# Dyson’s model in infinite dimensions is irreducible

Hirofumi Osada; Ryosuke Tsuboi  Hirofumi Osada; Ryosuke Tsuboi

# Dyson’s model in infinite dimensions is irreducible

Hirofumi Osada; Ryosuke Tsuboi  Hirofumi Osada; Ryosuke Tsuboi

###### Abstract

Dyson’s model in infinite dimensions is a system of Brownian particles
interacting via a logarithmic potential with an inverse temperature of
$\beta=2$. The stochastic process is given as a solution to an infinite-
dimensional stochastic differential equation. Additionally, a Dirichlet form
with the sine2 point process as a reference measure constructs the stochastic
process as a functional of the associated configuration-valued diffusion
process. In this paper, we prove that Dyson’s model in infinite dimensions is
irreducible.

Keywords: Dyson’s model, random matrices, irreducibility, diffusion process,
interacting Brownian motion, infinite-dimensional stochastic differential
equations, logarithmic potential, Gaussian unitary ensembles

MSC2020: 60B20, 60H10, 60J40, 60J60, 60K35

To appear in
the Festschrift in honor of Professor Fukushima

##  1 Introduction

This paper considers an infinite-dimensional stochastic differential equation
(ISDE) of the form

$\displaystyle
X_{t}^{i}-X_{0}^{i}=B_{t}^{i}+\frac{\beta}{2}\int_{0}^{t}\lim_{r\to\infty}\sum_{|X_{u}^{i}-X_{u}^{j}|<r,\
j\not=i}^{\infty}\frac{1}{X_{u}^{i}-X_{u}^{j}}du\quad(i\in\mathbb{Z}).$ (1.1)

For $\beta=2$, the ISDE was introduced by Spohn [28], who called it Dyson’s
model. Spohn derived (1.1) for $\beta=2$ as an informal limit of Dyson’s
Brownian motion in finite dimensions. Here, Dyson’s Brownian motion is a
solution of a finite-dimensional stochastic differential equation (SDE) such
that

$\displaystyle
X_{t}^{N,i}-X_{0}^{N,i}=B_{t}^{i}+\frac{\beta}{2}\int_{0}^{t}\sum_{j\not=i}^{N}\frac{1}{X_{u}^{N,i}-X_{u}^{N,j}}du-\frac{\beta}{2N}\int_{0}^{t}\frac{1}{X_{u}^{N,i}}du.$
(1.2)

If $\beta=2$, then SDE (1.2) describes the dynamics of the eigenvalues of
Gaussian unitary ensembles of order $N\in\mathbb{N}$ [4, 17]. Spohn [28]
constructed the limit dynamics as the $L^{2}$-Markovian semi-group given by
the Dirichlet form

$\displaystyle\mathcal{E}(f,g)=\int_{\mathfrak{S}}\mathbb{D}[f,g]d\mu$ (1.3)

on $L^{2}(\mathfrak{S},\mu)$, where $\mathfrak{S}$ is the configuration space
over $\mathbb{R}$, $\mathbb{D}$ is the standard carré du champ on
$\mathfrak{S}$, and $\mu$ is the sine2 random point field. Furthermore, the
domain of the Dirichlet form is taken to be the closure of the polynomials on
$\mathfrak{S}$.

Let $\mu$ be the sineβ-random point field. If $\beta=2$, then $\mu$ becomes a
determinantal random point field whose $m$-point correlation function
$\rho^{m}$ with respect to the Lebesgue measure is given by

$\displaystyle\rho^{m}(\mathbf{x})=\det[\mathcal{K}_{\mathrm{sin},2}(x^{i},x^{j})]_{i,j=1}^{m}.$

Here, $\mathcal{K}$ is the sine kernel given by

$\displaystyle\mathcal{K}(x,y)=\frac{\sin\\{\theta\sqrt{2}(x-y)\\}}{\pi(x-y)}.$

Spohn [28] proved the closability of $\mathcal{E}$ on
$L^{2}(\mathfrak{S},\mu)$ with a predomain consisting of polynomials on
$\mathfrak{S}$ for $\beta=2$.

In [18], the first author proved that
$(\mathcal{E},\mathcal{D}_{\circ}^{\mu})$ is closable on
$L^{2}(\mathfrak{S},\mu)$, and that its closure is a quasi-regular Dirichlet
form. Here, $\mathcal{D}_{\circ}$ is the set consisting of local and smooth
functions on $\mathfrak{S}$ and $\mathcal{D}_{\circ}^{\mu}$ is given by

$\displaystyle\mathcal{D}_{\circ}^{\mu}=\\{f\in\mathcal{D}_{\circ}\,;\,\mathcal{E}_{1}(f,f)<\infty\\}.$

Thus, Osada constructed the $L^{2}$-Markovian semi-group as well as the
diffusion

$\displaystyle\mathfrak{X}(t)=\sum_{i\in\mathbb{Z}}\delta_{X^{i}(t)}$

associated with the Dirichlet form $(\mathcal{E},\mathcal{D})$ on
$L^{2}(\mathfrak{S},\mu)$. We call $\mathfrak{X}$ the unlabeled dynamics or
unlabeled diffusion because the state space of the process is $\mathfrak{S}$.
The unlabeled diffusion can be constructed for $\beta=1,4$ [22], and the
associated labeled process $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ satisfies
ISDE (1.1) for $\beta=1,2,4$ [21]. These cases have been proved as examples of
the general theory developed in various papers [20, 21, 22, 23]. In [21], the
meaning of a solution to an ISDE is a weak solution; the uniqueness of a weak
solution of an ISDE and the Dirichlet form is left open in [21, 22]. (See [9]
for the concept of strong and weak solutions of stochastic differential
equations).

Tsai [30] solved ISDE (1.1) for all $\beta\in[1,\infty)$. He proved the
existence of a strong solution and the path-wise uniqueness of this solution.
The method used by Tsai depends on an artistic coupling specific to Dyson’s
model. A non-equilibrium solution is obtained in the sense that the ISDE is
solved by starting at each point in an explicitly given subset
$\mathfrak{S}_{0}\subset\mathfrak{S}$ such that $\mu(\mathfrak{S}_{0})=1$.

The $\mu$-reversibility of the associated unlabeled diffusion is left open in
[30]. Combining [21] and [30], we find that the unlabeled process given by the
solution of (1.2) obtained in [30] is reversible with respect to $\mu$ for
$\beta=1,4$. For a general $\beta>0$, note that the reversible probability
measure of the unlabeled diffusion given by the solution to ISDE (1.1) is
expected to be a sineβ-random point field. This remains an open problem,
except for $\beta=1,2,4$ [22].

One of the authors and Tanemura [25] also proved the existence of a strong
solution and the path-wise uniqueness of this solution for $\beta=1,2,4$.
Their method can be applied to quite a wide range of examples. Using the
result in [25], Kawamoto et al. proved the uniqueness of Dirichlet forms [12].
They checked the infinite system of finite-dimensional SDEs with consistency
(IFC) condition in [13], which plays an important role in the theory developed
in [25]. Kawamoto and the second author of [13] derive a solution to the ISDE
from $N$-particle systems [10, 11].

The goal of this paper is to prove that the solution of (1.3) for $\beta=2$ is
irreducible (Theorem 1.1). In the remainder of this paper, we consider the
case $\beta=2$. Hence, we take $\mu$ to be the sine2 random point field.

By definition, the configuration $\mathfrak{S}$ over $\mathbb{R}$ is given by

$\displaystyle\mathfrak{S}=\Big{\\{}\mathfrak{s}=\sum_{i}\delta_{s^{i}}\,;\,\mathfrak{s}(K)<\infty\
\text{ for any compact }K\Big{\\}}.$

We endow $\mathfrak{S}$ with the vague topology. Under the vague topology,
$\mathfrak{S}$ is a Polish space. A probability measure on
$(\mathfrak{S},\mathscr{B}(\mathfrak{S}))$ is called a random point field
(also called a point process). Let

$\displaystyle\mathfrak{S}_{\mathrm{s,i}}=\Big{\\{}\mathfrak{s}\in\mathfrak{S}\,;\,\mathfrak{s}(\\{s\\})\leq
1\text{ for all }s\in\mathbb{R},\,\mathfrak{s}(\mathbb{R})=\infty\Big{\\}}.$

In [22, 19], we proved that the sine2 random point field $\mu$ satisfies

$\displaystyle\mathrm{Cap}((\mathfrak{S}_{\mathrm{s,i}})^{c})=0.$ (1.4)

Furthermore, $\mu$ is translation invariant and tail trivial [24, 16]. Hence,
by the individual ergodic theorem, we have that, for $\mu$-a.s.
$\mathfrak{s}$,

$\displaystyle\lim_{R\to\infty}\frac{\mathfrak{s}([-R,R])}{R}=\int_{\mathfrak{S}}\mathfrak{s}([-1,1])d\mu.$

Then, we set

$\displaystyle\mathfrak{S}_{n}=\Big{\\{}\mathfrak{s}\in\mathfrak{S}\,;\,\frac{1}{n}\leq\frac{\mathfrak{s}([-R,R])}{R}\leq
n\text{ for all }R\in\mathbb{N}\Big{\\}}.$ (1.5)

Using the argument in the proof of Theorem 1 in [18, p.127], we see that

$\displaystyle\mathrm{Cap}\Big{(}\big{(}\bigcup_{n=1}^{\infty}\mathfrak{S}_{n}\big{)}^{c}\Big{)}=0.$
(1.6)

We write $\mathbf{s}=({s^{i}})_{i\in\mathbb{Z}}\in\mathbb{R}^{\mathbb{Z}}$,
and we set

$\displaystyle\mathbb{R}_{<}^{\mathbb{Z}}=\\{\mathbf{s}=({s^{i}})_{i\in\mathbb{Z}}\in\mathbb{R}^{\mathbb{Z}}\,;\,{s^{i}}<{s^{i+1}}\
\text{ for all }\ i\,\\}.$

Let $\mathfrak{u}$ be a map on $\mathbb{R}_{<}^{\mathbb{Z}}$ such that
$\mathfrak{u}(\mathbf{s})=\sum_{i\in\mathbb{Z}}\delta_{s^{i}}$. Let
$\mathfrak{l}\\!:\\!\mathfrak{S}_{\mathrm{s,i}}\\!\to\\!\mathbb{R}_{<}^{\mathbb{Z}}$
be a function such that $\mathfrak{u}\circ\mathfrak{l}=\mathrm{id.}$. We call
$\mathfrak{u}$ an unlabeling map and $\mathfrak{l}$ a labeling map. There
exist many labeling maps. We can take $\mathfrak{l}$ in such a way that
$\mathfrak{l}^{0}(\mathfrak{s})=\min\\{{s^{i}};{s^{i}}\geq
0,\,\mathfrak{s}=\sum_{i\in\mathbb{Z}}\delta_{{s^{i}}}\\}$ and
$\mathfrak{l}^{i}(\mathfrak{s})<\mathfrak{l}^{i+1}(\mathfrak{s})$ for all
$i\in\mathbb{Z}$, where
$\mathfrak{l}(\mathfrak{s})=(\mathfrak{l}^{i}(\mathfrak{s}))_{i\in\mathbb{Z}}$.
This choice of $\mathfrak{l}$ is just for convenience and has no specific
meaning. Let

$\displaystyle\mathrm{W}=C([0,\infty);\mathbb{R}_{<}^{\mathbb{Z}}).$ (1.7)

Let
$\mathfrak{l}_{\mathrm{path}}=\\{\mathfrak{l}_{\mathrm{path}}(\mathfrak{w})_{t}\\}_{t\in[0,\infty)}$
be the label path map generated by $\mathfrak{l}$ (see [25, pp. 1148-1149] and
(2.6) in [13]). By definition, $\mathfrak{l}_{\mathrm{path}}$ is the map from
$C([0,\infty);\mathfrak{S}_{\mathrm{s,i}})$ to $\mathrm{W}$ such that
$\mathfrak{l}_{\mathrm{path}}(\mathfrak{w})_{0}=\mathfrak{l}(\mathfrak{w}_{0})$,
where $\mathfrak{w}=\\{\mathfrak{w}_{t}\\}_{t\in[0,\infty)}\in
C([0,\infty);\mathfrak{S}_{\mathrm{s,i}})$.

Let $\mathbf{X}=(X^{i})_{i\in\mathbb{Z}}$ be a solution to ISDE (1.1) with
$\beta=2$ defined on a filtered space
$(\Omega,\mathscr{F},P,\\{\mathscr{F}_{t}\\})$. We set

$\displaystyle\mu^{\infty}=\mu\circ{\mathfrak{l}}^{-1}$ (1.8)

and assume that

$\displaystyle\mu^{\infty}=P(\mathbf{X}_{0}\in\cdot).$ (1.9)

The associated unlabeled process

$\displaystyle\mathfrak{X}=\sum_{i\in\mathbb{Z}}\delta_{X^{i}}$

is a $\mu$-reversible diffusion given by the Dirichlet form
$(\mathcal{E},\mathcal{D})$ in (1.3) [25]. Note also that the labeled process
$\mathbf{X}=\mathfrak{l}_{\mathrm{path}}(\mathfrak{X})$ obtained by the
Dirichlet form in [21, 25] coincides with the solution obtained by Tsai [30].

From (1.4) and $\mathbf{X}=\mathfrak{l}_{\mathrm{path}}(\mathfrak{X})$ we find
that

$\displaystyle P(\mathbf{X}\in\mathrm{W})=1.$

We set $\mathbf{w}=(w^{i})_{i\in\mathbb{Z}}\in\mathrm{W}$ and

$\displaystyle P^{\infty}=P\circ\mathbf{X}^{-1},\quad
P^{\infty}_{\mathbf{x}}=P^{\infty}(\cdot|\mathbf{w}_{0}=\mathbf{x}).$ (1.10)

######  Theorem 1.1.

$\\{P_{\mathbf{x}}^{\infty}\\}$ is irreducible. That is, if $\mathbf{A}$ and
$\mathbf{B}\in\mathscr{B}(\mathbb{R}_{<}^{\mathbb{Z}})$ satisfy

$\displaystyle
P^{\infty}(\mathbf{w}_{0}\in\mathbf{A},\,\mathbf{w}_{t}\in\mathbf{B})=0,$
(1.11)

then $P^{\infty}(\mathbf{w}_{0}\in\mathbf{A})=0$ or
$P^{\infty}(\mathbf{w}_{t}\in\mathbf{B})=0$.

We do not know whether $\mathbf{X}$ has an invariant probability measure that
is absolutely continuous with respect to $\mu^{\infty}$. If this is the case,
then Theorem 1.1 implies that $\mathbf{X}$ is irreducible in the usual sense.

From Theorem 1.1, (1.8), and (1.10), we immediately have the following.

######  Corollary 1.2.

The solution of (1.1) with $\beta=2$ is irreducible in the sense that, if
$\mathbf{A}$ and $\mathbf{B}\in\mathscr{B}(\mathbb{R}_{<}^{\mathbb{Z}})$
satisfy

$\displaystyle
P(\mathbf{X}_{0}\in\mathbf{A},\,\mathbf{X}_{t}\in\mathbf{B})=0,$

then $P(\mathbf{X}_{0}\in\mathbf{A})=0$ or $P(\mathbf{X}_{t}\in\mathbf{B})=0$.

To prove Theorem 1.1, we prepare two results, Theorem 1.3 and Theorem 1.4.

For $\mathbf{x}\in\mathbb{R}^{\mathbb{Z}}$, we set
$\mathbf{x}^{m}=(x^{i})_{|i|<m}$ and $\mathbf{x}^{m*}=(x^{i})_{|i|\geq m}$. We
set

$\displaystyle\mathbb{R}_{<}^{m}=\\{\mathbf{x}^{m}=(x^{i})_{|i|<m};x^{i}<x^{i+1}\text{
for all }-m<i<m-1\\},$
$\displaystyle\mathbb{R}_{<}^{m*}=\\{\mathbf{x}^{m*}=(x^{i})_{|i|\geq
m};\,x^{i}<x^{i+1}\text{ for all }i<-m,\,m\leq i\\}.$

Let $\mathbf{X}^{m}=(X^{i})_{|i|<m}$ and $\mathbf{X}^{m*}=(X^{i})_{|i|\geq
m}$. Let $\mathbf{w}^{m*}=(w^{i})_{|i|\geq m}$ for
$\mathbf{w}=(w^{i})_{i\in\mathbb{Z}}$. We introduce the regular conditional
probabilities such that

$\displaystyle P_{\mathbf{w}}^{m}=$ $\displaystyle
P(\mathbf{X}^{m}\in\,\cdot\,|\mathbf{X}^{m*}=\mathbf{w}^{m*}),$ (1.12)
$\displaystyle P_{\mathbf{x},\mathbf{w}}^{m}=$ $\displaystyle
P(\mathbf{X}^{m}\in\cdot|\mathbf{X}_{0}^{m}=\mathbf{x}^{m},\mathbf{X}^{m*}=\mathbf{w}^{m*}).$

By construction, $\mathbf{X}^{m}$ under $\\{P_{\mathbf{x},\mathbf{w}}^{m}\\}$
is a time-inhomogeneous diffusion. The heat equations describing the
transition probability density are given by (3.20) and (3.21).

######  Theorem 1.3.

Let $P^{m*}=P\circ(\mathbf{X}^{m*})^{-1}$. For each $m\in\mathbb{N}$,
$\\{P_{\mathbf{x},\mathbf{w}}^{m}\\}$ is irreducible for $P^{m*}$-a.s.
$\mathbf{w}$. That is, if $\mathbf{A}$ and
$\mathbf{B}\in\mathscr{B}(\mathbb{R}_{<}^{m})$ satisfy

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A},\,\mathbf{w}^{m}_{t}\in\mathbf{B})=0\quad\text{for
$P^{m*}$-a.s.\,$\mathbf{w}$},$ (1.13)

then $P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A})=0$ for $P^{m*}$-a.s.
$\mathbf{w}$ or $P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{t}\in\mathbf{B})=0$ for
$P^{m*}$-a.s. $\mathbf{w}$.

######  Theorem 1.4.

Let $P^{m}=P\circ(\mathbf{X}^{m})^{-1}$. For each $m\in\mathbb{N}$, the
process $P^{m}$ is irreducible. That is, if $\mathbf{A}$ and
$\mathbf{B}\in\mathscr{B}(\mathbb{R}_{<}^{m})$ satisfy

$\displaystyle
P^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A},\,\mathbf{w}^{m}_{t}\in\mathbf{B})=0,$
(1.14)

then $P^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A})=0$ or
$P^{m}(\mathbf{w}^{m}_{t}\in\mathbf{B})=0$.

Let $\Phi\\!:\\!\mathbb{R}^{d}\\!\to\\!\mathbb{R}\cup\\{\infty\\}$ and
$\Psi\\!:\\!\mathbb{R}^{d}\times\mathbb{R}^{d}\\!\to\\!\mathbb{R}\cup\\{\infty\\}$
be measurable functions. A stochastic process given by a solution
$\mathbf{X}=(X^{i})_{i}$ of the ISDE

$\displaystyle
X_{t}^{i}-X_{0}^{i}=B_{t}^{i}+\frac{1}{2}\int_{0}^{t}\nabla\Phi(X_{u}^{i})du+\frac{1}{2}\int_{0}^{t}\sum_{j\neq\i}\nabla\Psi(X_{u}^{i},X_{u}^{j})du$

is called an interacting Brownian motion (in infinite dimensions) with
potential $(\Phi,\Psi)$. Here, $(\nabla\Psi)(x,y)=\nabla_{x}\Psi(x,y)$. The
study of interacting Brownian motions was initiated by Lang [14, 15], who
solved the ISDE for $(0,\Psi)$ with $\Psi\in C_{0}^{3}(\mathbb{R}^{d})$ such
that $\Psi$ is of Ruelle’s class in the sense that it is super stable and
regular. Fritz [6] constructed non-equilibrium solutions for the same
potentials as in [14, 15] with a further restriction that the dimension $d\leq
4$. Tanemura solved the ISDE for the hard-core potential [29], while
Fradon–Roelly–Tanemura solved the ISDE for the hard-core potential with long
range interactions, but still of Ruelle’s class [5]. Various ISDEs with
logarithmic interaction potentials have also been solved [8, 12, 21, 23, 25,
26, 30].

There are fewer results for the irreducibility and ergodicity of solutions of
interacting Brownian motions. Albeverio–Ma–Röckner [2] proved the equivalence
of the ergodicity of Dirichlet forms and the extremal property of the
associated (grand canonical or canonical) Gibbs measures with potentials of
Ruelle’s class [27]. Corwin-Sun [3] proved the ergodicity of the Airy line
ensembles, for which the dynamics are related to the Airy2 random point field.
A general result concerning the ergodicity of Dirichlet forms can be found in
[7].

The remainder of this paper is organized as follows. In Section 2, we recall
the concept of the $m$-labeled process and the Lyons–Zheng decomposition for
interacting Brownian motions. In Section 3, we prove Theorem 1.3 and Theorem
1.4. Finally, in Section 4, we prove Theorem 1.1.

##  2 The $m$-labeled process and the Lyons–Zheng decomposition

We introduce the $m$-labeled process
$\mathbf{X}^{[m]}=(\mathbf{X}^{m},\mathfrak{X}^{m*})$, where

$\displaystyle\mathfrak{X}^{m*}_{t}=\sum_{|j|\geq m}\delta_{X_{t}^{j}}.$

The process $\mathbf{X}^{[m]}$ is given by the Dirichlet form
$(\mathcal{E}^{[m]},\mathcal{D}^{[m]})$ on
$L^{2}(\mathbb{R}_{<}^{m}\times\mathfrak{S},\mu^{[m]})$ such that

$\displaystyle\mathcal{E}^{[m]}(f,g)=\int_{\mathbb{R}_{<}^{m}\times\mathfrak{S}}\mathbb{D}^{[m]}[f,g]\,d\mu^{[m]}.$

Here, $\mathbb{D}^{[m]}$ is the standard carré du champ on
$\mathbb{R}_{<}^{m}\times\mathfrak{S}$ [25], $\mathcal{D}^{[m]}$ is the
closure of

$\displaystyle\\{f_{1}\otimes f_{2}\,\in
C_{0}^{\infty}\otimes\mathcal{D}_{\circ};\,\mathcal{E}_{1}^{[m]}(f_{1}\otimes
f_{2},f_{1}\otimes f_{2})<\infty\\},$

and $\mu^{[m]}$ is the $m$-reduced Campbell measure such that

$\displaystyle\mu^{[m]}(A\times
B)=\int_{A}\rho^{m}(\mathbf{x}^{m})\mu_{\mathbf{x}^{m}}(B)d\mathbf{x}^{m}.$

Moreover, $\rho^{m}$ is the $m$-point correlation function of $\mu$, and
$\mu_{\mathbf{x}^{m}}$ is a reduced Palm measure conditioned at
$\mathbf{x}^{m}=(x^{i})_{|i|<m}$ given by

$\displaystyle\mu_{\mathbf{x}^{m}}=\mu(\,\cdot\,-\sum_{|i|<m}\delta_{x^{i}}|\mathfrak{l}^{i}(\mathfrak{s})=x^{i},\,\text{
for all }|i|<m).$ (2.1)

The standard definition of the reduced Palm measure $\mu_{\mathbf{x}^{m}}$ is

$\displaystyle\mu_{\mathbf{x}^{m}}=\mu(\,\cdot\,-\sum_{|i|<m}\delta_{x^{i}}|\mathfrak{s}(\\{x^{i}\\})\geq
1\text{ for all }|i|<m).$

Because the state space of process $\mathbf{X}$ is
$\mathbb{R}_{<}^{\mathbb{Z}}$, we take $\mu_{\mathbf{x}^{m}}$ given by (2.1).
We set

$\displaystyle
P_{\mathbf{x},\mathfrak{s}}^{[m]}=P(\mathbf{X}^{[m]}\in\cdot|\mathbf{X}_{0}^{[m]}=(\mathbf{x},\mathfrak{s})).$

From [20, 25], we have that $\\{P_{\mathbf{x},\mathfrak{s}}^{[m]}\\}$ is a
diffusion associated with the Dirichlet form
$(\mathcal{E}^{[m]},\mathcal{D}^{[m]})$ on
$L^{2}(\mathbb{R}_{<}^{m}\times\mathfrak{S},\mu^{[m]})$. By construction,
$\\{P_{\mathbf{x},\mathfrak{s}}^{[m]}\\}$ is $\mu^{[m]}$-symmetric and
$\mu^{[m]}$ is an invariant measure of
$\\{P_{\mathbf{x},\mathfrak{s}}^{[m]}\\}$. One of the most critical properties
of $\\{P_{\mathbf{x},\mathfrak{s}}^{[m]}\\}$ is its consistency. To explain
the consistency, we prepare some notations.

Let $\mathfrak{l}_{\mathrm{path}}$ be the path label introduced in Section 1.
We write
$\mathfrak{l}_{\mathrm{path}}=(\mathfrak{l}_{\mathrm{path}}^{i})_{i\in\mathbb{Z}}$.
We set $\mathfrak{l}_{\mathrm{path}}^{[m]}$ from
$\mathfrak{l}_{\mathrm{path}}$ as follows:

$\displaystyle\mathfrak{l}_{\mathrm{path}}^{[m]}(\mathfrak{w})_{t}=((\mathfrak{l}_{\mathrm{path}}^{i}(\mathfrak{w})_{t})_{|i|<m},\sum_{|j|\geq
m}\delta_{\mathfrak{l}_{\mathrm{path}}^{j}(\mathfrak{w})_{t}}).$

For an $\mathbb{R}^{[m]}$-valued path $\mathbf{w}^{[m]}$ such that
$\mathbf{w}_{t}^{[m]}=((w_{t}^{i})_{|i|<m},\sum_{|j|\geq
m}\delta_{w_{t}^{j}})$, we set

$\displaystyle\mathfrak{u}_{\mathrm{path}}^{[m]}(\mathbf{w}^{[m]})_{t}=\sum_{i\in\mathbb{Z}}\delta_{w_{t}^{i}}.$

Clearly,
$\mathfrak{u}_{\mathrm{path}}^{[m]}(\mathbf{w}^{[m]})_{t}=\mathfrak{u}(\mathbf{w}_{t})$,
where $\mathfrak{u}$ is the unlabeling map defined in Section 1. Additionally,
$\mathfrak{u}(\mathbf{x},\mathfrak{s})=\sum_{i}\delta_{x^{i}}+\mathfrak{s}$
for $\mathbf{x}=(x^{i})$.

We have the following consistency.

######  Lemma 2.1.

For each $m\in\\{0\\}\cup\mathbb{N}$,

$\displaystyle P_{\mathbf{x},\mathfrak{s}}^{[m]}$
$\displaystyle\circ(\mathfrak{u}_{\mathrm{path}}^{[m]})^{-1}=P_{\mathfrak{u}(\mathbf{x},\mathfrak{s})}^{[0]},$
$\displaystyle P_{\mathfrak{s}}^{[0]}$
$\displaystyle\circ(\mathfrak{l}_{\mathrm{path}}^{[m]})^{-1}=P_{\mathfrak{l}_{m}(\mathfrak{s})}^{[m]}.$

###### Proof.

Applying Theorem 2.4 in [20] to Dyson’s model, we obtain Lemma 2.1. ∎∎

Let $\mathbf{x}^{[m]}=(\mathbf{x}^{m},\mathfrak{x}^{m*})$ for
$\mathbf{x}=(x^{i})_{i\in\mathbb{Z}}\in\mathbb{R}_{<}^{\mathbb{Z}}$, where
$\mathfrak{x}^{m*}=\sum_{|i|\geq m}\delta_{x^{i}}$.

######  Lemma 2.2.

For each $m\in\\{0\\}\cup\mathbb{N}$

$\displaystyle
P^{\infty}_{\mathbf{x}}\circ(\mathfrak{u}_{\mathrm{path}}^{[m]})^{-1}=P_{\mathbf{x}^{[m]}}^{[m]}.$

###### Proof.

Lemma 2.2 follows from (1.10) and Lemma 2.1. ∎∎

Note that $w^{j}$ under $P^{[m]}$ is a solution to SDE (1.1) for $|j|<m$.
Thus, the martingale term of the Fukushima decomposition of $w^{j}$ describes
Brownian motion. Hence, applying the Lyons–Zheng decomposition to $w^{j}$
under $P^{[m]}$, we obtain

$\displaystyle
w_{t}^{j}-w_{u}^{j}=\frac{1}{2}\\{B_{t}^{j}-B_{u}^{j}+\hat{B}_{t}^{j}-\hat{B}_{u}^{j}\\}\quad\text{
for }0\leq t,u\leq T,$ (2.2)

where $\hat{B}_{t}^{j}=B_{T-t}^{j}$. Because $P^{[m]}$ is a symmetric
diffusion, $\hat{B}_{t}^{j}$ describes Brownian motion. Furthermore,
$\\{B_{t}^{j}\\}_{|j|<m}$ is a sequence of independent Brownian motions under
$P^{[m]}$. Because $\hat{B}_{t}^{j}$ is a time reversal of $B_{t}^{j}$,
$\\{\hat{B}_{t}^{j}\\}_{|j|<m}$ is a sequence of independent Brownian motions
under $P^{[m]}$. We refer to Section 9 in [13] for the proof of the
Lyons–Zheng decomposition of this form. Because of the consistency in Lemma
2.2, we have that (2.2) holds for all $j\in\mathbb{Z}$ under $P^{\infty}$.
Furthermore, $\\{B_{t}^{j}\\}_{j\in\mathbb{Z}}$ and
$\\{\hat{B}_{t}^{j}\\}_{j\in\mathbb{Z}}$ are sequences of independent Brownian
motions under $P^{\infty}$. Thus, $\\{B_{t}^{j}-B_{u}^{j}\\}_{j\in\mathbb{Z}}$
and $\\{\hat{B}_{t}^{j}-\hat{B}_{u}^{j}\\}_{j\in\mathbb{Z}}$ are sequences of
increments of independent Brownian motions. Collecting these statements
together, we obtain the following.

######  Lemma 2.3.

1 For each $j\in\mathbb{Z}$, we have (2.2) for $P^{\infty}$-a.s.
2 $\\{B_{t}^{j}-B_{u}^{j}\\}_{j\in\mathbb{Z}}$ and
$\\{\hat{B}_{t}^{j}-\hat{B}_{u}^{j}\\}_{j\in\mathbb{Z}}$ are sequences of
increments of independent Brownian motions under $P^{\infty}$.

##  3 Proof of Theorem 1.3 and Theorem 1.4

Let $\mathfrak{d}^{\mu}$ be the logarithmic derivative of $\mu$. By
definition, $\mathfrak{d}^{\mu}$ is a function defined on
$\mathbb{R}\times\mathfrak{S}$ such that $\mathfrak{d}^{\mu}\in
L_{\mathrm{loc}}^{1}(\mu^{[1]})$ and

$\displaystyle\int_{\mathbb{R}\times\mathfrak{S}}\mathfrak{d}^{\mu}(s,\mathfrak{s})\varphi(s,\mathfrak{s})d\mu^{[1]}=-\int_{\mathbb{R}\times\mathfrak{S}}\nabla\varphi(s,\mathfrak{s})d\mu^{[1]}$

for all $\varphi\in C_{0}^{\infty}(\mathbb{R})\otimes\mathcal{D}_{\circ}^{b}$,
where $\mathcal{D}_{\circ}^{b}$ is the set consisting of bounded, local, and
smooth functions on $\mathfrak{S}$ [25]. We write
$\mathfrak{s}=\sum_{i}\delta_{s^{i}}$. In [21], it is proved that $\mu$ has a
logarithmic derivative such that

$\displaystyle\mathfrak{d}^{\mu}(s,\mathfrak{s})$
$\displaystyle=2\lim_{R\to\infty}\sum_{{s^{i}}\in
S_{R}}\frac{1}{s-{s^{i}}}\quad\text{ in }L_{\mathrm{loc}}^{2}(\mu^{[1]})$
(3.1)
$\displaystyle=2\lim_{R\to\infty}\sum_{|s-{s^{i}}|<R}\frac{1}{s-{s^{i}}}\quad\text{
in }L_{\mathrm{loc}}^{2}(\mu^{[1]}).$

The sums in (3.1) converge because $\mu$ is translation invariant, $d=1$, and
the variance of $\mathfrak{s}([-R,R])$ under $\mu$ increases logarithmically
as $R\to\infty$. The second equality in (3.1) comes from $d=1$.

Note that the Ginibre random point field $\mu_{\mathrm{gin}}$ satisfies the
following [21]:

$\displaystyle\mathfrak{d}^{\mu_{\mathrm{gin}}}(s,\mathfrak{s})$
$\displaystyle=-2s+2\lim_{R\to\infty}\sum_{{s^{i}}\in
S_{R}}\frac{s-{s^{i}}}{|s-{s^{i}}|^{2}}\quad\text{ in
}L_{\mathrm{loc}}^{2}(\mu_{\mathrm{gin}}^{[1]})$ (3.2)
$\displaystyle=2\lim_{R\to\infty}\sum_{|s-{s^{i}}|<R}\frac{s-{s^{i}}}{|s-{s^{i}}|^{2}}\quad\text{
in }L_{\mathrm{loc}}^{2}(\mu_{\mathrm{gin}}^{[1]}).$

The Ginibre random point field $\mu_{\mathrm{gin}}$ is the counterpart of
$\mu$ in $\mathbb{R}^{2}$, because $\mu_{\mathrm{gin}}$ is rotation- and
translation-invariant, and the interaction potential of $\mu_{\mathrm{gin}}$
is the logarithmic potential with an inverse temperature of $\beta=2$. Compare
(3.1) and (3.2). The first equalities in (3.1) and (3.2) have different
expressions according to the dimension $d$.

Recall that $\beta=2$. Then, the ISDE in question is given by

$\displaystyle
X_{t}^{i}-X_{0}^{i}=B_{t}^{i}+\int_{0}^{t}\lim_{r\to\infty}\sum_{|X_{u}^{i}-X_{u}^{j}|<r,\,j\not=i}\frac{1}{X_{u}^{i}-X_{u}^{j}}du\quad(i\in\mathbb{Z}).$
(3.3)

Using (3.1) and (3.3), we have

$\displaystyle
X_{t}^{i}-X_{0}^{i}=B_{t}^{i}+\frac{1}{2}\int_{0}^{t}\mathfrak{d}^{\mu}(X_{u}^{i},\sum_{j\neq
i}\delta_{X_{u}^{j}})du\quad(i\in\mathbb{Z}).$ (3.4)

From (3.1), (3.3), and (3.4), it is easy to see that, $i\in\mathbb{Z}$,

$\displaystyle X_{t}^{i}-X_{0}^{i}=B_{t}^{i}$
$\displaystyle+\int_{0}^{t}\sum_{|j|<m,\,j\neq
i}^{m}\frac{1}{X_{u}^{i}-X_{u}^{j}}du+\int_{0}^{t}\lim_{r\to\infty}\sum_{|X_{u}^{i}-X_{u}^{j}|<r\atop
m\leq|j|,\,j\not=i}\frac{1}{X_{u}^{i}-X_{u}^{j}}du$ $\displaystyle=B_{t}^{i}$
$\displaystyle+\int_{0}^{t}\sum_{|j|<m,\,j\neq
i}^{m}\frac{1}{X_{u}^{i}-X_{u}^{j}}du+\int_{0}^{t}\lim_{n\to\infty}\sum_{m\leq|j|\leq
n,\,j\not=i}\frac{1}{X_{u}^{i}-X_{u}^{j}}du.$

Taking this equation into account, we set
$b_{\mathbf{w}}^{m}=(b_{\mathbf{w}}^{m,i})_{|i|<m}$ such that

$\displaystyle b_{\mathbf{w}}^{m,i}(\mathbf{x}^{m},t)=\sum_{j\neq
i\atop|j|<m}\frac{1}{x^{i}-x^{j}}+\lim_{n\to\infty}\sum_{m\leq|j|\leq
n}\frac{1}{x^{i}-w_{t}^{j}}.$ (3.5)

Let $\mathbf{x}^{m}=(x^{i})_{|i|<m}$, $\mathbf{x}=(x^{i})_{i\in\mathbb{Z}}$,
and $\mathbf{y}=(y^{i})_{i\in\mathbb{Z}}$. For
$\mathbf{y}\in\mathbb{R}_{<}^{\mathbb{Z}}$, we set

$\displaystyle\mathbb{R}_{<}^{m}(\mathbf{y})=\\{\mathbf{x}^{m}\in\mathbb{R}_{<}^{m}\,;\,y^{-m}<x^{-m+1},\
x^{m-1}<y^{m}\\}.$

Let $\mathscr{O}_{T,\mathbf{w}}^{m}$ be a time-dependent open set in
$\mathbb{R}_{<}^{m}$ such that

$\displaystyle\mathscr{O}_{T,\mathbf{w}}^{m}=\\{(\mathbf{x}^{m},t)\in\mathbb{R}_{<}^{m}\times[0,T]\,;\,\mathbf{x}^{m}\in\mathbb{R}_{<}^{m}(\mathbf{w}_{t})\\}.$

For $(\mathbf{x}^{m},t)$, we set $\mathbf{x}^{m}_{t}=(x_{t}^{i})_{|i|<m}$ such
that $(\mathbf{x}^{m}_{t},t)=(\mathbf{x}^{m},t)$. For $\epsilon\geq 0$,
$m,T\in\mathbb{N}$, and $\mathbf{w}\in\mathrm{W}$, we set

$\displaystyle\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}=\\{(\mathbf{x}^{m},t)\in\mathscr{O}_{T,\mathbf{w}}^{m}\,;\,$
$\displaystyle|x_{t}^{i}-x_{t}^{i+1}|>\epsilon,\,-m<i<m-1$ (3.6)
$\displaystyle|x_{t}^{-m+1}-w_{t}^{-m}|>\epsilon,\
|x_{t}^{m-1}-w_{t}^{m}|>\epsilon\\}.$

Suppose that $\epsilon>0$ and that $\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}$
is nonempty and connected. For $P^{\infty}$-a.s. $\mathbf{w}$, we find a
connected open set $\mathscr{Q}_{T,\mathbf{w}}^{m,\epsilon}$ in
$\mathbb{R}_{<}^{m}\times[0,T]$ with smooth boundary such that

$\displaystyle\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}\subset\mathscr{Q}_{T,\mathbf{w}}^{m,\epsilon}\subset\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon/2}.$
(3.7)

######  Lemma 3.1.

For each $T,m\in\mathbb{N}$, and $P^{\infty}$-a.s. $\mathbf{w}$, the following
hold.
1 $b_{\mathbf{w}}^{m}(\mathbf{x}^{m},t)$ is Hölder continuous in $t$ in
$\mathscr{Q}_{T,\mathbf{w}}^{m,\epsilon}$ for each $\mathbf{x}^{m}$.
2 $b_{\mathbf{w}}^{m}(\mathbf{x}^{m},t)$ is Lipschitz continuous in
$\mathbf{x}^{m}$ in $\mathscr{Q}_{T,\mathbf{w}}^{m,\epsilon}$.

###### Proof.

Let
$(\mathbf{x}^{m},t),(\mathbf{x}^{m},u)\in\mathscr{Q}_{T,\mathbf{w}}^{m,\epsilon}$
and fix $i$ such that $|i|<m$. Then from (3.5)

$\displaystyle
b_{\mathbf{w}}^{m,i}(\mathbf{x}^{m},t)-b_{\mathbf{w}}^{m,i}(\mathbf{x}^{m},u)$
$\displaystyle=\sum_{|j|\geq m}\frac{1}{x^{i}-w_{t}^{j}}-\sum_{|j|\geq
m}\frac{1}{x^{i}-w_{u}^{j}}$ (3.8) $\displaystyle=\sum_{|j|\geq
m}\frac{w_{t}^{j}-w_{u}^{j}}{(x^{i}-w_{t}^{j})(x^{i}-w_{u}^{j})}.$

From Lemma 2.3, we can deduce for $P^{\infty}$-a.s. that

$\displaystyle
w_{t}^{j}-w_{u}^{j}=\frac{1}{2}\\{B_{t}^{j}-B_{u}^{j}+\hat{B}_{t}^{j}-\hat{B}_{u}^{j}\\},$
(3.9)

where $\\{B_{t}^{j}-B_{u}^{j}\\}_{j\in\mathbb{Z}}$ and
$\\{\hat{B}_{t}^{j}-\hat{B}_{u}^{j}\\}_{j\in\mathbb{Z}}$ are sequences of
increments of independent Brownian motions under $P^{\infty}$. From (3.8) and
(3.9), we have that

$\displaystyle b_{\mathbf{w}}^{m,i}(\mathbf{x}^{m},t)-$ $\displaystyle
b_{\mathbf{w}}^{m,i}(\mathbf{x}^{m},u)=\frac{1}{2}\sum_{|j|\geq
m}\frac{B_{t}^{j}-B_{u}^{j}+\hat{B}_{t}^{j}-\hat{B}_{u}^{j}}{(x^{i}-w_{t}^{j})(x^{i}-w_{u}^{j})}$
(3.10) $\displaystyle=\frac{1}{2}\sum_{|j|\geq
m}\frac{B_{t}^{j}-B_{u}^{j}}{(x^{i}-w_{t}^{j})(x^{i}-w_{u}^{j})}+\frac{1}{2}\sum_{|j|\geq
m}\frac{\hat{B}_{t}^{j}-\hat{B}_{u}^{j}}{(x^{i}-w_{t}^{j})(x^{i}-w_{u}^{j})}.$

Let $\mathrm{W}$ be as in (1.7). To control the denominator in (3.10), we set

$\displaystyle
A_{n}=\Big{\\{}\mathbf{w}\in\mathrm{W}\,;\,\big{\\{}\min_{t\in[a,b]}|x^{i}-w_{t}^{j}|\big{\\}}\geq\frac{|j|}{n}\text{
for all }|j|\geq m\Big{\\}}.$ (3.11)

Using (3.11), we deduce that, for $P^{\infty}$-a.s. $\mathbf{w}\in A_{n}$,

$\displaystyle\sup_{t\in[a,b]}\Big{\\{}\sum_{|j|\geq
m}\frac{|j|}{|x^{i}-w_{t}^{j}|^{3}}\Big{\\}}$
$\displaystyle\leq\Big{\\{}\sum_{|j|\geq
m}\frac{|j|}{\min_{t\in[a,b]}|x^{i}-w_{t}^{j}|^{3}}\Big{\\}}$ (3.12)
$\displaystyle\leq\Big{\\{}\sum_{|j|\geq
m}\frac{|j|}{(\frac{|j|}{n})^{3}}\Big{\\}}\quad\text{ by }\eqref{:31d}$
$\displaystyle=n^{3}\Big{\\{}\sum_{|j|\geq
m}\frac{1}{|j|^{2}}\Big{\\}}<\infty.$

We set
$Q(\mathbf{x}^{m})=\\{\mathbf{w}\in\mathrm{W}\,;\,\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon/2}\cap(\\{\mathbf{x}^{m}\\}\times[0,T])\neq\emptyset\\}$.
Then using (1.6), (3.6), and (3.7), we deduce

$\displaystyle
P^{\infty}(\\{\bigcup_{n\in\mathbb{N}}A_{n}\\}^{c}\,;\,Q(\mathbf{x}^{m}))=0.$
(3.13)

Hence, from (3.12) and (3.13), we obtain, for $P^{\infty}$-a.s. $\mathbf{w}\in
Q(\mathbf{x}^{m})$,

$\displaystyle
c_{\ref{;31h}}(\mathbf{w}):=\sup_{t\in[a,b]}\Big{\\{}\sum_{|j|\geq
m}\frac{|j|}{|x^{i}-w_{t}^{j}|^{3}}\Big{\\}}<\infty.$ (3.14)

Using Young’s inequality and (3.14), we have

$\displaystyle\sum_{|j|\geq
m}\Big{|}\frac{B_{t}^{j}-B_{u}^{j}}{(x^{i}-w_{t}^{j})(x^{i}-w_{u}^{j})}\Big{|}$
(3.15) $\displaystyle\leq$ $\displaystyle\Big{(}\sup_{t\in[a,b]}\sum_{|j|\geq
m}\frac{|j|}{|x^{i}-w_{t}^{j}|^{3}}\Big{)}^{1/3}\Big{(}\sup_{u\in[a,b]}\sum_{|j|\geq
m}\frac{|j|}{|x^{i}-w_{u}^{j}|^{3}}\Big{)}^{1/3}\Big{(}\sum_{|j|\geq
m}\frac{|B_{t}^{j}-B_{u}^{j}|^{3}}{|j|^{2}}\Big{)}^{1/3}$ $\displaystyle=$
$\displaystyle c_{\ref{;31h}}(\mathbf{w})^{2/3}\Big{(}\sum_{|j|\geq
m}\frac{|B_{t}^{j}-B_{u}^{j}|^{3}}{|j|^{2}}\Big{)}^{1/3}.$

Similarly, for $P^{\infty}$-a.s. $\mathbf{w}\in Q(\mathbf{x}^{m})$, we have
that

$\displaystyle\sum_{|j|\geq
m}\Big{|}\frac{\hat{B}_{t}^{j}-\hat{B}_{u}^{j}}{(x^{i}-w_{t}^{j})(x^{i}-w_{u}^{j})}\Big{|}\leq
c_{\ref{;31h}}(\mathbf{w})^{2/3}\Big{(}\sum_{|j|\geq
m}\frac{|\hat{B}_{t}^{j}-\hat{B}_{u}^{j}|^{3}}{|j|^{2}}\Big{)}^{1/3}.$ (3.16)

Recall that $c_{1}(\mathbf{w})<\infty$ for $P^{\infty}$-a.s. $\mathbf{w}\in
Q(\mathbf{x}^{m})$ from (3.14). Note that $\\{B_{t}^{j}\\}_{j\in\mathbb{Z}}$
and $\\{\hat{B}_{t}^{j}\\}_{j\in\mathbb{Z}}$ are sequences of independent
Brownian motions. Then, we obtain Lemma 3.1 1 from (3.10), (3.15), and (3.16).

Let $(\mathbf{x}^{m},t)$ and
$(\mathbf{y}^{m},t)\in\mathscr{Q}_{T,\mathbf{w}}^{m,\epsilon}$. From (3.5), we
have that

$\displaystyle
b_{\mathbf{w}}^{m,i}(\mathbf{x}^{m},t)-b_{\mathbf{w}}^{m,i}(\mathbf{y}^{m},t)$
(3.17) $\displaystyle=$ $\displaystyle\sum_{j\neq
i\atop|j|<m}\frac{1}{x^{i}-x^{j}}-\sum_{j\neq
i\atop|j|<m}\frac{1}{y^{i}-y^{j}}+\sum_{|j|\geq
m}\frac{1}{x^{i}-w_{t}^{j}}-\sum_{|j|\geq m}\frac{1}{y^{i}-w_{t}^{j}}$
$\displaystyle=$ $\displaystyle\sum_{j\neq
i\atop|j|<m}\frac{1}{x^{i}-x^{j}}-\sum_{j\neq
i\atop|j|<m}\frac{1}{y^{i}-y^{j}}+\sum_{|j|\geq
m}\frac{y^{i}-x^{i}}{(x^{i}-w_{t}^{j})(y^{i}-w_{t}^{j})}.$

Then, using (1.6) and (3.17), we obtain 2. ∎∎

We define the probability measure on $\mathbb{R}_{<}^{m}$ by

$\displaystyle
P_{\mathbf{w}}^{m}(t)=P^{\infty}(\mathbf{w}^{m}_{t}\,\in\cdot\,|\mathbf{w}^{m*}_{0}).$
(3.18)

Let $\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}$ be as in (3.6). Let
$\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}(t)$ be the cross section of
$\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}$ such that

$\displaystyle\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}(t)=\\{\mathbf{x}^{m}\in\mathbb{R}_{<}^{m}\,;\,(\mathbf{x}^{m},t)\in\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}\\}.$
(3.19)

Proof of Theorem 1.3.  We consider the time-inhomogeneous heat equation on
$\mathscr{Q}_{T,\mathbf{w}}^{m,\epsilon}$ such that the associated backward
equation is given by

$\displaystyle\Big{\\{}\frac{\partial}{\partial
t}+\frac{1}{2}\sum_{|i|<m}(\frac{\partial}{\partial x^{i}})^{2}$
$\displaystyle+\sum_{|i|,|j|<m,i\neq
j}\frac{1}{x^{i}-x^{j}}\frac{\partial}{\partial x^{i}}$ (3.20)
$\displaystyle+\sum_{|i|<m\leq|j|}\frac{1}{x^{i}-w_{t}^{j}}\frac{\partial}{\partial
x^{i}}\Big{\\}}p(t,x,u,y)=0$

and the forward equation is given by

$\displaystyle\Big{\\{}\frac{\partial}{\partial
u}-\frac{1}{2}\sum_{|i|<m}(\frac{\partial}{\partial x^{i}})^{2}$
$\displaystyle-\sum_{|i|,|j|<m,i\neq
j}\frac{1}{x^{i}-x^{j}}\frac{\partial}{\partial x^{i}}$ (3.21)
$\displaystyle-\sum_{|i|<m\leq|j|}\frac{1}{x^{i}-w_{u}^{j}}\frac{\partial}{\partial
x^{i}}\Big{\\}}p(t,x,u,y)=0.$

From Lemma 3.1, we have constants $c_{2}$ and $\alpha$ such that $0<\alpha<1$
and

$\displaystyle|b_{\mathbf{w}}^{m,i}(\mathbf{x}^{m},t)-b_{\mathbf{w}}^{m,i}(\mathbf{y}^{m},u)|\leq
c_{\ref{;38}}\\{|\mathbf{x}^{m}-\mathbf{y}^{m}|+|t-u|^{\alpha}\\}.$ (3.22)

From (3.22), we can apply a general theorem of heat equations to determine
that the fundamental solution (the transition probability density) of (3.20)
and (3.21) on $\mathscr{O}_{T,\mathbf{w}}^{m,\epsilon}$ under a Dirichlet
boundary condition on the boundary is positive and continuous. Taking
$\epsilon\to 0$ and using the obvious inequality such that the heat kernel
dominates that with the Dirichlet boundary condition, we find that the heat
kernel $p(t,x,u,y)=p_{T,\mathbf{w}}^{m,0}(t,x,u,y)$ on
$\mathscr{O}_{T,\mathbf{w}}^{m,0}(t)\times\mathscr{O}_{T,\mathbf{w}}^{m,0}(u)$
is a positive density of the transition probability with respect to the
Lebesgue measure. Using (1.13), we find

$\displaystyle\int_{\mathbf{A}\times\mathbf{B}}p(0,x,t,y)dxdy=0.$ (3.23)

From (3.23) and positivity of $p$, we deduce that $\mathbf{A}$ or $\mathbf{B}$
have Lebesgue measure zero. Hence, either of the following hold:

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A})=\int_{\mathbf{A}\times\mathscr{O}_{T,\mathbf{w}}^{m,0}(t)}p(0,x,t,y)dxdy=0$
(3.24)

or

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{t}\in\mathbf{B})=\int_{\mathscr{O}_{T,\mathbf{w}}^{m,0}(0)\times\mathbf{B}}p(0,x,t,y)dxdy=0.$
(3.25)

We thus obtain Theorem 1.3. ∎

Proof of Theorem 1.4.  Using (1.14) and Fubini’s theorem, we deduce (1.13).
Then, applying Theorem 1.3, we have that

$P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A})=0$ for $P^{m*}$-a.s.
$\mathbf{w}$

or

$\displaystyle\text{ $P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{t}\in\mathbf{B})=0$
for $P^{m*}$-a.s.\,$\mathbf{w}$}.$

Integrating these with respect to $P^{m*}$, we conclude that Theorem 1.4 holds
from (1.12). ∎

##  4 Proof of Theorem 1.1

Let $\mathrm{W}^{m*}=C([0,\infty);\mathbb{R}_{<}^{m*})$. Let
$\varpi^{m*}\\!:\\!\mathrm{W}\\!\to\\!\mathrm{W}^{m*}$ be the projection such
that
$\mathbf{w}=(w^{i})_{i\in\mathbb{Z}}\mapsto\mathbf{w}^{m*}=(w^{i})_{|i|\geq
m}$. Let

$\displaystyle\mathbf{T}=\\{\mathbf{t}=(t_{1},\ldots,t_{l})\,;\,0<t_{k}<t_{k+1}\,(1\leq
k<l),\,\ l\in\mathbb{N}\\}.$

We set
$\varpi_{\mathbf{t}}^{m*}(\mathbf{w})=\mathbf{w}^{m*}_{\mathbf{t}}=(w_{\mathbf{t}}^{i})_{|i|\geq
m}$, where $w_{\mathbf{t}}^{i}=(w_{t_{1}}^{i},\ldots,w_{t_{l}}^{i})$, and

$\displaystyle\mathscr{C}_{\mathrm{path}}^{\infty*}=\bigvee_{\mathbf{t}\in\mathbf{T}}\bigcap_{m=1}^{\infty}\sigma[\varpi_{\mathbf{t}}^{m*}].$

We know that $\mu$ is tail trivial [24, 16]. That is, $\mu(A)\in\\{0,1\\}$ for
each $A\in\mathcal{T}(\mathfrak{S})$, where

$\displaystyle\mathcal{T}(\mathfrak{S})=\bigcap_{R=1}^{\infty}\sigma[\pi_{R}^{c}].$

The tail triviality of $\mu$ can be refined to the triviality of
$\mathscr{C}_{\mathrm{path}}^{\infty*}$ with respect to $P^{\infty}$ using
Lemma 4.1. The triviality of $\mathscr{C}_{\mathrm{path}}^{\infty*}$ with
respect to $P^{\infty}$ is one of the critical properties in the proof of the
uniqueness of solutions to ISDEs in [25]. We require a rather difficult
argument for the proof of this fact.

######  Lemma 4.1.

$\mathscr{C}_{\mathrm{path}}^{\infty*}$ is trivial with respect to
$P^{\infty}$. That is,

$\displaystyle\text{$P^{\infty}(\mathscr{A})\in\\{0,1\\}$ for each
$\mathscr{A}\in\mathscr{C}_{\mathrm{path}}^{\infty*}$}.$

###### Proof.

Lemma 4.1 follows directly from Theorem 5.3 in [25]. ∎∎

Proof of Theorem 1.1.  Recall that
$\varpi_{t}^{m*}(\mathbf{w})=(w_{t}^{i})_{m\leq|i|}$ for
$\mathbf{w}=(w^{i})_{i\in\mathbb{Z}}$. We set

$\displaystyle\mathscr{F}_{0,t}^{m*}=\sigma[\varpi_{0}^{m*},\varpi_{t}^{m*}].$

Let $\varpi^{m}\\!:\\!\mathrm{W}\\!\to\\!\mathrm{W}^{m}$ be the projection
such that
$\mathbf{w}=(w^{i})_{i\in\mathbb{Z}}\mapsto\mathbf{w}^{m}=(w^{i})_{|i|<m}$,
where $\mathrm{W}^{m}=C([0,\infty);\mathbb{R}_{<}^{m})$. We set
$\varpi_{u}(\mathbf{w})=\mathbf{w}_{u}$,
$\varpi_{u}^{m}(\mathbf{w})=\mathbf{w}_{u}^{m}$, and
$\varpi_{u}^{m*}(\mathbf{w})=\mathbf{w}_{u}^{m*}$. For $\mathscr{A}$ and
$\mathscr{B}\subset\mathrm{W}$, we set

$\displaystyle\mathbf{A}_{0}=\varpi_{0}(\mathscr{A}),$
$\displaystyle\mathbf{A}_{0}^{m}=\varpi_{0}^{m}(\mathscr{A}),$
$\displaystyle\mathbf{A}_{0}^{m*}=\varpi_{0}^{m*}(\mathscr{A}),$
$\displaystyle\mathbf{B}_{t}=\varpi_{t}(\mathscr{B}),$
$\displaystyle\mathbf{B}_{t}^{m}=\varpi_{t}^{m}(\mathscr{B}),$
$\displaystyle\mathbf{B}_{t}^{m*}=\varpi_{t}^{m*}(\mathscr{B}).$

Let $\mathbf{A}$ and $\mathbf{B}$ be as in the statement of Theorem 1.1. We
take $\mathscr{A}=\varpi_{0}^{-1}(\mathbf{A})$ and
$\mathscr{B}=\varpi_{t}^{-1}(\mathbf{B})$. Then we find
$\mathbf{A}=\mathbf{A}_{0}$ and $\mathbf{B}=\mathbf{B}_{t}$. Noting
$\mathbf{A}_{0}^{m*},\mathbf{B}_{t}^{m*}\in\mathscr{F}_{0,t}^{m*}$ and using
(1.12), we deduce that, for $P^{m*}$-a.s. $\mathbf{w}$,

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m},\mathbf{w}^{m}_{t}\in\mathbf{B}_{t}^{m}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})$
(4.1) $\displaystyle=$ $\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m},\mathbf{w}^{m}_{t}\in\mathbf{B}_{t}^{m},\mathbf{w}^{m*}_{0}\in\mathbf{A}_{0}^{m*},\mathbf{w}^{m*}_{t}\in\mathbf{B}_{t}^{m*}|\mathscr{F}_{0,t}^{m*})$
$\displaystyle=$ $\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}_{0}\in\mathbf{A}_{0},\mathbf{w}_{t}\in\mathbf{B}_{t}|\mathscr{F}_{0,t}^{m*})$
$\displaystyle=$ $\displaystyle
P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0},\mathbf{w}_{t}\in\mathbf{B}_{t}|\mathscr{F}_{0,t}^{m*})\quad\text{
by \eqref{:12q}}.$

From (4.1) and (1.11), we have

$\displaystyle\int_{\mathrm{W}}P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m},\mathbf{w}^{m}_{t}\in\mathbf{B}_{t}^{m}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})P^{m*}(d\mathbf{w})$
$\displaystyle=$
$\displaystyle\int_{\mathrm{W}}P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0},\mathbf{w}_{t}\in\mathbf{B}_{t}|\mathscr{F}_{0,t}^{m*})P^{m*}(d\mathbf{w})\quad\text{
by \eqref{:52p}}$ $\displaystyle=$ $\displaystyle
P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0},\mathbf{w}_{t}\in\mathbf{B}_{t})=0\quad\text{
by \eqref{:11b}}.$

Using this, we obtain, for $P^{m*}$-a.s. $\mathbf{w}$,

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m},\,\mathbf{w}^{m}_{t}\in\mathbf{B}_{t}^{m}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})=0.$

From this, we easily deduce, for $P^{m*}$-a.s. $\mathbf{w}$,

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m},\,\mathbf{w}^{m}_{t}\in\mathbf{B}_{t}^{m})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})=0.$
(4.2)

Using (4.2) and Theorem 1.3, we deduce

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})=0\quad\text{
for $P^{m*}$-a.s.\,$\mathbf{w}$}$ (4.3) or $\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{t}\in\mathbf{B}_{t}^{m})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})=0\quad\text{
for $P^{m*}$-a.s.\,$\mathbf{w}$}.$ (4.4)

Suppose (4.3). Then, using $P^{m*}=P^{\infty}\circ(\varpi^{m*})^{-1}$, we
obtain

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})=0\quad\text{for
$P^{\infty}$-a.s.\,$\mathbf{w}$}.$ (4.5)

Taking $\mathscr{B}=\mathrm{W}$ in (4.1) and using
$P^{m*}=P^{\infty}\circ(\varpi^{m*})^{-1}$, we obtain

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})=P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}|\mathscr{F}_{0,t}^{m*})\quad\text{
for $P^{\infty}$-a.s.\,$\mathbf{w}$}.$

Hence, we deduce

$\displaystyle
P_{\mathbf{w}}^{m}(\mathbf{w}^{m}_{0}\in\mathbf{A}_{0}^{m})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})$
(4.6) $\displaystyle=$ $\displaystyle
P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})\quad\text{
for $P^{\infty}$-a.s.\,$\mathbf{w}$}.$

From (4.5) and (4.6), we obtain

$\displaystyle
P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})=0\quad\text{
for $P^{\infty}$-a.s.\,$\mathbf{w}$}$ (4.7)

Integrating (4.7) with respect to $P^{\infty}$, we obtain

$\displaystyle\int_{\mathrm{W}}P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})P^{\infty}(d\mathbf{w})=0.$
(4.8)

Next, suppose (4.4). Then, similarly as (4.8), we obtain

$\displaystyle\int_{\mathrm{W}}P^{\infty}(\mathbf{w}_{t}\in\mathbf{B}_{t}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})P^{\infty}(d\mathbf{w})=0.$
(4.9)

Thus, we see either (4.8) or (4.9) holds for each $m\in\mathbb{N}$. Hence,
(4.8) holds for infinitely many $m\in\mathbb{N}$ or (4.9) holds for infinitely
many $m\in\mathbb{N}$.

Note that the sequence of $\sigma$-fields
$\\{\mathscr{F}_{0,t}^{m*}\\}_{m\in\mathbb{N}}$ is decreasing. Furthermore,
the sequences of sets

$\\{(\varpi_{0}^{m*})^{-1}(\mathbf{A}_{0}^{m*})\\}_{m\in\mathbb{N}}$ and
$\\{(\varpi_{t}^{m*})^{-1}(\mathbf{B}_{t}^{m*})\\}_{m\in\mathbb{N}}$

are increasing and the limits

$\widetilde{\mathscr{A}}:=\bigcup_{m=1}^{\infty}(\varpi_{0}^{m*})^{-1}(\mathbf{A}_{0}^{m*})$
and
$\widetilde{\mathscr{B}}:=\bigcup_{m=1}^{\infty}(\varpi_{t}^{m*})^{-1}(\mathbf{B}_{t}^{m*})$

are $\mathscr{C}_{\mathrm{path}}^{\infty*}$-measurable. The sets
$\widetilde{\mathscr{A}}$ and $\widetilde{\mathscr{B}}$ contain $\mathscr{A}$
and $\mathscr{B}$, respectively. Hence, using the martingale convergence
theorem and the Lebesgue convergence theorem, we find that, $P^{\infty}$-a.s.
and in $L^{1}(\mathrm{W},P^{\infty})$,

$\displaystyle\lim_{m\to\infty}$ $\displaystyle
P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{B}_{t}^{m*}}(\mathbf{w}^{m*}_{t})$
(4.10) $\displaystyle=$ $\displaystyle
P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}|\bigcap_{m=1}^{\infty}\mathscr{F}_{0,t}^{m*})1_{\widetilde{\mathscr{B}}}(\mathbf{w}),$
$\displaystyle\lim_{m\to\infty}$ $\displaystyle
P^{\infty}(\mathbf{w}_{t}\in\mathbf{B}_{t}|\mathscr{F}_{0,t}^{m*})1_{\mathbf{A}_{0}^{m*}}(\mathbf{w}^{m*}_{0})$
(4.11) $\displaystyle=$ $\displaystyle
P^{\infty}(\mathbf{w}_{t}\in\mathbf{B}_{t}|\bigcap_{m=1}^{\infty}\mathscr{F}_{0,t}^{m*})1_{\widetilde{\mathscr{A}}}(\mathbf{w}).$

From Lemma 4.1 and
$\bigcap_{m=1}^{\infty}\mathscr{F}_{0,t}^{m*}\subset\mathscr{C}_{\mathrm{path}}^{\infty*}$,
we deduce $P^{\infty}(\widetilde{\mathscr{A}})\in\\{0,1\\}$. Furthermore,
$\\{\mathbf{w};\mathbf{w}_{0}\in\mathbf{A}_{0}\\}\subset\widetilde{\mathscr{A}}$
by construction. Hence,

$\displaystyle\int_{\mathscr{A}}P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}|\bigcap_{m=1}^{\infty}\mathscr{F}_{0,t}^{m*})dP^{\infty}=$
$\displaystyle
P^{\infty}(\mathscr{A})P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0}).$ (4.12)

Similarly, we have

$\displaystyle\int_{\mathscr{B}}P^{\infty}(\mathbf{w}_{t}\in\mathbf{B}_{t}|\bigcap_{m=1}^{\infty}\mathscr{F}_{0,t}^{m*})dP^{\infty}=$
$\displaystyle
P^{\infty}(\mathscr{B})P^{\infty}(\mathbf{w}_{t}\in\mathbf{B}_{t}).$ (4.13)

Suppose $P^{\infty}(\widetilde{\mathscr{A}})=0$. Then
$P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0})=0$ because
$\\{\mathbf{w};\mathbf{w}_{0}\in\mathbf{A}_{0}\\}\subset\widetilde{\mathscr{A}}$.
Suppose $P^{\infty}(\widetilde{\mathscr{A}})=1$. If, in addition, (4.8) holds
for infinitely many $m\in\mathbb{N}$, then from (4.8), (4.10), and (4.12), we
deduce $P^{\infty}(\mathbf{w}_{0}\in\mathbf{A}_{0})=0$.

Similarly, $P^{\infty}(\widetilde{\mathscr{B}})=0$ implies
$P^{\infty}(\mathbf{w}_{t}\in\mathbf{B}_{t})=0$. If
$P^{\infty}(\widetilde{\mathscr{B}})=1$ and (4.9) holds for infinitely many
$m\in\mathbb{N}$, then $P^{\infty}(\mathbf{w}_{t}\in\mathbf{B}_{t})=0$ from
(4.9), (4.11), and (4.13).

Combining these and recalling $\mathbf{A}=\mathbf{A}_{0}$ and
$\mathbf{B}=\mathbf{B}_{t}$ complete the proof. ∎

##  5  Acknowledgment

This work was supported by JSPS KAKENHI Grant Numbers JP16H06338, JP20K20885,
JP21H04432, and JP18H03672. We thank Stuart Jenkinson, PhD, from Edanz Group
(https://jpen-author-services.edanz.com/ac) for editing a draft of this
manuscript.

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Hirofumi Osada
Faculty of Mathematics, Kyushu University,
Fukuoka 819-0395, Japan.
<EMAIL_ADDRESS>
Ryosuke Tsuboi

Financial Institutions BU, Financial Information Systems 2233G, Hitachi, Ltd.
Tokyo 104-0045, Japan
<EMAIL_ADDRESS>