic Dyson models of types C and D given by Theorem 1.2 are deduced to the
noncolliding BES(3) and BES(1) in the double limit $t_{\ast}\to\infty$,
$r\to\infty$, respectively. In other words, the present elliptic Dyson models
of types C and D can be regarded as the elliptic extensions of the
noncolliding BES(3) and BES(1), respectively. Trigonometric and elliptic
extensions of noncolliding BES(D) with general $D\geq 1$ [14, 23] will be
studied. Moreover, infinite-particle limits of the elliptic determinantal
processes should be studied [16, 17, 22].

Connection between the present elliptic determinantal processes and
probabilistic discrete models with elliptic weights [1, 3, 35, 36, 37] will be
also an interesting future problem.

##  Appendix A The Jacobi theta functions and related functions

###  A.1 Notations and formulas of the Jacobi theta functions

Let

$\displaystyle z=e^{v\pi i},\qquad q=e^{\tau\pi i},$

where $v,\tau\in\mathbb{C}$ and $\Im\tau>0$. The Jacobi theta functions are
defined as follows [32, 43]

$\displaystyle\vartheta_{0}(v;\tau)=-ie^{\pi
i(v+\tau/4)}\vartheta_{1}\left(v+\frac{\tau}{2};\tau\right)=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{n^{2}}z^{2n}=1+2\sum_{n=1}^{\infty}(-1)^{n}e^{\tau\pi
in^{2}}\cos(2n\pi v),$
$\displaystyle\vartheta_{1}(v;\tau)=i\sum_{n\in\mathbb{Z}}(-1)^{n}q^{(n-1/2)^{2}}z^{2n-1}=2\sum_{n=1}^{\infty}(-1)^{n-1}e^{\tau\pi
i(n-1/2)^{2}}\sin\\{(2n-1)\pi v\\},$
$\displaystyle\vartheta_{2}(v;\tau)=\vartheta_{1}\left(v+\frac{1}{2};\tau\right)=\sum_{n\in\mathbb{Z}}q^{(n-1/2)^{2}}z^{2n-1}=2\sum_{n=1}^{\infty}e^{\tau\pi
i(n-1/2)^{2}}\cos\\{(2n-1)\pi v\\},$
$\displaystyle\vartheta_{3}(v;\tau)=e^{\pi
i(v+\tau/4)}\vartheta_{1}\left(v+\frac{1+\tau}{2};\tau\right)=\sum_{n\in\mathbb{Z}}q^{n^{2}}z^{2n}=1+2\sum_{n=1}^{\infty}e^{\tau\pi
in^{2}}\cos(2n\pi v).$ (A.1)

(Note that the present functions $\vartheta_{\mu}(v;\tau)$, $\mu=1,2,3$ are
denoted by $\vartheta_{\mu}(\pi v,q)$, and $\vartheta_{0}(v;\tau)$ by
$\vartheta_{4}(\pi v,q)$ in [43].) For $\Im\tau>0$, $\vartheta_{\mu}(v;\tau)$,
$\mu=0,1,2,3$ are holomorphic for $|v|<\infty$ and satisfy the partial
differential equation

$\displaystyle\frac{\partial\vartheta_{\mu}(v;\tau)}{\partial\tau}=\frac{1}{4\pi
i}\frac{\partial^{2}\vartheta_{\mu}(v;\tau)}{\partial v^{2}}.$ (A.2)

They have the quasi-periodicity; for instance, $\vartheta_{1}$ satisfies

$\displaystyle\vartheta_{1}(v+1;\tau)=-\vartheta_{1}(v;\tau),\qquad\vartheta_{1}(v+\tau;\tau)=-e^{-\pi
i(2v+\tau)}\vartheta_{1}(v;\tau).$

By the definition (A.1), when $\Im\tau>0$,

$\displaystyle\vartheta_{1}(0;\tau)=\vartheta_{1}(1;\tau)=0,\qquad\vartheta_{1}(x;\tau)>0,\qquad
x\in(0,1),$ $\displaystyle\vartheta_{0}(x;\tau)>0,\qquad x\in\mathbb{R}.$
(A.3)

We see the asymptotics

$\displaystyle\vartheta_{0}(v;\tau)\sim 1,\qquad\vartheta_{1}(v;\tau)\sim
2e^{\tau\pi i/4}\sin(\pi v),\qquad\vartheta_{2}(v;\tau)\sim 2e^{\tau\pi
i/4}\cos(\pi v),$ $\displaystyle\vartheta_{3}(v;\tau)\sim
1\qquad\text{in}\quad\Im\tau\to+\infty\qquad(\text{i.e.},\ q=e^{\tau\pi i}\to
0).$ (A.4)

The following functional equalities are known as Jacobi’s imaginary
transformations [32, 43]

$\displaystyle\vartheta_{0}(v;\tau)=e^{\pi i/4}\tau^{-1/2}e^{-\pi
iv^{2}/\tau}\vartheta_{2}\left(\frac{v}{\tau};-\frac{1}{\tau}\right),$
$\displaystyle\vartheta_{1}(v;\tau)=e^{3\pi i/4}\tau^{-1/2}e^{-\pi
iv^{2}/\tau}\vartheta_{1}\left(\frac{v}{\tau};-\frac{1}{\tau}\right),$
$\displaystyle\vartheta_{3}(v;\tau)=e^{\pi i/4}\tau^{-1/2}e^{-\pi
iv^{2}/\tau}\vartheta_{3}\left(\frac{v}{\tau};-\frac{1}{\tau}\right).$ (A.5)

###  A.2 Basic properties of $\boldsymbol{A_{{\cal N}}^{2\pi
r}(t_{\ast}-t,x)}$

For $0<t_{\ast}<\infty$, $0<r<\infty$, $0<{\cal N}<\infty$, the function
$A_{{\cal N}}^{2\pi r}(t_{\ast}-t,x)$ is defined by (1.1), which is written as

$\displaystyle A_{{\cal N}}^{2\pi r}(t_{\ast}-t,x)=\frac{1}{2\pi
r}\frac{\vartheta_{1}^{\prime}\big{(}x/2\pi r;i{\cal N}(t_{\ast}-t)/2\pi
r^{2}\big{)}}{\vartheta_{1}\big{(}x/2\pi r;i{\cal N}(t_{\ast}-t)/2\pi
r^{2}\big{)}},\qquad t\in[0,t_{\ast}),$ (A.6)

where $\vartheta_{1}^{\prime}(v;\tau)=\partial\vartheta_{1}(v;\tau)/\partial
v$. As a function of $x\in\mathbb{R}$, $A_{{\cal N}}^{2\pi r}(t_{\ast}-t,x)$
is odd,

$\displaystyle A_{{\cal N}}^{2\pi r}(t_{\ast}-t,-x)=-A_{{\cal N}}^{2\pi
r}(t_{\ast}-t,x),$ (A.7)

and periodic with period $2\pi r$

$\displaystyle A_{{\cal N}}^{2\pi r}(t_{\ast}-t,x+2m\pi r)=A_{{\cal N}}^{2\pi
r}(t_{\ast}-t,x),\qquad m\in\mathbb{Z}.$

It has only simple poles at $x=2m\pi r$, $m\in\mathbb{Z}$, and simple zeroes
at $x=(2m+1)\pi r$, $m\in\mathbb{Z}$. Independently of the values of
$t\in[0,t_{\ast})$ and $0<{\cal N}<\infty$, $A_{{\cal N}}^{2\pi
r}(t_{\ast}-t,x)$ behaves as

$\displaystyle A_{{\cal N}}^{2\pi
r}(t_{\ast}-t,x)\sim\begin{cases}\dfrac{1}{x},&\text{as $x\downarrow
0$},\vspace{1mm}\\\ -\dfrac{1}{2\pi r-x},&\text{as $x\uparrow 2\pi
r$}.\end{cases}$ (A.8)

Using Jacobi’s imaginary transformation (A.5), we can show by (A.4) that

$\displaystyle A^{2\pi r}_{{\cal
N}}(t_{\ast}-t,x)\sim\begin{cases}-\dfrac{x-\pi r}{{\cal
N}(t_{\ast}-t)},&\text{if $x>0$},\vspace{1mm}\\\ -\dfrac{x+\pi r}{{\cal
N}(t_{\ast}-t)},&\text{if $x<0$},\end{cases}\qquad\text{as $t\uparrow
t_{\ast}$}.$ (A.9)

###  A.3 Dedekind modular function

The Dedekind modular function $\eta(\tau)$ is defined by (1.10), that is,

$\displaystyle\eta(\tau)=e^{\tau\pi i/12}\prod_{n=1}^{\infty}(1-e^{2n\tau\pi
i}),\qquad\Im\tau>0.$

For $t\in[0,t_{\ast})$, $0<{\cal N}<\infty$, define

$\displaystyle\eta_{{\cal
N}}^{1}(t_{\ast}-t)=\frac{\pi^{2}}{\omega_{1}}\left.\left(\frac{1}{12}-2\sum_{n=1}^{\infty}\frac{nq^{2n}}{1-q^{2n}}\right)\right|_{\omega_{1}=\pi
r,\,q=e^{-{\cal N}(t_{\ast}-t)/2r^{2}}}.$ (A.10)

Then the following equality is established [16, 17]

$\displaystyle\frac{d\log\eta(\tau^{{R}}(t))}{dt}={\cal
N}^{{R}}\frac{\eta_{{\cal N}^{{R}}}^{1}(t_{\ast}-t)}{2\pi r}.$ (A.11)

###  A.4 Transition probability densities of Brownian motions in an interval

Using the definitions of the Jacobi theta functions (A.1) and Jacobi’s
imaginary transformations (A.5), we can obtain the following expressions for
the transition probability densities $p^{[0,\pi r]}_{\rm aa}(t,y|x)$ and
$p^{[0,\pi r]}_{\rm rr}(t,y|x)$ defined by (1.11) and (1.12)

$\displaystyle p^{[0,\pi r]}_{\rm aa}(t,y|x)$ $\displaystyle=p_{\rm
BM}(t,y|x)\vartheta_{3}\left(\frac{i(y-x)r}{t};\frac{2\pi
ir^{2}}{t}\right)-p_{\rm
BM}(t,y|-x)\vartheta_{3}\left(\frac{i(y+x)r}{t};\frac{2\pi ir^{2}}{t}\right)$
$\displaystyle=\frac{1}{2\pi r}\left\\{\vartheta_{3}\left(\frac{y-x}{2\pi
r};\frac{it}{2\pi r^{2}}\right)-\vartheta_{3}\left(\frac{y+x}{2\pi
r};\frac{it}{2\pi r^{2}}\right)\right\\}$ $\displaystyle=\frac{1}{\pi
r}\sum_{n\in\mathbb{Z}}e^{-n^{2}t/2r^{2}}\sin\left(\frac{ny}{r}\right)\sin\left(\frac{nx}{r}\right),$
(A.12)

and

$\displaystyle p^{[0,\pi r]}_{\rm rr}(t,y|x)$ $\displaystyle=p_{\rm
BM}(t,y|x)\vartheta_{3}\left(\frac{i(y-x)r}{t};\frac{2\pi
ir^{2}}{t}\right)+p_{\rm
BM}(t,y|-x)\vartheta_{3}\left(\frac{i(y+x)r}{t};\frac{2\pi ir^{2}}{t}\right)$
$\displaystyle=\frac{1}{2\pi r}\left\\{\vartheta_{3}\left(\frac{y-x}{2\pi
r};\frac{it}{2\pi r^{2}}\right)+\vartheta_{3}\left(\frac{y+x}{2\pi
r};\frac{it}{2\pi r^{2}}\right)\right\\}$ $\displaystyle=\frac{1}{\pi
r}\sum_{n\in\mathbb{Z}}e^{-n^{2}t/2r^{2}}\cos\left(\frac{ny}{r}\right)\cos\left(\frac{nx}{r}\right),$
(A.13)

for $x,y\in[0,\pi r]$, $t\geq 0$. It should be noted that $\int_{0}^{\pi
r}p^{[0,\pi r]}_{\rm rr}(t,y|u)dy\equiv 1$, $\forall\,t\geq 0$, $u\in[0,\pi
r]$, while

$\displaystyle p^{\rm survival}_{u}(t)\equiv\int_{0}^{\pi r}p^{[0,\pi r]}_{\rm
aa}(t,y|u)dy\simeq\frac{4}{\pi}e^{-t/2r^{2}}\sin\left(\frac{u}{r}\right)\to
0\qquad\text{as $t\to\infty$},\qquad u\in(0,\pi r).$

When both boundaries at 0 and $\pi r$ are absorbing, $p^{\rm survival}_{u}(t)$
gives the probability for a Brownian motion to survive up to time $t$ in the
interval $(0,\pi r)$, when it starts from $u\in(0,\pi r)$. In other words,
with probability $1-p^{\rm survival}_{u}(t)$, the Brownian motion was absorbed
before time $t$ and has been fixed at one of the boundaries.

###  A.5 Weierstrass $\boldsymbol{\wp}$ function and zeta function
$\boldsymbol{\zeta}$

The Weierstrass $\wp$ function and zeta function $\zeta$ are defined by

$\displaystyle\wp(z|2\omega_{1},2\omega_{3})=\frac{1}{z^{2}}+\sum_{(m,n)\in\mathbb{Z}^{2}\setminus\\{(0,0)\\}}\left[\frac{1}{(z-\Omega_{m,n})^{2}}-\frac{1}{{\Omega_{m,n}}^{2}}\right],$
$\displaystyle\zeta(z|2\omega_{1},2\omega_{3})=\frac{1}{z}+\sum_{(m,n)\in\mathbb{Z}^{2}\setminus\\{(0,0)\\}}\left[\frac{1}{z-\Omega_{m,n}}+\frac{1}{\Omega_{m,n}}+\frac{z}{{\Omega_{m,n}}^{2}}\right],$
(A.14)

where $\omega_{1}$ and $\omega_{3}$ are fundamental periods with
$\tau=\omega_{3}/\omega_{1}$ and $\Omega_{m,n}=2m\omega_{1}+2n\omega_{3}$. Put
(4.16), then the following functional equation holds (see [43, Section 20.41]
and [16, Lemma 2.1])

$\displaystyle(\zeta_{{\cal N}^{{R}}}(t_{\ast}-s,z+u)-\zeta_{{\cal
N}^{{R}}}(t_{\ast}-s,z)-\zeta_{{\cal N}^{{R}}}(t_{\ast}-s,u))^{2}$
$\displaystyle\qquad{}=\wp_{{\cal N}^{{R}}}(t_{\ast}-s,z+u)+\wp_{{\cal
N}^{{R}}}(t_{\ast}-s,z)+\wp_{{\cal N}^{{R}}}(t_{\ast}-s,u).$ (A.15)

### Acknowledgements

The author would like to thank the anonymous referees whose comments
considerably improved the presentation of the paper. A part of the present
work was done during the participation of the author in the ESI workshop on
“Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and
Physics” (March 20–24, 2017). The present author expresses his gratitude for
the hospitality of Erwin Schrödinger International Institute for Mathematics
and Physics (ESI) of the University of Vienna and for well-organization of the
workshop by Christian Krattenthaler, Masatoshi Noumi, Simon Ruijsenaars,
Michael J. Schlosser, Vyacheslav P. Spiridonov, and S. Ole Warnaar. He also
thanks Soichi Okada, Masatoshi Noumi, Simon Ruijsenaars, and Michael J.
Schlosser for useful discussion. This work was supported in part by the Grant-
in-Aid for Scientific Research (C) (No. 26400405), (B) (No. 26287019), and (S)
(No. 16H06338) of Japan Society for the Promotion of Science.

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