Infinite-dimensional Stochastic Differential Equations with Symmetry

TL;DR

This paper reviews recent advances in the study of infinite-dimensional stochastic differential equations exhibiting symmetry, highlighting examples from random matrix theory to illustrate key concepts and progress.

Contribution

It provides a comprehensive overview of recent developments in symmetric infinite-dimensional SDEs, connecting them with applications in random matrix theory.

Findings

Examples from random matrix theory are discussed.

Recent progress in symmetric infinite-dimensional SDEs is summarized.

The paper highlights key theoretical advancements.

Abstract

We review recent progress in the study of infinite-dimensional stochastic differential equations with symmetry. This paper contains examples arising from random matrix theory.

Equations

$$d{X}_t^i=\sigma (X_t^i,{\mathbf{X}}_t^{i♢})d{B}_t^i+b(X_t^i,{\mathbf{X}}_t^{i♢})dt(i\in \mathbb{N}).$$

$$d{X}_t^i=\sigma (X_t^i,{\mathsf{X}}_t^{i♢})d{B}_t^i+b(X_t^i,{\mathsf{X}}_t^{i♢})dt(i\in \mathbb{N}).$$

$$d{X}_t^i=d{B}_t^i+\frac{\beta }{2}\sum _{j=1,j\ne i}^{\infty }\nabla \Psi (X_t^i-X_t^j)dt.$$

$${\mu }_{r,\mathsf{t}}^m(\cdot )=\mu ({\pi }_r(\mathsf{s})\in \cdot |\mathsf{s}(S_r)=m,{\pi }_r^c(\mathsf{s})={\pi }_r^c(\mathsf{t}))$$

$${\mathcal{H}}_r=\sum _{s_i\in S_r}\Phi (s_i)+\sum _{s_i,s_j\in S_r,i<j}\Psi (s_i,s_j).$$

$${\mu }_{r,\mathsf{t}}^m=c_{r,\mathsf{t}}^m{e}^{-{\mathcal{H}}_r-{\sum }_{x_i\in S_r,t_j\in S_r^c}\Psi (x_i,t_j)}d{\Lambda }_r^m.$$

$$\Psi (x,y)=-\beta \log |x-y|.$$

$${c_{r,\mathsf{t}}^m}^{-1}{e}^{-{\mathcal{H}}_r}d{\Lambda }_r^m\le {\mu }_{r,\mathsf{t}}^m\le c_{r,\mathsf{t}}^m{e}^{-{\mathcal{H}}_r}d{\Lambda }_r^m$$

$${\mathcal{E}}^{\mu }(f,g)={\int }_{\mathsf{S}}\mathbb{D}[f,g]d\mu ,\mathbb{D}[f,g]=\frac{1}{2}\sum _{i}a(s_i,{\mathsf{s}}^{i♢})\frac{\partial \check{f}}{\partial s_i}\cdot \frac{\partial \check{g}}{\partial s_i}.$$

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

$${\int }_{{\mathbb{R}}^d\times \mathsf{S}}{\nabla }_{x}fd{\mu }^{[1]}=-{\int }_{{\mathbb{R}}^d\times \mathsf{S}}f{\mathsf{d}}^{\mu }d{\mu }^{[1]}.$$

$${\nabla }_{x}a(x,\mathsf{s})+a(x,\mathsf{s}){\nabla }_x\log {\mu }^{[1]}(x,\mathsf{s})=2b(x,\mathsf{s}).$$

$$W_{\mathrm{sol}}$$

$${\mathbf{S}}_0$$

$${\sigma }^i,b^i:W_{\mathrm{sol}}\to W(({\mathbb{R}}^d{)}^{\mathbb{N}})$$

$$d{X}_t^i={\sigma }^i(\mathbf{X}{)}_{t}d{B}_t^i+b^i(\mathbf{X}{)}_{t}dt(i\in \mathbb{N}),$$

$${\mathbf{X}}_0=\mathbf{s}=(s_i{)}_{i\in \mathbb{N}}\in {\mathbf{S}}_0,\mathbf{X}\in W_{\mathrm{sol}}.$$

$$d{Y}_t^{m,i}={\sigma }^i({\mathbf{Y}}^m,{\mathbf{X}}^{m\ast }{)}_{t}d{B}_t^i+b^i({\mathbf{Y}}^m,{\mathbf{X}}^{m\ast }{)}_{t}dt(i=1,\dots ,m),$$

$${\mathbf{Y}}_0^m=(s_1,\dots ,s_m),({\mathbf{Y}}^m,{\mathbf{X}}^{m\ast })\in W_{\mathrm{sol}}$$

$${\bar{P}}_{\mathbf{s},\mathbf{B}}={\bar{P}}_{\mathbf{s}}(\mathbf{X}\in \cdot |\mathbf{B})$$

$$\text{Tpath((Rd)N)=∩m=1∞σ[Xm∗]},$$

$${\mathcal{T}}_{\mathrm{path}}^{(1)}[{\bar{P}}_{\mathbf{s},\mathbf{B}}]=\{\mathbf{A}\in {\mathcal{T}}_{\mathrm{path}}(({\mathbb{R}}^d{)}^{\mathbb{N}});{\bar{P}}_{\mathbf{s},\mathbf{B}}(\mathbf{A})=1\}$$

$${\tilde{\mathcal{T}}}_{\mathrm{path}}(\mathsf{S})=\underset{\mathbf{t}\in \mathbf{T}}{\bigvee }\bigcap _{r=1}^{\infty }\sigma [{\pi }_r^c({\mathsf{X}}_{\mathbf{t}})],{\tilde{\mathcal{T}}}_{\mathrm{path}}(({\mathbb{R}}^d{)}^{\mathbb{N}})=\underset{\mathbf{t}\in \mathbf{T}}{\bigvee }\bigcap _{n=1}^{\infty }\sigma [{\mathbf{X}}_{\mathbf{t}}^{n\ast }].$$

$$\displaystyle\mathcal{T}(\mathsf{S})\xrightarrow[{\rm\textsf{Q1}},\,{\rm\textsf{Q2}}]{(\textrm{Step I})}$$

$$\mu$$

$$\{{\mathbf{X}}_{\mathbf{s}}{\}}_{\mathbf{s}\in \mathfrak{l}(\mathsf{H})}=\{\{{\mathbf{X}}_{\mathbf{s},t}{\}}_{t\in [0,\infty )}{\}}_{\mathbf{s}\in \mathfrak{l}(\mathsf{H})}.$$

$$d{X}_t^i=d{B}_t^i+\frac{\beta }{2}\underset{r\to \infty }{\lim }\sum _{|X_t^i-X_t^j|<r,j\ne i}\frac{1}{X_t^i-X_t^j}dt,$$

$$d{X}_t^i=d{B}_t^i+\frac{\beta }{2}\underset{r\to \infty }{\lim }\{(\sum _{j\ne i,|X_t^j|<r}\frac{1}{X_t^i-X_t^j})-\frac{1}{\pi }{\int }_{-r}^0\frac{\sqrt{-x}}{-x}dx\}dt,$$

$$d{X}_t^i=d{B}_t^i+\frac{a}{2X_t^i}dt+\frac{\beta }{2}\sum _{j\ne i}^{\infty }\frac{1}{X_t^i-X_t^j}dt\text{ a≥1}.$$

$$d{X}_t^i=d{B}_t^i+\underset{r\to \infty }{\lim }\sum _{|X_t^i-X_t^j|<r,j\ne i}\frac{X_t^i-X_t^j}{|X_t^i-X_t^j{|}^2}dt,$$

$$d{X}_t^i=d{B}_t^i-X_t^i+\underset{r\to \infty }{\lim }\sum _{|X_t^j|<r,j\ne i}\frac{X_t^i-X_t^j}{|X_t^i-X_t^j{|}^2}dt.$$

$$\displaystyle{\bf E}\big{[}\exp\big{\{}\sum_{m=1}^{M}\langle f_{t_{m}},\mathsf{X}_{t_{m}}\rangle\big{\}}\big{]}=\mathop{\rm Det}_{\begin{subarray}{c}(s,t)\in\mathbf{t}^{2},\,(x,y)\in\mathbb{R}^{2}\end{subarray}}\Big{[}\delta_{st}\delta(x-y)+{\bf K}_{\mathrm{Airy}}(s,x;t,y)\chi_{t}(y)\Big{]}.$$

$${\mathbf{K}}_{\mathrm{Ai}}(s,x;t,y)=\{\begin{array}{ll}{\int }_0^{\infty }du{e}^{-u(t-s)/2}\mathrm{Ai}(u+x)\mathrm{Ai}(u+y),& t\ge s\\ -{\int }_{-\infty }^{0}du{e}^{-u(t-s)/2}\mathrm{Ai}(u+x)\mathrm{Ai}(u+y),& t<s.\end{array}$$

Full text

AMC2010: 60H110, 60J60, 60K35, 60B20, 15B52

Key Words: random matrices, infinitely many particle systems, interacting Brownian motions, Dirichlet forms, logarithmic potentials

1 Introduction

We consider $(\mathbb{R}^{d})^{\mathbb{N}}$-valued infinite-dimensional stochastic differential equations (ISDEs) of $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ of the form

[TABLE]

Here $\mathbf{B}=(B^{i})_{i\in\mathbb{N}}$ is $(\mathbb{R}^{d})^{\mathbb{N}}$-valued standard Brownian motion. For $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ we set $\mathbf{X}^{i\diamondsuit}=(X^{j})_{j\in\mathbb{N}\backslash\{i\}}$. Coefficients $\sigma(x,\mathbf{y})$ and $b(x,\mathbf{y})$ are defined on a subset $\mathbf{S}_{\mathbf{0}}$ of $\mathbb{R}^{d}\times(\mathbb{R}^{d})^{\mathbb{N}}$ independent of $i\in\mathbb{N}$. By definition $\sigma(x,\mathbf{y})$ is $\mathbb{R}^{d^{2}}$-valued, and $b(x,\mathbf{y})$ is $\mathbb{R}^{d}$-valued. We assume that $\sigma(x,\mathbf{y})$ and $b(x,\mathbf{y})$ are symmetric in $\mathbf{y}=(y_{i})_{i\in\mathbb{N}}$ for each $x\in\mathbb{R}^{d}$. Therefore, the set of (1.1) is referred to as “ISDEs with symmetry”. In the present article, we review recent results in this regard. Using a Dirichlet form technique and an analysis on tail $\sigma$-fields of configuration spaces, we prove the existence and pathwise uniqueness of strong solutions of the ISDEs of (1.1). We emphasize that the coefficients are defined only on a thin subset in $(\mathbb{R}^{d})^{\mathbb{N}}$ and the state space of the solution $\mathbf{X}$ is in this subset. Solving the ISDEs of (1.1) includes identifying such a subset.

Let $\mathsf{S}=\{\mathsf{s}=\sum_{i}\delta_{s_{i}};\ \mathsf{s}(K)<\infty\text{ for any compact }K\subset\mathbb{R}^{d}\}$ be the configuration space over $\mathbb{R}^{d}$. $\mathsf{S}$ is a Polish space equipped with the vague topology. With the symmetry of $\sigma(x,\mathbf{y})$ and $b(x,\mathbf{y})$ in $\mathbf{y}$, we regard $\sigma$ and $b$ as functions on $\mathbb{R}^{d}\times\mathsf{S}$. We denote these by the same symbol such that $\sigma(x,\mathbf{y})=\sigma(x,\mathsf{y})$ and $b(x,\mathbf{y})=b(x,\mathsf{y})$, where $\mathsf{y}=\sum_{i}\delta_{y_{i}}$ for $\mathbf{y}=(y_{i})$. Then we rewrite the ISDEs of (1.1) for $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ as:

[TABLE]

Here $\mathsf{X}^{i\diamondsuit}=\sum_{j\not=i}^{\infty}\delta_{X^{j}}$, which is the $\mathsf{S}$-valued process $\{\sum_{j\not=i}^{\infty}\delta_{X_{t}^{j}}\}$. $\mathbf{X}$ is called the labeled dynamics, and the associated unlabeled dynamics $\mathsf{X}$ is given by $\mathsf{X}=\sum_{i\in\mathbb{N}}\delta_{X^{i}}$.

If $\sigma$ is a unit matrix and $b$ is given by a pair interaction $\Psi(x-y)$, (1.1) becomes

[TABLE]

Here $\beta>0$ is inverse temperature. Having $\Psi$ of Ruelle class and $\Psi\in C_{0}^{3}(\mathbb{R}^{d})$, Lang lang.1 ; lang.2 then solved (1.3), Fritz Fr constructed non-equilibrium solutions for $d\leq 4$, and Tanemura tane.2 provided solutions for hard core Brownian balls. The stochastic dynamics $\mathbf{X}$ given by the solution of (1.3) are called the interacting Brownian motions (IBM).

These solutions are strong solutions in the sense that $\mathbf{X}$ are functionals of the given Brownian motions $\mathbf{B}$ and initial starting points $\mathbf{s}$. The method used in these studies are based on the classic It$\hat{\mathrm{o}}$ scheme. Hence, if $\Psi$ is of long range such as a polynomial decay, then it is difficult to apply this scheme. Tsai tsai.14 solved (1.3) for the Dyson model. He used very cleverly a specific monotonicity of the logarithmic potential and its one-dimensional structure. As for the weak solution, we present a robust method based on the Dirichlet form technique from o.isde . We present a general theory to give $\mu$-pathwise unique strong solutions applicable to the logarithmic interaction from o-t.tail .

Thus, our demonstration is divided into two steps. In the first step, we obtain weak solutions of ISDEs (1.1). That is, we construct solutions $(\mathbf{X},\mathbf{B})$ satisfying (1.1) (see Section 2–Section 4). In the second step, we prove the existence of strong solutions and the $\mu$-pathwise uniqueness. For this, we perform a fine analysis of the tail $\sigma$-field of $\mathsf{S}$ (see Section 5 and Section 6). In Section 7, we give ISDEs arising from random matrix theory. In Section 8, we present the algebraic construction of the dynamics, and the coincidence of the algebraic dynamics with solutions of ISDEs.

2 Unlabeled dynamics: quasi-Gibbs property

We next construct a natural $\mu$-reversible unlabeled diffusion, where $\mu$ is a point process. The key point is the quasi-Gibbs property of $\mu$, which we proceed to describe.

Let $S_{r}=\{x\in\mathbb{R}^{d};|x|<r\}$. Let $\pi_{r},\pi_{r}^{c}\!:\!\mathsf{S}\!\to\!\mathsf{S}$ be projections such that $\pi_{r}(\mathsf{s})=\mathsf{s}(\cdot\cap S_{r})$, $\pi_{r}^{c}(\mathsf{s})=\mathsf{s}(\cdot\cap S_{r}^{c})$. For a point process $\mu$, we set

[TABLE]

Let $\Phi\!:\!\mathbb{R}^{d}\!\to\!\mathbb{R}\cup\{\infty\}$ and $\Psi\!:\!(\mathbb{R}^{d})^{2}\!\to\!\mathbb{R}\cup\{\infty\}$ be potentials. We set

[TABLE]

A point process $\mu$ is called a canonical Gibbs measure if $\mu$ satisfies Dobrushin-Lanford-Ruelle (DLR) equation, that is, for $\mu$-a.s.​ $\mathsf{t}=\sum_{j}\delta_{t_{j}}$

[TABLE]

Here $\Lambda^{m}_{r}=\Lambda(\cdot|\mathsf{s}(S_{r})=m)$ and $\Lambda_{r}$ is the Poisson PP with intensity $1_{S_{r}}dx$.

Point processes appear in random matrix theory in the form sine, Airy, Bessel, and Ginibre point processes having logarithmic potentials

[TABLE]

However, the DLR equation (2.4) does not make sense for a logarithmic potential. Hence we introduce the notion of quasi-Gibbs measures:

Definition 1

$\mu$ is $(\Phi,\Psi)$-quasi-Gibbs measure if $\exists\ c^{m}_{r,\mathsf{t}}$ such that

[TABLE]

By definition a canonical Gibbs measure is a quasi-Gibbs measure. We refer to o.rm ; o.rm2 for a sufficient condition for quasi-Gibbs property. We assume:

A1 $\mu$ is a quasi-Gibbs measure with upper semi-continuous $(\Phi,\Psi)$. Furthermore, $a(x,\mathsf{s})=\sigma^{t}(x,\mathsf{s})\sigma(x,\mathsf{s})$ is bounded and uniformly elliptic.

A2 There exists a $1<p\leq\infty$ such that the $k$-point correlation function $\rho^{k}$ of $\mu$ is in $L_{\mathrm{loc}}^{p}((\mathbb{R}^{d})^{m})$ for each $k\in\mathbb{N}$.

For a given point process $\mu$ we introduce a Dirichlet form such that

[TABLE]

Here we set $\mathsf{s}^{i\diamondsuit}=\sum_{j\not=i}\delta_{s_{j}}$ for $\mathsf{s}=\sum_{i}\delta_{s_{i}}$, and $f(\mathsf{s})=\check{f}(s_{1},s_{2},\ldots)$, where $\check{f}$ is symmetric in $(s_{1},s_{2},\ldots)$. Note that $\mathbb{D}[f,g]$ is a function of $\mathsf{s}$ by construction.

Theorem 2.1 (o.dfa ; o.rm ; o-t.tail )

*Let $\mathcal{D}_{\circ}$ be the set of local, smooth functions on $\mathsf{S}$. Set $\mathcal{D}_{\circ}^{\mu}=\{f\in\mathcal{D}_{\circ}\cap L^{2}(\mu)\,;\,\mathcal{E}^{\mu}(f,f)<\infty\}$.

i Assume A1. Then $(\mathcal{E}^{\mu},\mathcal{D}_{\circ}^{\mu})$ is closable on $L^{2}(\mu)$.

ii Assume A1 and A2. Then there exists a diffusion $\mathsf{X}_{t}=\sum_{i}\delta_{X_{t}^{i}}$ associated with the closure $(\mathcal{E}^{\mu},\mathcal{D}^{\mu})$ of $(\mathcal{E}^{\mu},\mathcal{D}_{\circ}^{\mu})$ on $L^{2}(\mu)$.*

The local boundedness of the correlation functions is used for the quasi-regularity of the Dirichlet form. Once quasi-regularity is established, the existence of $\mu$-reversible diffusion is immediate from the general theory m-r ; FOT.2 .

Unlabeled dynamics are also obtained in akr ; y.96 with a different frame work. It is now proved these are the same dynamics as in o-t.sm ; o-t.core . We remark that ergodicity of unlabeled dynamics with grand canonical Gibbs measures with small enough activity constant is obtained in akr .

3 Labeled dynamics: A scheme of Dirichlet spaces

We next lift the unlabeled dynamics $\mathsf{X}$ in Theorem 2.1 to a labeled dynamics $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ solving (1.1). For this we present a natural scheme of Dirichlet spaces describing the labeled dynamics $\mathbf{X}$. We assume a pair of mild assumptions:

A3 $\{X^{i}\}$ do not collide with each other (non-collision)

A4 each tagged particle $X^{i}$ never explode (non-explosion)

Let $\mathsf{S}_{s,i}=\{\mathsf{s}\in\mathsf{S}\,;\,\mathsf{s}(\{x\})=0\text{ for all }x\in S,\ \,\mathsf{s}(S)=\infty\}$. Then A3 is equivalent to $\mathrm{Cap}^{\mu}(\mathsf{S}_{s,i}^{c})=0$. A4 follows from $\rho^{1}(x)=O(e^{|x|^{\alpha}})$, $\alpha<2$.

We call $\mathfrak{u}$ the unlabeling map if $\mathfrak{u}((s_{i}))=\sum_{i}\delta_{s_{i}}$. We call $\mathfrak{l}$ a label if $\mathfrak{l}$ is defined for $\mu$-a.s.  $\mathsf{s}$, and $\mathfrak{u}\circ\mathfrak{l}(\mathsf{s})=\mathsf{s}$. For a unlabeled dynamics satisfying A3 and A4, the particles can keep the initial label $\mathfrak{l}(\mathsf{s})$. Thus we can construct a map $\mathfrak{l}_{\mathrm{path}}$ to $C([0,\infty);(\mathbb{R}^{d})^{\mathbb{N}})$ such that $\{\mathfrak{u}(\mathfrak{l}_{\mathrm{path}}(\mathsf{X}_{t}))\}_{t\in[0,\infty)}=\mathsf{X}$. Hence we obtain:

Theorem 3.1 (o.tp )

Assume A1–A4. Then there exists a labeled dynamics $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ such that $\mathsf{X}=\sum_{i\in\mathbb{N}}\delta_{X^{i}}$ and that $\mathbf{X}_{0}=\mathfrak{l}(\mathsf{X}_{0})$.

Remark that $(\mathbb{R}^{d})^{\mathbb{N}}$ has no good measures. Then no Dirichlet forms on $(\mathbb{R}^{d})^{\mathbb{N}}$ associated with the labeled dynamics $\mathbf{X}$. We hence introduce the scheme of spaces $(\mathbb{R}^{d})^{m}\times\mathsf{S}$ with Campbell measures $\mu^{[m]}$ such that $d\mu^{[m]}=\rho^{m}(\mathbf{x}_{m})\mu_{\mathbf{x}_{m}}(d\mathsf{s})d\mathbf{x}_{m}$, where $\rho^{m}$ is a $m$-point correlation function of $\mu$ and $\mu_{\mathbf{x}_{m}}$ is the reduced Palm measure conditioned at $\mathbf{x}_{m}$. For $a=\sigma^{t}\sigma$, let $\mathbb{D}^{[m]}$ be the square field on $(\mathbb{R}^{d})^{m}\times\mathsf{S}$ defined similarly as $\mathbb{D}$ on $\mathsf{S}$ given by (2.5) in Section 2. Let

[TABLE]

Let $\mathcal{D}_{\circ}^{[1],\mu}=\{f\in C_{0}^{\infty}(\mathbb{R}^{d})\otimes\mathcal{D}_{\circ};\mathcal{E}^{[m]}(f,f)<\infty,\,f\in L^{2}(\mu^{[m]})\}$.

Theorem 3.2 (o.tp )

Assume A1 and A2. Then $(\mathcal{E}^{[m]},\mathcal{D}_{\circ}^{[1],\mu})$ is closable on $L^{2}(\mu^{[m]})$, and its closures is quasi-regular. Hence the associated diffusion $(\mathbf{X}^{[m]},\mathsf{X}^{[m]*})$ exists. Here we write $(\mathbf{X}^{[m]},\mathsf{X}^{[m]*})=(X^{[m],1},\ldots,X^{[m],m},\sum_{i=m+1}^{\infty}\delta_{X^{[m],i}})$.

Let $(\mathcal{E}^{\mu},\mathcal{D}^{\mu},L^{2}(\mu))$ be the original Dirichlet form. Let $\mathsf{X}=\sum_{i=1}^{\infty}\delta_{X^{i}}$ be the associated unlabeled diffusion. We fix a label $\mathfrak{l}$. Let $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ be the labeled dynamics given by $\mathfrak{l}$. We set $(\mathbf{X}^{m},\mathsf{X}^{m*})=(X^{1},\ldots,X^{m},\sum_{i=m+1}^{\infty}\delta_{X^{i}})$.

Theorem 3.3 (o.tp )

Assume A1–A4. Assume $(\mathbf{X}^{[m]},\mathsf{X}^{[m]*})$ and $(\mathbf{X}^{m},\mathsf{X}^{m*})$ start at the same initial point. Then $(\mathbf{X}^{[m]},\mathsf{X}^{[m]*})=(\mathbf{X}^{m},\mathsf{X}^{m*})$ in distribution for each $m\in\mathbb{N}$.

Instead of the huge space $(\mathbb{R}^{d})^{\mathbb{N}}$, we use a scheme of countably infinite good infinite-dimensional spaces $\{(\mathbb{R}^{d})^{m}\times\mathsf{S}\}_{m\in\{0\}\cup\mathbb{N}}$. Using the diffusion $\mathsf{X}$ on the original unlabeled space $\mathsf{S}$, we construct a scheme of the coupled diffusions $(\mathbf{X}^{m},\mathsf{X}^{m*})$ on $(\mathbb{R}^{d})^{m}\times\mathsf{S}$ associated with the scheme of Dirichlet spaces $(\mathcal{E}^{[m]},L^{2}(\mu^{[m]}))\text{ on }(\mathbb{R}^{d})^{m}\times\mathsf{S}$. This construction is key for the ISDE-representation below.

4 ISDE-representation: Logarithmic derivative

Definition 2 (o.isde )

Let $\nabla_{x}$ be the nabla on $\mathbb{R}^{d}$. $\mathsf{d}^{\mu}\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d}\times\mathsf{S},\mu^{[1]})^{d}$ is called the logarithmic derivative of $\mu$ if, for all $f\in C_{0}^{\infty}(\mathbb{R}^{d})\otimes L^{\infty}(\mathsf{S})$,

[TABLE]

Let $a(x,\mathsf{s})=\sigma^{t}(x,\mathsf{s})\sigma(x,\mathsf{s})$ as before. We set $\nabla_{x}a$ such that $\nabla_{x}a=\sum_{j=1}^{d}\frac{\partial}{\partial x_{j}}a_{ij}$. We introduce a “geometric” differential equation on $\mathsf{d}^{\mu}(x,\mathsf{s})=:\nabla_{x}\log\mu^{[1]}(x,\mathsf{s})$:

[TABLE]

A5 $\mu$ has a logarithmic derivative $\mathsf{d}^{\mu}$.

A6 The logarithmic derivative $\mathsf{d}^{\mu}$ satisfies (4.7).

Theorem 4.1 (o.isde )

Assume A1–A6. Then there exists an $\mathsf{S}_{0}\subset\mathsf{S}$ such that $\mu(\mathsf{S}_{0})=1$ and that, for each $\mathbf{s}\in\mathfrak{u}^{-1}(\mathsf{S}_{0})$, ISDE (1.1) has a solution $(\mathbf{X},\mathbf{B})$ satisfying $\mathbf{X}_{0}=\mathbf{s}$ and $\mathbf{X}_{t}\in\mathfrak{u}^{-1}(\mathsf{S}_{0})$ for all $t$.

From the coupling in Theorem 3.3 and Fukushima decomposition (It$\hat{\mathrm{o}}$ formula), we prove that $\mathbf{X}=(X^{i})_{i\in\mathbb{N}}$ satisfies the ISDEs of (1.1). We use the $m$-labeled process $(\mathbf{X}^{m},\mathsf{X}^{m*})$, to apply It$\hat{\mathrm{o}}$ formula to coordinate functions $x_{1},\ldots,x_{m}$.

5 Strong solutions of ISDEs and pathwise uniqueness

We lift the weak solutions $(\mathbf{X},\mathbf{B})$ to pathwise unique strong solutions. For this purpose, we introduce a scheme consisting of an infinite system of finite-dimensional SDEs with consistency (IFC), and perform an analysis of the tail $\sigma$-field of the path space $W((\mathbb{R}^{d})^{\mathbb{N}})=C([0,T];(\mathbb{R}^{d})^{\mathbb{N}})$. The key idea is the following interpretations:

$\bullet$ a single ISDE $\Longleftrightarrow$ a scheme of IFC.

$\bullet$ the tail $\sigma$-field of $W((\mathbb{R}^{d})^{\mathbb{N}})$ $\Longleftrightarrow$ the boundary condition of the ISDEs.

The method is robust and may be applied to many other models. We consider non-Markovian ISDEs because the argument is general. Let $(W_{\mathrm{sol}},\mathbf{S}_{\mathbf{0}},\{\sigma^{i}\},\{b^{i}\})$ be

[TABLE]

We consider the ISDEs on $(\mathbb{R}^{d})^{\mathbb{N}}$ of the form: $\mathbf{X}=\{(X_{t}^{i})_{i\in\mathbb{N}}\}_{t\in[0,T]}\in W((\mathbb{R}^{d})^{\mathbb{N}})$

[TABLE]

Note that (1.1) is a special case of (5.8). We assume:

P1 The ISDEs (5.8) has a weak solution $(\mathbf{X},\mathbf{B})$.  (not a strong solution!)

From a weak solution $(\mathbf{X},\mathbf{B})$, we define a new SDE of $\mathbf{Y}^{m}=(Y_{t}^{m,i})_{i=1}^{m}$ such that

[TABLE]

for each $\mathbf{X}\in W_{\mathrm{sol}}^{\mathbf{s}}$, $\mathbf{s}=(s_{i})_{i=1}^{\infty}\in\mathbf{S}_{\mathbf{0}}$, and $m\in\mathbb{N}$. Here $\mathbf{X}^{m*}:=(X^{n})_{n>m}$ is interpreted as a part of the coefficients of the SDE (5.9) and $W_{\mathrm{sol}}^{\mathbf{s}}=\{\mathbf{X}\in W_{\mathrm{sol}};\mathbf{X}_{0}=\mathbf{s}\}$. Indeed, we regard (5.9) as finite-dimensional SDEs of $\mathbf{Y}^{m}$. (5.9) become automatically time-inhomogeneous SDEs. We have therefore obtained a scheme of finite-dimensional SDEs of $\{\mathbf{Y}^{m}\}_{m\in\mathbb{N}}$. We assume:

P2 The SDE (5.9) has a unique, strong solution for each $\mathbf{s}\in\mathbf{S}_{\mathbf{0}}$, $\mathbf{X}\in W_{\mathrm{sol}}^{\mathbf{s}}$, and $m\in\mathbb{N}$.

Let $\bar{{P}}_{\mathbf{s}}$ be the distribution of solution $(\mathbf{X},\mathbf{B})$ on $W_{\mathrm{sol}}^{\mathbf{s}}\times W((\mathbb{R}^{d})^{\mathbb{N}})$. Let

[TABLE]

P3 $\mathcal{T}_{\mathrm{path}}((\mathbb{R}^{d})^{\mathbb{N}})$ is $\bar{{P}}_{\mathbf{s},\mathbf{B}}$-trivial for each $\mathbf{s}\in\mathbf{S}_{\mathbf{0}}$ and $P_{\mathrm{Br}}^{\infty}$-a.s.​ $\mathbf{B}$.

Theorem 5.1 (o-t.tail )

*Assume P1–P3. Then

i $\mathbf{X}$ is a strong solution of the ISDEs (5.8) for each $\mathbf{s}\in\mathbf{S}_{0}$.

ii Let $\mathbf{X}_{\mathbf{s}}$ and $\mathbf{X}_{\mathbf{s}}^{\prime}$ be strong solutions of the ISDEs (5.8) starting at $\mathbf{s}\in\mathbf{S}_{0}$ defined on the same space of Brownian motions $\mathbf{B}$. Then $\mathbf{X}_{\mathbf{s}}=\mathbf{X}_{\mathbf{s}}^{\prime}$ for $P_{\mathrm{Br}}^{\infty}$-a.s.​ $\mathbf{B}$ if and only if $\mathcal{T}_{\mathrm{path}}^{(1)}[\bar{{P}}_{\mathbf{s},\mathbf{B}}]=\mathcal{T}_{\mathrm{path}}^{(1)}[\bar{{P}}_{\mathbf{s},\mathbf{B}}^{\prime}]$ for $P_{\mathrm{Br}}^{\infty}$-a.s.​ $\mathbf{B}$.*

Idea of the proof i: Let $(\mathbf{X},\mathbf{B})$ be a weak solution of ISDE given by P1, and fix it. Let $\mathbf{Y}^{m}$ be the unique strong solution of (5.9) given by P2. By construction $\mathbf{Y}^{m}$ is $\sigma[\mathbf{B}]\bigvee\sigma[\mathbf{X}^{m*}]$-measurable. Because the solution (5.9) is unique, we see that $\mathbf{Y}^{m}=\mathbf{X}^{m}:=(X^{1},\ldots,X^{m})$. Let $\mathbf{Y}$ be the limit $\mathbf{Y}=\lim_{m\to\infty}\mathbf{Y}^{m}$. Then $\mathbf{Y}=\mathbf{X}$ and $\mathbf{Y}$ is $\sigma[\mathbf{B}]\bigvee\mathcal{T}_{\mathrm{path}}((\mathbb{R}^{d})^{\mathbb{N}})$-measurable. Because $\mathcal{T}_{\mathrm{path}}((\mathbb{R}^{d})^{\mathbb{N}})$ is $\bar{{P}}_{\mathbf{s},\mathbf{B}}$-trivial by P3, $\mathbf{Y}$ depends only on $\mathbf{s}$ and $\mathbf{B}$. This means $\mathbf{Y}=\mathbf{X}$ is a strong solution. ∎

In o-t.tail we intoduce a notion of IFC solution, with which we generalize Theorem 5.1.

6 Tail triviality: Application to interacting Brownian motions.

We return to the Markovian-type ISDEs of (1.1). We assume A1–A6. We apply Theorem 5.1 to ISDEs of (1.1) by checking P1–P3. P1 follows from Theorem 4.1. Controlling the capacity of $(\mathcal{E}^{\mu},\mathcal{D}^{\mu},L^{2}(\mu))$ we obtain P2. Because $\mathbf{X}\in W_{\mathrm{sol}}$ and $W_{\mathrm{sol}}$ is a nice subset of $W((\mathbb{R}^{d})^{\mathbb{N}})$, we can assume P2 for the solution of (1.1). Dirichlet form theory proves that $\mathbf{X}$ stays in $W_{\mathrm{sol}}$. Indeed, such a condition is reduced to a calculation of capacity related to the unlabeled Dirichlet space FOT.2 ; m-r . Roughly speaking, P2 is satisfied if $\nabla_{x}^{m}\mathsf{d}^{\mu}\in\mathcal{D}_{\mathrm{loc}}^{\mu^{[1]}}$ for a suitable $m$, see (o-t.tail, , Sections 8,9).

Theorem 6.1 (o-t.tail )

Assume Q1–Q3 below. Then P3 holds.

Q1* $\mu$ is tail trivial. That is, $\mu(A)\in\{0,1\}$ for all $A\in\mathcal{T}(\mathsf{S}):=\cap_{r=1}^{\infty}\sigma[\pi_{r}^{c}]$. *

Q2* $P_{\mu}\circ\mathsf{X}_{t}^{-1}\prec\mu$ for all $t$.  (absolute continuity condition). *

Q3* $P_{\mu}(\cap_{r=1}^{\infty}\{\mathsf{m}_{r}(\mathsf{X})<\infty\})=1$, *

where $\mathsf{m}_{r}=\inf\{m\in\mathbb{N};X^{i}\in C([0,T];S_{r}^{c})\text{ for }m<\forall i\in\mathbb{N}\}$ for $\mathsf{X}=\sum_{i\in\mathbb{N}}\delta_{X^{i}}$.

Remark 1

i Determinantal point processes satisfy Q1 (see o-o.tt ).

ii Q2 is obvious because the unlabeled dynamics is $\mu$-reversible.

iii Q3 is satisfied if the one-point correlation function $\rho^{1}$ satisfies$\rho^{1}(x)=O(e^{|x|^{\alpha}})$ $(|x|\to\infty)$ for some $\alpha<2$.

Let $\mathbf{T}=\{\mathbf{t}=(t_{1},\ldots,t_{m})\,;\,t_{i}\in[0,T],m\in\mathbb{N}\}$ and $\mathbf{X}_{\mathbf{t}}^{n*}=(\mathbf{X}_{t_{1}}^{n*},\ldots,\mathbf{X}_{t_{m}}^{n*})$. Let

[TABLE]

Hence by definition, $\tilde{\mathcal{T}}_{\mathrm{path}}(\mathsf{S})$ is the cylindrical tail $\sigma$-field of the unlabeled path space and $\tilde{\mathcal{T}}_{\mathrm{path}}((\mathbb{R}^{d})^{\mathbb{N}})$ is the cylindrical tail $\sigma$-field of the labeled path space $W((\mathbb{R}^{d})^{\mathbb{N}})$. We deduce the triviality of $\mathcal{T}_{\mathrm{path}}((\mathbb{R}^{d})^{\mathbb{N}})$ from that of $\mathsf{S}$. We do this step-by-step following the scheme:

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We denote by $\mathcal{B}(\SS)^{\mu}$ the completion of the $\sigma$-field $\mathcal{B}(\SS)$ with respect to $\mu$.

Definition 3

For $\mathbf{s}\in(\mathbb{R}^{d})^{\mathbb{N}}$, we set $\mathbf{X}_{\mathbf{s}}=\{\mathbf{X}_{\mathbf{s},t}\}_{t\in[0,\infty)}$ such that $\mathbf{X}_{\mathbf{s},0}=\mathbf{s}$. We set

[TABLE]

i We call $(\{\mathbf{X}_{\mathbf{s}}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H})},\mathbf{B})$ a $\mu$-solution of (1.1) if $\mathsf{H}\in\mathcal{B}(\SS)^{\mu}$ satisfying $\mu(\mathsf{H})=1$ and $\mathfrak{u}^{-1}(\mathsf{H})\subset\mathbf{S}_{\mathbf{0}}$, and if $(\mathbf{X}_{\mathbf{s}},\mathbf{B})$ is a solution of ISDE (1.1) for each $\mathbf{s}\in\mathfrak{l}(\mathsf{H})$.

ii We call $(\{\mathbf{X}_{\mathbf{s}}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H})},\mathbf{B})$ a $\mu$-strong solution if it is a $\mu$-solution such that $(\mathbf{X}_{\mathbf{s}},\mathbf{B})$ is a strong solution for each $\mathbf{s}\in\mathfrak{l}(\mathsf{H})$.

Definition 4

We say that the $\mu$-strong uniqueness holds if the following holds.

i The $\mu$-uniqueness in law holds. That is, $\mathbf{X}_{\mathbf{s}}=\mathbf{X}_{\mathbf{s}}^{\prime}$ in law for each $\mathbf{s}\in\mathfrak{l}(\mathsf{H}\cap\mathsf{H}^{\prime})$ for any pair of $\mu$-solutions $(\{\mathbf{X}_{\mathbf{s}}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H})},\mathbf{B})$ and $(\{\mathbf{X}_{\mathbf{s}}^{\prime}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H}^{\prime})},\mathbf{B}^{\prime})$ satisfying Q2.

ii A $\mu$-solution $(\{\mathbf{X}_{\mathbf{s}}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H})},\mathbf{B})$ satisfying Q2 is a $\mu$-strong solution $(\{\mathbf{X}_{\mathbf{s}}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H}^{\prime})},\mathbf{B})$ for some $\mathsf{H}^{\prime}\subset\mathsf{H}$.

iii The $\mu$-pathwise uniqueness holds. That is, $P^{\mathbf{B}}(\mathbf{X}_{\mathbf{s}}=\mathbf{X}_{\mathbf{s}}^{\prime})=1$ for each $\mathbf{s}\in\mathfrak{l}(\mathsf{H}\cap\mathsf{H}^{\prime})$, where $\{\mathbf{X}_{\mathbf{s}}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H})}$ and $\{\mathbf{X}_{\mathbf{s}}^{\prime}\}_{\mathbf{s}\in\mathfrak{l}(\mathsf{H}^{\prime})}$ are any pair of $\mu$-strong solutions defined for the same Brownian motion $\mathbf{B}$ satisfying Q2.

Theorem 6.2 (o-t.tail )

Make the same assumptions as for Theorem 4.1. Assume P2, Q1, and Q3. Then (1.1) has a $\mu$-strong solution $\mathbf{X}$ such that the associated unlabeled dynamics $\mathbf{X}$ is $\mu$-reversibile, and the $\mu$-strong uniqueness holds.

7 Examples arising from random matrix theory.

The first three examples are particle systems in $\mathbb{R}$ ($[0,\infty)$ for Bessel), whereas the last example is in $\mathbb{R}^{2}$. All examples have logarithmic interaction potential.

**Sine, Airy, and Bessel IBM o.isde ; tsai.14 ; o-t.airy ; o-h.bes : ** Let $d=1$.

[TABLE]

The equilibrium states of these dynamics are sine, Airy, and Bessel point processes ($\beta=1,2,4$ (sine, Airy), $\beta=2$ (Bessel)). These point processes correspond to bulk, soft edge, and hard edge scaling limits respectively. The relationships to inverse temperature are: $\beta=1\Rightarrow$GOE, $\beta=2\Rightarrow$GUE, and $\beta=4\Rightarrow$GSE, respectively.

**Ginibre IBM o.isde : ** Let $d=2$ and $\beta=2$. We consider two ISDEs.

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The equilibrium state $\mu_{{}^{\mathrm{gin}}}$ is the Ginibre point process, which has various rigidities such as small variance shirai.06 , number rigidity GP , and dichotomy in its reduced Palm measures o-s.abs . The drift coefficients are equal on and tangential to the support of $\mu_{{}^{\mathrm{gin}}}$, yielding the coincidence of the solutions of (7.10) and (7.11). This dynamical rigidity reflects rigidity of $\mu_{{}^{\mathrm{gin}}}$.

Another dynamical rigidity is the sub-diffusivity o.sub : $\lim_{\epsilon\to 0}\epsilon X_{t/\epsilon^{2}}^{i}=0$. All translation invariant IBMs in $\mathbb{R}^{d}$ ($d\geq 2$) with Ruelle-class potentials with hard core are diffusive o.p . Therefore the sub-diffusivity is in contrast with Ruelle-class potentials, and indicates a dynamical rigidity as a slow down in tagged particles.

8 Algebraic construction and finite particle approxiations

An algebraic construction is known for stochastic dynamics related to point processes appearing in random matrix theory in $\mathbb{R}$ with $\beta=2$, which is given by space-time correlation functions e.g. j.02 ; KT07b ; KT11 ; KT11-b . For example, as for the Airy2 point process, the multi-time, moment generating function is

[TABLE]

Here $\mathbf{t}=\{t_{1},\ldots,t_{M}\}$, $\chi_{t}=e^{f_{t}}-1$, and ${\bf K}_{\mathrm{Ai}}$ is the extended Airy kernel

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Theorem 8.1 (o-t.sm ; o-t.core )

The algebraic construction and the ISDEs define the same stochastic dynamics for sine2, Airy2, and Bessel2.

By algebraic method, the finite particle approximation for sine2, Airy2, and Bessel2 is proved o-t.sm . By analytic method, the same is proved for these point processes with $\beta=1,2,4$ and also the Ginibre point process k-o.sg ; k-o.fpa . The latter approach is robust and valid for many other examples.

Acknowledgements.

H.O. is supported in part by a Grant-in-Aid for Scenic Research (KIBAN-A, No.​​ 24244010; KIBAN-A, No.​​ 16H02149; KIBAN-S, No.​​ 16H06338) from the Japan Society for the Promotion of Science.