# Orthogonal Intertwiners for Infinite Particle Systems
In The Continuum

Stefan Wagner Mathematisches Institut, Ludwig-Maximilians-Universität,
Theresienstr. 39, 80333 München, Germany.Munich Center for Quantum Science and
Technology (MCQST), Schellingstr. 4, 80799 München.

###### Abstract

This article presents intertwining relations for infinite particle systems in
the continuum, motivated by the generalization of stochastic duality. The
study focuses on a system of sticky Brownian motions, also known as Howitt-
Warren martingale problem, and correlated Brownian motions and shows that
infinite-dimensional orthogonal polynomials intertwine the dynamics of
infinitely many particles and their $n$-particle evolution. The proof is based
on two assumptions about the model: information about the reversible measures
for the $n$-particle dynamics and consistency. The article shows that
compatible families (according to Le Jan and Raimond, 2004), including the two
examples mentioned, are consistent. Additionally, explicit formulas for the
polynomials are used, including a new explicit formula for the orthogonal
polynomials with respect to the Pascal process. As an application of the
intertwining relations, new reversible measures for the infinite-particle
dynamics are obtained.

_Keywords_ : Correlated Brownian motions; sticky Brownian motions;
intertwining relations; orthogonal polynomials; duality; consistency; point
processes.

_Mathematics Subject Classification_ : 60J60, 60J25, 82C22.

###### Contents

  1. 1 Introduction
  2. 2 Correlated Brownian Motions
    1. 2.1 The Model
    2. 2.2 Unlabeled Dynamics
    3. 2.3 Multiple Wiener-Itô Integrals for the Poisson Process
    4. 2.4 Main Result
  3. 3 Sticky Brownian Motions
    1. 3.1 The Model
    2. 3.2 Pascal Process and Infinite-Dimensional Meixner Polynomials
    3. 3.3 Main Result
  4. 4 Strategy for the Proof
    1. 4.1 Consistency
    2. 4.2 Main Result: a Broader Perspective
  5. 5 Proofs
    1. 5.1 Unlabeled Dynamics and Consistency
    2. 5.2 Intertwining Relations
    3. 5.3 Infinite-Dimensional Meixner Polynomials: an Explicit Formula
    4. 5.4 Reversible Measures for the $n$-Particle Dynamics

##  1 Introduction

Stochastic duality is a powerful tool that establishes a connection between
two Markov processes using a common observable, known as the duality function
(see e.g. [JK14]). When the two processes are identical, we speak of self-
duality. Duality is particularly useful when one of the processes is easier to
study. In the case of self-duality, simplification may arise from using a
simpler initial condition in the dual process.

Stochastic duality has found widespread application in diverse fields,
including interacting particle systems (see e.g. [Lig05]), population genetics
models (see e.g. [EK86], [Eth06], [DG11]), and (stochastic) partial
differential equations (see e.g. [Mue15]). In [KMP82], duality serves as a
tool for understanding the non-equilibrium steady state in the KMP model of
heat conduction. Self-duality, specifically with orthogonal polynomials, is
particularly valuable in studying fluctuation fields, the Boltzmann Gibbs
principle, and cumulants in non-equilibrium steady states. This approach is
elaborated upon in [DP91], [ACR18], [ACR21b] and [FRS22].

In the realm of interacting particle systems, dualities and self-dualities are
typically explored in the context of lattices. Here, the particle
configuration is described by the number of particles present at each site of
the lattice. Self-duality with respect to orthogonal polynomials has been
demonstrated in three prominent classical discrete interacting particle
systems: the exclusion process, inclusion process, and independent random
walks, as explored in [GKR07], [GKRV09], [FG19], [Gro19], [CFG+19], and
[FRS22]. Self-duality means that the time evolution of an orthogonal
polynomial of degree $n$ can be expressed by the time evolution of $n$ dual
particles. This is crucial because it allows us to characterize properties of
a system of infinitely many particles by analyzing only a finite system. One
shared property of these systems is consistency, which means that the system’s
time evolution commutes with the action of randomly removing a particle from
the system. In [CGR21], the relationship between self-duality and consistency
for reversible systems was investigated.

In [FJRW21], a advance was made in the study of consistent particle systems.
The authors developed a general method that shows how the orthogonal dualities
of these systems rely solely on the conditions of consistency and
reversibility. This method does not require an explicit formula of the
polynomials and works not only for particle systems on discrete spaces but
also on Borel spaces like the real line $\mathbb{R}$. This approach produces
then self-intertwiners in terms of infinite-dimensional orthogonal
polynomials.

However, the authors of [FJRW21] only consider particle systems with a finite
number of particles. The arguments presented in [FJRW21] can apply to systems
with infinitely many particles, subject to certain limitations. Specifically,
to extend the analysis, it is necessary to have a reversible measure for the
dynamics for such systems. Additionally, the concept of consistency alone is
insufficient because removing a single particle does not link infinite and
finite dynamics. However, it is worth noting that the intertwining relation
[FJRW21, Equation (3.5)] in terms of generalized falling factorial
polynomials, together with reversibility, implies the intertwining relation
[FJRW21, Equation (3.12)] in terms of orthogonal polynomials.

In this article, we present a novel approach for infinite particle systems
with particles on a non-discrete space that overcomes these obstacles. Our
alternative setup has two key features: firstly, only reversible measures for
the unlabeled $n$-particle dynamics are required, which are more manageable to
obtain than reversible measures for infinite dynamics. Secondly, we define
consistency in a manner that is compatible with an infinite number of
particles. Given these requirements, we find that infinite-dimensional
polynomials of degree $n$ intertwine the dynamics of an infinite number of
particles with the dynamics of $n$ particles where $n<\infty$. Our proof
relies on explicit formulas for the infinite-dimensional orthogonal
polynomials. Intertwiners for infinitely many particles have practical
applications. In particular, we obtain new reversible measures for the
dynamics of infinitely many correlated or sticky Brownian motions.

We illustrate our procedure by presenting examples of strongly consistent
systems, called compatibility by Le Jan and Raimond [LR04, Definition 1.1].
Such systems possess the property that the time-evolution of the process
commutes with the removal of any deterministic particle. This characteristic,
related to Kolmogorov’s extension theorem, enables the construction of an
infinite dynamics in a straightforward manner. Furthermore, there is a one-to-
one correspondence between strongly consistent families and stochastic flows
(see [LR04]). These flows can be interpreted as independent particles in
randomly chosen environments. Brownian flows, such as the Arratia flow [Arr79,
BGS15], the Harris flow [Har84], or the Howitt-Warren flow [HW09], are
prominent examples of strongly consistent models.

We focus on two examples. The first model is a system of correlated Brownian
motions. We obtain intertwining relations in terms of infinite-dimensional
orthogonal polynomials with respect to the distribution of the Poisson
process. Their connection to the multiple Wiener-Itô integral was extensively
studied in e.g. [Ogu72] or [Sur83]. The second example refers to a system of
sticky Brownian motions, which were introduced by Howitt and Warren via a
martingale problem (see [HW09]). They are observed as a scaling limit of
random walks in random environments, a generalized exclusion process, or
through a condensation rescaling of symmetric inclusion process (see e.g.
[RS15], [CGR20], and [ACR21a]). Sticky interactions are used to model colloids
in materials science (see [HC17]). In addition, these interactions have
recently received attention in [DDP23]. We focus on a special case of the
Howitt-Warren martingale problem called uniform sticky Brownian motions and
studied in [BR20] and [BW23]. The multiparticle interactions in this model are
entirely determined by two-particle interactions. The dynamics of this model
consist of independent Brownian motions if they are apart. If they meet, they
exhibit a slowly reflecting interaction behavior, as per Feller’s boundary
classification [Fel52]. Specifically, the time during which they are equal has
a positive Lebesgue measure with positive probability, but does not contain
any interval. For this model, we investigate intertwining relations in terms
of infinite-dimensional Meixner polynomials that are orthogonal with respect
to the distribution of the Pascal process. For this latter family, we obtain a
new explicit formula. Both families of infinite-dimensional orthogonal
polynomials belong to the Meixner class ([Mei34], [Lyt03b]).

The article is organized as follows. In the first two sections, we introduce
the two models: correlated Brownian motions (Section 2) and sticky Brownian
motions (Section 3). We introduce infinite-dimensional polynomials and present
our main theorems. In Section 4, we generalize the concept of consistency for
infinitely many particles and develop a general approach that works with
models beyond the two we present. In Section 5, we prove that strongly
consistent systems, including our examples, satisfy the consistency condition.
We then establish the intertwining relations using the explicit formula for
infinite-dimensional polynomials, and in particular, prove the explicit
formula for Meixner polynomials. Finally, we examine the reversible measures
for $n$-particle dynamics of correlated and sticky Brownian motion.

##  2 Correlated Brownian Motions

This section focuses on intertwining relations in terms of the multiple
Wiener-Itô integral with respect to the Poisson process. While previous works
have focused on independent particle systems (see e.g. [Sur83], [KLR08] and
[KKO+09]) and consistent finite particle systems (see [FJRW21]), we aim to
find intertwiners for infinite particle systems with interaction.

We consider correlated Brownian motions, a simple example within the
established setting of [LR04]. Notably, this example stands out for its
simplicity, as the interaction of the Brownian motions does not depend on
local interactions when the particles meet, unlike other correlated Brownian
motion models such as the Harris flow [Har84] or those presented in [LR14].

###  2.1 The Model

We define a family of real-valued stochastic processes $(X_{k,t})_{t\geq 0}$,
$k\in\mathbb{N}$ to be a family of _correlated Brownian motions with pairwise
correlation $0\leq a\leq 1$_ starting at a (deterministic) sequence
$x=(x_{k})_{k\in\mathbb{N}}$ of real numbers if

  * •

The family $(X_{k,t})_{k\in\mathbb{N},t\geq 0}$ is a Gaussian process.

  * •

For all $t\geq 0$ and $k\in\mathbb{N}$:
$\mathbb{E}\left[X_{k,t}\right]=x_{k}$.

  * •

For all $t,s\geq 0$ and $k\in\mathbb{N}$:
$\mathrm{Cov}\left[X_{k,t},X_{k,s}\right]=\min{\left\\{t,s\right\\}}$.

  * •

For all $t,s\geq 0$ and $k,l\in\mathbb{N}$ with $k\neq l$:
$\mathrm{Cov}\left[X_{k,t},X_{l,s}\right]=a\min{\left\\{t,s\right\\}}$.

If $a=0$, the $(X_{1,t})_{t\geq 0},(X_{2,t})_{t\geq 0},\ldots$ are independent
Brownian motions. On the other hand, if $a=1$, they are modifications of each
other up to an additive constant: for any $k\in\mathbb{N}$ and $t\geq 0$, we
have $X_{k,t}=X_{k,1}-x_{1}+x_{k}$ almost surely.

An explicit construction can be done as follows: Let $(B_{t})_{t\geq 0}$,
$(B_{1,t})_{t\geq 0},(B_{2,t})_{t\geq 0},\ldots$ be independent Brownian
motions all starting at zero. From there, define

$\displaystyle X_{k,t}:=\sqrt{1-a}B_{k,t}+\sqrt{a}B_{t}+x_{k},\qquad
k\in\mathbb{N},t\geq 0.$ (2.1)

A family of correlated Brownian motions satisfies the Markov property. We
define the _$n$ -particle semigroup_

$\displaystyle
P_{t}^{[n]}f_{n}(x):=\mathbb{E}\left[f_{n}(\sqrt{1-a}B_{1,t}+\sqrt{a}B_{t}+x_{1},\ldots,\sqrt{1-a}B_{n,t}+\sqrt{a}B_{t}+x_{n})\right]$
(2.2)

for $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ where
$f_{n}:\mathbb{R}^{n}\to\mathbb{R}$, $n\in\mathbb{N}$ is bounded and
measurable.

######  Remark 2.1.

The family of $n$-particle semigroups of correlated Brownian motions is
strongly consistent (called _compatibility_ by [LR04]). More precisely, a
family of Markov semigroups $(P_{t}^{[n]})_{t\geq 0}$ defined on
$\mathbb{R}^{n}$, $n\in\mathbb{N}$, is called _strongly consistent_ if the
equation

$\displaystyle
P_{t}^{[n]}f_{n}(x_{1},\ldots,x_{n})=P_{t}^{[l]}g_{l}(x_{i_{1}},\ldots,x_{i_{l}}),\qquad
x_{1},\ldots,x_{n}\in\mathbb{R}$ (2.3)

holds for all $l\leq n$, $i_{1},\ldots,i_{l}\in{\left\\{1,\ldots,n\right\\}}$
pairwise different and $g_{l}:\mathbb{R}^{l}\to\mathbb{R}$ bounded and
measurable, where $f_{n}:\mathbb{R}^{n}\to\mathbb{R}$,
$(x_{1},\ldots,x_{n})\mapsto g_{l}(x_{i_{1}},\ldots,x_{i_{l}})$.

We remark that or a general strongly consistent family $(P_{t}^{[n]})_{t\geq
0}$, $n\in\mathbb{N}$, the existence of a Markov family
${\left(\Omega,\mathcal{F},(X_{t})_{t\geq
0},(\mathbb{P}_{x})_{x\in\mathbb{R}^{\infty}}\right)}$,
$X_{t}=(X_{k,t})_{k\in\mathbb{N}}$ describing the evolution of infinitely many
particles is implied by Kolmogorov’s theorem (see e.g. [LR04, Section 1.5.3]).
Thereby, we denote by $\mathbb{R}^{\infty}$ the set of sequences of real
numbers, and equip it with its cylindrical $\sigma$-algebra. $\mathbb{P}_{x}$
is a probability measure on the probability space $(\Omega,\mathcal{F})$ for
each $x\in\mathbb{R}^{\infty}$ and $X_{k,t}$ is a real-valued random variable
for $k\in\mathbb{N}$, $t\geq 0$. $X_{0}=x$ holds $\mathbb{P}_{x}$-almost
surely for all $x\in\mathbb{R}^{\infty}$, $x\mapsto\mathbb{P}_{x}[X_{t}\in A]$
is measurable for all measurable $A\subset\mathbb{R}^{\infty}$ and $t\geq 0$,
and $(X_{t})_{t\geq 0}$ satisfies the Markov property with respect to its
natural filtration. This family is such that each finite subconfiguration of
$l$-particles evolves according to $P_{t}^{[l]}$. More precisely, for each
family $i_{1},\ldots,i_{l}\in\mathbb{N},l\in\mathbb{N}$ of pairwise different
indices, $t\geq 0$ and $x=(x_{k})_{k\in\mathbb{N}}\in\mathbb{R}^{\infty}$, the
distribution of ${\left(X_{i_{1},t},\ldots,X_{i_{l},t}\right)}$ under
$\mathbb{P}_{x}$ is equal to
$P_{t}^{[l]}((x_{i_{1}},\ldots,x_{i_{l}}),\,\cdot\,)$.

The correspondence between strongly consistent families and stochastic flows,
as outlined in [LR04], is evident in our example of correlated Brownian
motions. Indeed, let $Z$ be a random variable that follows the standard normal
distribution and let $K_{t}$ be the random probability kernel such that
$K_{t}(v,\,\cdot\,)$ is equal to the normal distribution with expected value
$\sqrt{at}Z+v$ and variance $(1-a)t$ for $v\in\mathbb{R}$ and $t\geq 0$. By
using (2.2), we obtain

$\displaystyle
P_{t}^{[n]}f_{n}(x_{1},\ldots,x_{n})=\mathbb{E}\left[\int\cdots\int
f_{n}(y_{1},\ldots,y_{n})K_{t}(x_{1},\mathrm{d}y_{1})\cdots
K_{t}(x_{1},\mathrm{d}y_{1})\right]$

for all $t\geq 0$ and $x_{1},\ldots,x_{n}\in\mathbb{R}$.

###  2.2 Unlabeled Dynamics

Next, we convert the notation for labeled particle systems to the modern point
process notation modeling particle configurations on $\mathbb{R}$ using
counting measures (see e.g. [LP17]), which employs an unlabeled notation. Let
$\mathbf{N}$ denote the space of _counting measures_ , i.e., the space of
countable sums of measures that assign values in $\mathbb{N}_{0}$ to every
measurable $B\subset\mathbb{R}$. We equip $\mathbf{N}$ with the smallest
$\sigma$-algebra $\mathcal{N}$ such that $\mathbf{N}\ni\mu\mapsto\mu(B)$ is
measurable for each measurable $B\subset\mathbb{R}$. Since $\mathbb{R}$ is a
Borel space, every $\mu\in\mathbf{N}$ is of the form
$\mu=\sum_{k=1}^{n}\delta_{x_{k}}$ with
$n\in\mathbb{N}\cup{\left\\{\infty\right\\}}$, $x_{k}\in\mathbb{R}$, where
$\delta_{x}$ denotes the Dirac measure given $x\in\mathbb{R}$, or is equal to
the zero measure, which we interpret as the empty configuration, see [Kal17,
Section 1.1] or [LP17, Chapter 6]. In particular, $\mu(\mathbb{R})$
corresponds to the total number of particles. We denote by
$\mathbf{N}_{<\infty}:={\left\\{\mu\in\mathbf{N}:\mu(\mathbb{R})<\infty\right\\}}$
the set of _finite configurations_ and by
$\mathbb{N}_{n}:={\left\\{\mu\in\mathbf{N}:\mu(\mathbb{R})=n\right\\}}$ the
set of configurations consisting of exactly $n\in\mathbb{N}_{0}$ particles.

For our purpose, a _Markov family with state space $\mathbf{N}$_ is a
collection $(\Omega$, $\mathcal{F}$, $(\eta_{t})_{t\geq 0}$,
$(\mathbb{P}_{\mu})_{\mu\in\mathbf{N}})$, consisting of a measurable space
$(\Omega,\mathcal{F})$, measurable mappings
$\eta_{t}:(\Omega,\mathcal{F})\to(\mathbf{N},\mathcal{N})$, $t\geq 0$ and
probability measures $\mathbb{P}_{\mu},\mu\in\mathbf{N}$ on
$(\Omega,\mathcal{F})$ such that

  * •

For each $\mu\in\mathbf{N}$, $\mathbb{P}_{\mu}[\eta_{0}=\mu]=1$.

  * •

For each $n\in\mathbb{N}\cup{\left\\{\infty\right\\}}$, $t\geq 0$ and
$A\in\mathcal{N}$, the mapping

$\displaystyle\mathbb{R}^{n}\ni x\mapsto\mathbb{P}_{\iota_{n}(x)}[\eta_{t}\in
A]$ (2.4)

is measurable where $\iota_{n}(x):=\sum_{k=1}^{n}\delta_{x_{k}}$,
$x=(x_{k})_{k=1}^{n}\in\mathbb{R}^{n}$.

  * •

The Markov property is implicitly assumed to be satisfied with respect to the
natural filtration $\mathcal{F}_{t}:=\sigma(\eta_{s}:0\leq s\leq t)$.

We use the notation $\mathbb{E}_{\mu}$ for the expected value with respect to
the probability measure $\mathbb{P}_{\mu}$.

######  Lemma 2.2.

For each $0\leq a\leq 1$, there exists a Markov family $(\Omega$,
$\mathcal{F}$, $(\eta_{t})_{t\geq 0}$,
$(\mathbb{P}_{\mu})_{\mu\in\mathbf{N}})$ with state space $\mathbf{N}$ such
that $\eta_{t}$, $t\geq 0$ is proper and describes the evolution of an
unlabeled system of correlated Brownian motions with pairwise correlation $a$.
More precisely, for each $\mu=\sum_{k=1}^{n}\delta_{x_{k}}$,
$x_{k}\in\mathbb{R}$, $n\in\mathbb{N}\cup{\left\\{\infty\right\\}}$ the
distribution of $(\eta_{t})_{t\geq 0}$ under $\mathbb{P}_{\mu}$ is equal to
the distribution of $\sum_{k=1}^{n}\delta_{X_{k,t}}$ where each
$(X_{k,t})_{t\geq 0}$ is starting at $x_{k}$.

A random measure $\zeta\in\mathbf{N}$ is said to be _proper_ if there exist
random variables $Z_{k}\in\mathbb{R}$,
$K\in\mathbb{N}_{0}\cup{\left\\{\infty\right\\}}$ such that

$\displaystyle\zeta=\sum_{k=1}^{K}\delta_{Z_{k}}$ (2.5)

is satisfied. For more details on proper point processes, we refer to [LP17].

In Lemma 2.2, when $n=0$, the summation considered to be zero. This implies
that the empty configuration remains empty at all times. The Markov family can
be constructed directly. Only a proof of the Markov property is required,
which can be found in Section 5.1 below.

######  Remark 2.3.

For all $n<\infty$ and $t\geq 0$, the measurability of
$\mathbf{N}_{n}\ni\mu\mapsto\mathbb{P}_{\mu}[\eta_{t}\in A]$ is equivalent to
the measurability of (2.4). For $n=\infty$ the requirement of (2.4) is in
general weaker compared to measurability of the mapping

$\displaystyle{\left\\{\mu\in\mathbf{N}:\mu(\mathbb{R})=\infty\right\\}}\ni\mu\mapsto\mathbb{P}_{\mu}[\eta_{t}\in
A].$ (2.6)

With this approach, we can easily convert the concept of labeled particles
into that of unlabeled particles without requiring a detailed analysis of the
intricate issue of measurability of (2.6). In particular, in subsequent
sections of the article, we always assume that the unlabeled process starts
with the distribution of a proper point process.

###  2.3 Multiple Wiener-Itô Integrals for the Poisson Process

For the sake of completeness, let us recapitulate the multiple Wiener-Itô
integral corresponding to the Poisson process, see e.g. [Ogu72] or [Sur83].
The _$k$ -th factorial measure_ (see e.g. [LP17, Eq. (4.5)]) of
$\mu=\sum_{i=1}^{m}\delta_{x_{i}}\in\mathbf{N}$, $x_{i}\in\mathbb{R}$,
$m\in\mathbb{N}_{0}\cup{\left\\{\infty\right\\}}$ is given by

$\displaystyle\mu^{(k)}=\sum_{\begin{subarray}{c}i_{1},\ldots,i_{k}=1\\\
\text{pairwise
different}\end{subarray}}^{m}\delta_{(x_{i_{1}},\ldots,x_{i_{k}})},$ (2.7)

where an empty sum is considered to be equal to the zero measure. The
_symmetrization_ of a function $f_{n}:\mathbb{R}^{n}\to\mathbb{R}$ is defined
by taking the average of $f_{n}$ over all permutations of the coordinates (see
e.g. [Las16, Eq. (27)]). In other words, if $\mathfrak{S}_{n}$ denotes the set
of permutations of the set ${\left\\{1,\ldots,n\right\\}}$, then the
symmetrization of $f_{n}$ is given by

$\displaystyle\widetilde{f_{n}}(x_{1},\ldots,x_{n}):=\frac{1}{n!}\sum_{s\in\mathfrak{S}_{n}}f_{n}(x_{s(1)},\ldots,x_{s(n)}),\qquad
x_{1},\ldots,x_{n}\in\mathbb{R}.$ (2.8)

A function $f_{n}$ is called symmetric if
$f_{n}(x_{1},\ldots,x_{n})=f_{n}(x_{s(1)},\ldots,x_{s(n)})$ for all
$s\in\mathfrak{S}_{n}$ and $x_{1},\ldots,x_{n}\in\mathbb{R}$, i.e., $f_{n}$ is
equal to its symmetrization.

Let $\lambda$ denote the Lebesgue measure on $\mathbb{R}$. The _multiple
Wiener-Itô integral of degree $n$_ with respect to the Poisson process with
intensity measure $\lambda$ (see [LP17, Section 12.2] or [Las16]) is defined
by

$\displaystyle I_{n}f_{n}(\mu):=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}$ $\displaystyle\hskip
40.00006pt\iint\widetilde{f_{n}}(x_{1},\ldots,x_{n})\lambda^{\otimes(n-k)}(\mathrm{d}(x_{k+1},\ldots,x_{n}))\mu^{(k)}(\mathrm{d}(x_{1},\ldots,x_{k}))$
(2.9)

for $\mu\in\mathbf{N}_{\mathrm{lf}}$ and $f_{n}\in\mathcal{C}_{n}$.
$\mathbf{N}_{\mathrm{lf}}$ denotes the set of _locally finite_
$\mu\in\mathbf{N}$, i.e., $\mu(B)<\infty$ for measurable, bounded
$B\subset\mathbb{R}$ while $\mathcal{C}_{n}$ denotes the space of bounded,
measurable functions $f_{n}:\mathbb{R}^{n}\to\mathbb{R}$ with bounded support.
Furthermore, when integrating with respect to $\mu^{(0)}$ or $\lambda^{\otimes
0}$, we treat the integrals as if they were absent. Specifically, we obtain
$I_{0}c(\mu)=c$, for $c\in\mathcal{C}_{0}:=\mathbb{R}$.

The orthogonality relation

$\displaystyle\int(I_{n}f_{n})(I_{m}g_{m})\>\mathrm{d}\pi_{\lambda}=\mathbbm{1}_{{\left\\{n=m\right\\}}}n!\int
f_{n}g_{m}\>\mathrm{d}\lambda^{\otimes n}$ (2.10)

holds for symmetric $f_{n}\in\mathcal{C}_{n}$, $f_{m}\in\mathcal{C}_{m}$ where
$n,m\in\mathbb{N}_{0}$ (see e.g. [Sur84], [Las16]). $\pi_{\lambda}$ denotes
the distribution of the Poisson point process with intensity measure
$\lambda$. Hence, $I_{n}$ can be uniquely extended to a bounded linear
operator mapping square-integrable, symmetric functions, denoted by
$L^{2}_{\mathrm{sym}}(\lambda^{\otimes n})$, to $L^{2}(\pi_{\lambda})$. We put
$L^{2}_{\mathrm{sym}}(\lambda^{\otimes 0}):=\mathbb{R}$.

Additionally, for each $F\in L^{2}(\pi_{\lambda})$, there exists a unique
sequence $(f_{n})_{n\in\mathbb{N}_{0}}$ that belongs to the _Fock space_
$\bigoplus_{n=0}^{\infty}\frac{1}{n!}L^{2}_{\mathrm{sym}}(\lambda^{\otimes
n})$, i.e., it satisfies $\sum_{n=0}^{\infty}\frac{1}{n!}\left\lVert
f_{n}\right\rVert_{L^{2}(\lambda^{\otimes n})}^{2}<\infty$. This sequence is
such that $F=\sum_{n=0}^{\infty}\frac{1}{n!}I_{n}f_{n}$, referred to as the
_chaos decomposition_.

Furthermore, it is worth noting that $I_{n}f_{n}$ has a close relationship
with _Charlier polynomials_ (see e.g. [Sur84],[LP11]) which are self-duality
functions of independent random walkers (see e.g. [RS18], [FG19]).

The study of non-Gaussian white noise naturally involves infinite-dimensional
orthogonal polynomials, as discussed in [Ber96]. In [Lyt03a], the relationship
between chaos decompositions using multiple stochastic integrals with power
jump martingales (e.g. [Sch00] or [NS00]) and polynomial chaos is thoroughly
examined. The use of chaos decompositions is significant in the analysis of
Lévy white noise and stochastic differential equations driven by Lévy white
noise, as explored in [DØP04],[LP06],[Mey08] and [Yab08]. In [FJRW21, Section
3.3], intertwiners based on infinite-dimensional orthogonal polynomials have
been used in the context of finite particle systems. To define the space of
infinite-dimensional polynomials, we consider polynomials with bounded
coefficients and bounded support to ensure their square-integrability, and
interpret the functions $f_{k}$ in the definition below as coefficients. The
set of polynomials of degree less or equal to $n\in\mathbb{N}_{0}$ is defined
by

$\displaystyle\mathcal{P}_{n}:={\left\\{\mathbf{N}_{\mathrm{lf}}\ni\mu\mapsto
u_{0}+\sum_{k=1}^{n}\int u_{k}\>\mathrm{d}\mu^{\otimes
k}:u_{0}\in\mathbb{R},u_{k}\in\mathcal{C}_{k},n\in\mathbb{N}_{0}\right\\}}.$
(2.11)

The _infinite-dimensional orthogonal polynomial of degree $n$_ is defined by
the orthogonal projection of ${\left(\mu\mapsto\int
f_{n}\>\mathrm{d}\mu^{\otimes n}\right)}$ onto $\mathcal{P}_{n-1}^{\perp}$ if
$n\geq 1$ and by the constant function equal to one if $n=0$. It coincides
with $I_{n}f_{n}$ $\pi_{\lambda}$-almost surely for all
$f_{n}\in\mathcal{C}_{n}$, $n\in\mathbb{N}_{0}$.

###  2.4 Main Result

According to [LP17, Corollary 3.7], there exists a proper Poisson process.
Thus, let $\zeta$ be a proper Poisson process with intensity measure given by
the Lebesgue measure $\lambda$. Without loss of generality, we can define
$\zeta$ on the measurable space $(\Omega,\mathcal{F})$ from Lemma 2.2,
equipped with a probability measure $\mathbb{P}$ and expected value
$\mathbb{E}$. Note that $\mathbb{P}\left[\zeta(\mathbb{R})=\infty\right]=1$
due to the fact that $\lambda(\mathbb{R})=\infty$.

######  Theorem 2.4.

Suppose $0\leq a\leq 1$. Then, for every $n\in\mathbb{N}$, the multiple
Wiener-Itô integral of degree $n$ intertwines the dynamics of infinitely many
correlated Brownian motions with pairwise correlation $a$ and their
$n$-particle evolution. In other words,

$\displaystyle\mathbb{E}_{\zeta}\left[I_{n}f_{n}(\eta_{t})\right]=I_{n}P_{t}^{[n]}f_{n}(\zeta)$
(2.12)

holds almost surely for all $t\geq 0$ and for all functions $f_{n}\in
L^{2}(\lambda^{\otimes n})$.

The proof of Theorem 2.4 relies on two key components. the notion of
consistency, which we introduce along with the abstract framework after
Theorem 2.4 in Section 4 below, and the following proposition, which follows
directly from the definition of correlated Brownian motions.

######  Proposition 2.5.

For each $n\in\mathbb{N}$, the measure $\lambda^{\otimes n}$ is reversible for
$n$ correlated Brownian motions with pairwise correlation $a$, i.e.,
$\int(P_{t}^{[n]}f_{n})g_{n}\>\mathrm{d}\lambda^{\otimes
n}=\int(P_{t}^{[n]}g_{n})f_{n}\>\mathrm{d}\lambda^{\otimes n}$ for $t\geq 0$,
$f_{n},g_{n}\in\mathcal{C}_{n}$.

Theorem 2.4 is applicable in finding reversible measures for infinitely many
particles. We say that the measure $\pi_{\lambda}$ is _reversible_ for
$(\eta_{t})_{t\geq 0}$ if

$\displaystyle\mathbb{E}\left[\mathbb{E}_{\zeta}\left[F(\eta_{t})\right]G(\zeta)\right]=\mathbb{E}\left[F(\zeta)\mathbb{E}_{\zeta}\left[G(\eta_{t})\right]\right]$
(2.13)

holds for all measurable, bounded functions $F,G:\mathbf{N}\to\mathbb{R}$ and
$t\geq 0$. Since $\zeta$ is proper and (2.4) is measurable,
$\mathbb{P}_{\zeta}\left[F(\eta_{t})\right]$ is measurable. Thus, the outer
expected values in (2.13) are well-defined.

By applying standard arguments, we obtain the following corollary.
Specifically, the family of $n$-particle Markov semigroups undergoes a unitary
transformation, resulting in the Markov semigroup of unlabeled infinite
dynamics. This transformation is the only significant step in the proof.

######  Corollary 2.6.

The measure $\pi_{\lambda}$, which denotes the distribution of the Poisson
process with intensity measure $\lambda$, is reversible for $(\eta_{t})_{t\geq
0}$.

###### Proof.

Consider the unitary operator

$\displaystyle\mathfrak{U}:\mathfrak{F}:=\bigoplus_{n=0}^{\infty}\frac{1}{n!}L^{2}_{\mathrm{sym}}(\lambda^{\otimes
n})\to
L^{2}(\pi_{\lambda}),\qquad(f_{n})_{n\in\mathbb{N}_{0}}\mapsto\sum_{n=0}^{\infty}\frac{1}{n!}I_{n}f_{n},$

and for each $t\geq 0$, define the operator
$P_{t}:\mathfrak{F}\to\mathfrak{F}$,
$(f_{n})_{n\in\mathbb{N}_{0}}\mapsto(P_{t}^{[n]}f_{n})_{n\in\mathbb{N}_{0}}$
where $P_{t}^{[0]}f_{0}:=f_{0}$, $f_{0}\in\mathbb{R}$. By Proposition 2.5, we
know that $P_{t}$ is a well-defined, bounded, self-adjoint operator for every
$t>0$. Hence, $\mathfrak{U}P_{t}\mathfrak{U}^{-1}$ is also self-adjoint.
Therefore, to complete the proof, it suffices to show that

$\displaystyle\mathfrak{U}P_{t}\mathfrak{U}^{-1}F(\zeta)=\mathbb{E}_{\zeta}\left[F(\eta_{t})\right]$
(2.14)

holds almost surely for all $F\in L^{2}(\pi_{\lambda})$. Theorem 2.4 implies
that (2.14) holds for $F=I_{n}f_{n}$, where $f_{n}\in L^{2}(\lambda^{\otimes
n})$.

In the proof of Theorem 2.4 below we show that the operator
$L^{2}(\pi_{\lambda})\to L^{2}(\Omega,\mathcal{F},\mathbb{P})$,
$F\mapsto\mathbb{E}_{\zeta}\left[F(\eta_{t})\right]$ obtained from the right-
hand side of (2.14) is a contraction, as can be seen in (5.5) below. Thus, by
means of an approximation argument, we conclude that (2.14) holds for all
$F\in L^{2}(\pi_{\lambda})$. ∎

##  3 Sticky Brownian Motions

The focus of this section is on sticky Brownian motions. Similar to correlated
Brownian motions, this model leads us to infinite-dimensional orthogonal
polynomials, specifically, infinite-dimensional Meixner polynomials. These
polynomials serve as intertwiners, and they are orthogonal with respect to the
distribution of the Pascal process. As an application of the intertwining
relations, we obtain a new result for a system of infinitely many sticky
Brownian motions: the distribution of the Pascal process is reversible.

Intertwining relations involving infinite-dimensional Meixner polynomials were
already studied in [FJRW21, Section 5]. The authors examined a version of the
symmetric inclusion processes in the continuum.

###  3.1 The Model

Feller considered in his boundary classification (see [Fel52]) a single
reflected Brownian motion that is sticky at the origin at a rate $\theta>0$.
However, it should be noted that this condition is more accurately described
as a slowly reflecting boundary, rather than true stickiness. Specifically,
the amount of time that the Brownian motion spends at zero has a strictly
positive Lebesgue measure with positive probability, but contains no interval.
In contrast, true stickiness is characterized by a process getting stuck at
zero over a random interval of time, as shown e.g. [KKR07] for an interest
rate process.

Based on that, a pair $(X_{1,t},X_{2,t})_{t\geq 0}$ of Brownian motions with
sticky interaction can be described, namely that $(X_{1,t})_{t\geq 0}$ and
$(X_{2,t})_{t\geq 0}$ are both Brownian motions and
$({\left|X_{1,t}-X_{2,t}\right|})_{t\geq 0}$ is the reflected sticky Brownian
motion at the origin according to Feller. They behave independently when they
are apart and interact when they meet. The pair $(X_{1,t},X_{2,t})_{t\geq 0}$
can be characterized by a martingale problem, as shown in [HW09].

Howitt and Warren (see [HW09]) generalized this concept to a family of $n$
diffusions on $\mathbb{R}$, which is commonly known as the _Howitt-Warren
martingale problem_. This yields a family of $n$ independent Brownian motions
that move separately when they are far apart and coalesce when individual
processes meet. A new non-negative quantity $\theta(i:j)$ is introduced, which
can be interpreted as the rate at which a group of $i+j$ particles splits into
a group of $i$ particles and a group of $j$ particles. Furthermore, the
concept of strong consistency (see Remark 2.1) corresponds to the condition
$\theta(i+1:j)+\theta(i:j+1)=\theta(i:j)$, which has been characterized in
[SSS14, Lemma A.4] by the existence of a finite measure $\nu$ on the interval
$[0,1]$, called _the characteristic measure_ , which satisfies

$\displaystyle\theta(i:j)=\int x^{i-1}(1-x)^{j-1}\nu(\mathrm{d}x),\qquad
i,j\in\mathbb{N}.$

For a full construction using the martingale problem, we refer to [HW09]. An
alternative formulation is available in [SSS14].

In this article, we examine uniform sticky Brownian motions with stickiness
$\theta>0$ and zero drift, which were studied by [BR20] in the context of
large deviation analysis. This model is defined by choosing the characteristic
measure $\nu=\frac{\theta}{2}\lambda_{[0,1]}$, where $\lambda_{[0,1]}$ is the
Lebesgue measure on $[0,1]$, leading to

$\displaystyle\theta(i:j)=\frac{\theta}{2}\frac{(i-1)!(j-1)!}{(i+j-2)!}.$

For this model, the multiparticle interactions are completely determined by
the two-particle interactions. The authors of [BW23] derive the Kolmogorov
backwards equation and demonstrate that, for this particular interaction, it
can be solved exactly using the Bethe ansatz. This result is used to obtain
the reversible measure for the $n$-particle dynamics. We provide the
definition of the uniform sticky Brownian motion from [SSS14, Definition 2.2]
for the sake of completeness.

######  Definition 3.1.

Let $n\in\mathbb{N}$. We say that $(X_{t})_{t\geq
0}=(X_{1},\ldots,X_{n})=(X_{1,t},\ldots,X_{n,t})_{t\geq 0}$ are _$n$ -particle
uniform sticky Brownian motions with stickiness $\theta>0$_ if the following
conditions are satisfied.

  1. (i)

$(X_{t})_{t\geq 0}$ is a continuous, square-integrable semimartingale.

  2. (ii)

The covariation of $X_{k}$ and $X_{l}$ is given by

$\displaystyle[X_{k},X_{l}]_{t}=\int_{0}^{t}\mathbbm{1}_{{\left\\{X_{k,s}=X_{l,s}\right\\}}}\>\mathrm{d}s,\qquad
t\geq 0$

for $k,l\in{\left\\{1,\ldots,n\right\\}}$.

  3. (iii)

For each $\Delta\subset{\left\\{1,\ldots,n\right\\}}$,
$f_{\Delta}(X_{t})-\theta\int_{0}^{t}\beta_{+}(g_{\Delta}(X_{s}))\>\mathrm{d}s$,
$t\geq 0$ is a martingale with respect to the natural filtration of
$(X_{t})_{t\geq 0}$ where

$\displaystyle f_{\Delta}(x):=\max_{k\in\Delta}x_{k},\qquad
g_{\Delta}(x):={\left|{\left\\{k\in\Delta:x_{k}=f_{\Delta}(x)\right\\}}\right|},\qquad
x:=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n},$

$\beta^{+}(1):=0$ and
$\beta^{+}(m):=1+\frac{1}{2}+\frac{1}{3}+\ldots\frac{1}{m-1}$, $m\geq 2$.

The existence of the $n$-particle uniform sticky Brownian motions starting at
an arbitrary initial value $x\in\mathbb{R}^{n}$ was proven in [HW09, Theorem
2.1.] together with the fact, that their distribution is unique. A family of
sticky Brownian motions satisfies the Markov property. Additionally, it is
shown that the corresponding $n$-particle semigroups form a strongly
consistent family, as described in Remark 2.1.

###  3.2 Pascal Process and Infinite-Dimensional Meixner Polynomials

Let $\alpha:=\theta\lambda$, where $\lambda$ denotes the Lebesgue measure on
$\mathbb{R}$ and $\theta>0$ is fixed. The Pascal process, also known as the
negative binomial process, is a well-studied point process (see [Ser90],
[Ber02] or [KP09] for further details). We briefly recall the construction of
a Pascal process. A _Pascal process_ with parameters $\alpha$ and $0<p<1$ is a
random measure $\zeta\in\mathbf{N}$ satisfying the following properties.

  1. (i)

The random variables $\zeta(A_{1}),\ldots,\zeta(A_{N})$ are independent if the
sets $A_{1},\ldots,A_{N}\subset\mathbb{R}$ are measurable and pairwise
disjoint.

  2. (ii)

For each measurable $A\subset\mathbb{R}$ such that $0<\alpha(A)<\infty$, the
random variable $\zeta(A)$ follows the negative binomial distribution with
parameters $p$ and $\alpha(A)$, i.e.,

$\displaystyle\mathbb{P}[\zeta(A)=k]=\alpha(A)(\alpha(A)+1)\cdots(\alpha(A)+k-1)\frac{p^{k}}{k!}(1-p)^{\alpha(A)},\qquad
k\in\mathbb{N}_{0}.$

If $\alpha(A)=0$, then $\zeta(A)=0$ almost surely. If $\alpha(A)=\infty$, then
$\zeta(A)=\infty$ almost surely.

The Pascal process is a compound Poisson process. Specifically, if we consider
a Poisson process $\xi$ on $\mathbb{R}\times\mathbb{N}$ with intensity measure
$\alpha\otimes\sum_{k=1}^{\infty}\frac{p^{k}}{k}\delta_{k}$, then
$\zeta(A):=\int_{A\times\mathbb{N}}y\xi(\mathrm{d}(x,y))$ defines a Pascal
process. In particular, the Pascal process can be constructed as a proper
point process, meaning it has the form (2.5). For more details on Pascal
processes, see also [Kin67] and [FJRW21].

As a reminder, the symmetrization $\widetilde{f_{n}}$ of a function $f_{n}$ is
defined as in (2.8). We put for each $\mu\in\mathbf{N}_{\mathrm{lf}}$ and
$f_{n}\in\mathcal{C}_{n}$

$\displaystyle\mathcal{M}^{p,\alpha}_{n}f_{n}(\mu)$
$\displaystyle:=\sum_{k=0}^{n}\binom{n}{k}{\left(1-\frac{1}{p}\right)}^{k-n}\iint\widetilde{f_{n}}(x_{1},\ldots,x_{n})$
$\displaystyle\hskip
40.00006pt\kappa_{n,k}(x_{1},\ldots,x_{k},\mathrm{d}(x_{k+1},\ldots,x_{n}))\mu^{(k)}(\mathrm{d}(x_{1},\ldots,x_{k})),$
(3.1)

the _infinite-dimensional Meixner polynomial of degree $n$_. The kernel
$\kappa_{n,n-1}:\mathbb{R}^{n-1}\times\mathcal{B}(\mathbb{R})\to[0,\infty)\cup{\left\\{\infty\right\\}}$
is defined by

$\displaystyle\kappa_{n,n-1}((x_{1},\ldots,x_{n-1}),\,\cdot\,):=\alpha+\delta_{x_{1}}+\ldots+\delta_{x_{n-1}},$

while
$\kappa_{n,k}:\mathbb{R}^{k}\times\mathcal{B}(\mathbb{R}^{n-k})\to[0,\infty)\cup{\left\\{\infty\right\\}}$
is defined by
$\kappa_{n,k}:=\kappa_{k+1,k}\otimes\kappa_{k+2,k+1}\otimes\cdots\otimes\kappa_{n,n-1}$.
In other words,

$\displaystyle\kappa_{n,k}((x_{1},\ldots,x_{k}),\mathrm{d}(x_{k+1},\ldots,x_{n}))=(\alpha+\delta_{x_{1}}+\ldots+\delta_{x_{n-1}})(\mathrm{d}x_{n})$
$\displaystyle\hskip
80.00012pt\cdots(\alpha+\delta_{x_{1}}+\ldots+\delta_{x_{k+1}})(\mathrm{d}x_{k+2})(\alpha+\delta_{x_{1}}+\ldots+\delta_{x_{k}})(\mathrm{d}x_{k+1}).$
(3.2)

We interpret the case $n=k$ in (3.2) as though the inner integral was not
present. If $k=0$, $\kappa_{n,0}$ reduces to a measure on $\mathbb{R}^{n}$,
and we interpret the outer integral with respect to $\mu^{(0)}$ as though it
was not present. If $n=0$, we put $\mathcal{M}^{p,\alpha}_{n}f_{0}:=f_{0}$ for
$f_{0}\in\mathcal{C}_{0}=\mathbb{R}$. Let $\mathcal{P}_{n}$ denote the set of
polynomials of degree less than or equal to $n$, as defined by (2.11). The
following proposition shows that (3.2) is a novel explicit formula for the
infinite-dimensional orthogonal polynomials with respect to the distribution
of the Pascal process. This formula plays a crucial role in the proof of
Theorem 3.5 below. The proof is provided in Section 5.3 below.

######  Proposition 3.2.

For each $n\in\mathbb{N}$ and $f_{n}\in\mathcal{C}_{n}$,

$\displaystyle\mathcal{M}^{p,\alpha}_{n}f_{n}=\text{orthogonal projection of
}{\left(\mu\mapsto\mu^{\otimes n}(f_{n})\right)}\text{ onto
}\mathcal{P}_{n-1}^{\perp}$ (3.3)

holds $\rho_{p,\alpha}$-almost surely.

######  Remark 3.3.

The infinite-dimensional Meixner polynomials, given by (3.2), can be regarded
as a natural extension of the monic univariate Meixner polynomials with
parameters $\alpha>0$ and $0<p<1$. The _monic Meixner polynomials_ are
expressed as follows (e.g. [KLS10]):

$\displaystyle\mathscr{M}^{p,a}_{n}(x)=\sum_{k=0}^{n}\binom{n}{k}{\left(1-\frac{1}{p}\right)}^{k-n}(\alpha+k)^{(n-k)}(x)_{k},\qquad
n,x\in\mathbb{N}_{0}.$ (3.4)

In the infinite-dimensional version of these polynomials, the _rising
factorial_ (also known as the _Pochhammer symbol_),
$(a+k)^{(n-k)}=(a+n-1)\cdots(a+k+1)(a+k)$, where $(a+k)^{(0)}=1$, turns into
an integration with respect to the kernel $\kappa_{n,k}$, while the _falling
factorial_ , $(x)_{k}=x(x-1)\cdots(x-k+1)$, where $(x)_{0}:=1$, occurs as an
integration with respect to the factorial measure $\mu^{(k)}$. These
relationships imply that if $f_{n}$ is wisely chosen, (3.2) reduces to a
product of univariate polynomials, as shown in Lemma 5.3 below.

We denote by $\Sigma_{n}$ the set of partitions $\sigma$ of the set
${\left\\{1,\ldots,n\right\\}}$. For a function
$f_{n}:\mathbb{R}^{n}\to\mathbb{R}$ denote by
$(f_{n})_{\sigma}:\mathbb{R}^{{\left|\sigma\right|}}\to\mathbb{R}$ the
function gained by identifying the variables belonging to the same
$A\in\sigma$ in the order of occurrence. Put

$\displaystyle\lambda_{n}:=\sum_{\sigma\in\Sigma_{n}}{\left(\prod_{A\in\sigma}({\left|A\right|}-1)!\right)}\alpha_{\sigma}$
(3.5)

where $\alpha_{\sigma}$ is the measure defined by $\int
f_{n}\>\mathrm{d}\alpha_{\sigma}=\int(f_{n})_{\alpha}\>\mathrm{d}\alpha^{\otimes{\left|\sigma\right|}}$.
Note that

$\displaystyle\lambda_{n}=\kappa_{n,0}=\lambda_{k}\otimes\kappa_{n,k}$ (3.6)

for all $n\geq k\geq 0$. The measure $\lambda_{n}$ is closely related to the
Pascal process in two ways. Firstly, the definition of the Pascal process
implies the following proposition.

######  Proposition 3.4.

The $n$-th factorial moment measure of the Pascal process with parameters
$p,\alpha$ is given by $(\frac{p}{1-p})^{n}\lambda_{n}$ for each
$n\in\mathbb{N}$, i.e.,
$\mathbb{E}\left[\zeta^{(n)}(A)\right]=(\frac{p}{1-p})^{n}\lambda_{n}(A)$ for
$A\subset\mathbb{R}^{n}$ measurable.

Secondly, the orthogonality relation

$\displaystyle\int(\mathcal{M}^{p,\alpha}_{n}f_{n})(\mathcal{M}^{p,\alpha}_{m}g_{m})\>\mathrm{d}\rho_{p,\alpha}=\mathbbm{1}_{{\left\\{n=m\right\\}}}\frac{p^{n}n!}{(1-p)^{2n}}\int
f_{n}g_{m}\>\mathrm{d}\lambda_{n}$ (3.7)

holds for symmetric $f_{n}\in\mathcal{C}_{n}$, $g_{m}\in\mathcal{C}_{m}$ (see
[Lyt03a], [FJRW21]). Hence, the linear operator $\mathcal{M}^{p,\alpha}_{n}$
can be extended continuously to symmetric, measurable, square-integrable
functions, which are denoted by $L^{2}_{\mathrm{sym}}(\lambda_{n})$. This
extension allows us to decompose any function $F\in L^{2}(\rho_{p,\alpha})$
into a series that converges in $L^{2}(\rho_{p,\alpha})$, given by
$F=\sum_{n=0}^{\infty}\frac{(1-p)^{n}}{n!}\mathcal{M}^{p,\alpha}_{n}f_{n}$.
The sequence $(f_{n})_{n\in\mathbb{N}_{0}}$ is contained in a space called the
_extended anyon Fock space_ , as defined in [Lyt03a], [Lyt03b], [BLR15]. This
space is defined as the set of sequences $(f_{n})_{n\in\mathbb{N}_{0}}$ that
satisfy the condition $\sum_{n=0}^{\infty}\frac{p^{n}}{n!}\left\lVert
f_{n}\right\rVert_{L^{2}_{\mathrm{sym}}(\lambda_{n})}^{2}<\infty$, where
$L^{2}_{\mathrm{sym}}(\lambda_{0})$ understood as $\mathbb{R}$.

###  3.3 Main Result

For each $x\in\mathbb{R}^{\infty}$, consider a family of stochastic processes
$(X_{k,t})_{k\in\mathbb{N}}$, $t\geq 0$ such that that for any pairwise
different $i_{1},\ldots,i_{n}\in\mathbb{N}$, the distribution of
$(X_{i_{1},t},\ldots,X_{i_{n},t})$, $t\geq 0$ is the same as that of the
$n$-particle uniform sticky Brownian motions with stickiness $\theta>0$
starting at $(x_{i_{1}},\ldots,x_{i_{n}})$. That sequence exists using strong
consistency and Kolmogorov’s theorem. Let $(P_{t}^{[n]})_{t\geq 0}$ denote the
Markov semigroup of the $n$-particle uniform sticky Brownian motions with
stickiness $\theta$. Consider a Markov family $(\Omega$, $\mathcal{F}$,
$(\eta_{t})_{t\geq 0}$, $(\mathbb{P}_{\mu})_{\mu\in\mathbf{N}})$, which is an
unlabeled version of $(X_{k,t})_{k\in\mathbb{N}}$, $t\geq 0$. This
construction is analogous to the one described in Lemma 2.2. Let $\zeta$
denote a Pascal process with parameters $p$ and $\alpha$ that is proper,
meaning it has the form (2.5). Recall that $\lambda$ denotes the Lebesgue
measure on $\mathbb{R}$.

######  Theorem 3.5.

Suppose $\theta>0$ and put $\alpha=\theta\lambda$. Then, for every
$n\in\mathbb{N}$ and $0<p<1$, the infinite-dimensional Meixner polynomial of
degree $n$ intertwines the dynamics of infinitely many uniform sticky Brownian
motions with stickiness $\theta$ and their $n$-particle evolution. In other
words,

$\displaystyle\mathbb{E}_{\zeta}\left[\mathcal{M}^{p,\alpha}_{n}f_{n}(\eta_{t})\right]=\mathcal{M}^{p,\alpha}_{n}P_{t}^{[n]}f_{n}(\zeta)$
(3.8)

holds almost surely for all $t\geq 0$ and for all functions $f_{n}\in
L^{2}(\lambda_{n})$.

Similar to Theorem 2.4, the proof is a direct consequence of consistency and
reversibility for the $n$-particle evolutions.

######  Proposition 3.6.

For unlabeled uniform sticky Brownian motions $(\eta_{t})_{t\geq 0}$ with
stickiness $\theta$, the push-forward measure of $\lambda_{n}$ under the
mapping $(x_{1},\ldots,x_{n})\to\delta_{x_{1}}+\ldots+\delta_{x_{n}}$ is
reversible.

To prove Proposition 3.6, we use [BW23, Theorem 4.17], which provides a
reversible measure for $n$ ordered uniform sticky Brownian motions.

The following corollary is a novel result providing a family of reversible
measures for a system of infinitely many sticky Brownian motions. The
reversibility of $\rho_{p,\alpha}$, the distribution of the Pascal process,
can be shown using the same arguments as in the proof of Corollary 2.6. In
this context, reversibility is defined in the analogue way as in (2.13).

######  Corollary 3.7.

Let $\theta>0$. Then, for each $0<p<1$, the distribution of the Pascal process
with parameters $p$ and $\alpha=\theta\lambda$ is a reversible measure for
$(\eta_{t})_{t\geq 0}$, which is an infinite system of unlabeled uniform
sticky Brownian motions with stickiness $\theta$.

##  4 Strategy for the Proof

We prove the intertwining relations for correlated Brownian motions and sticky
Brownian motions (Theorem 2.4 and Theorem 3.5) by using the same techniques.
First, in Section 4.1, we adapt the notation of consistency from e.g., [CGR21]
and [FJRW21] to infinite particle systems. We then establish that the two
models, correlated Brownian motions and sticky Brownian motions, are
consistent, based only on the fact that the family of $n$-particle dynamics is
a strongly consistent family according to Remark 2.1.

Next, in Section 4.2, we formulate a general theorem for both the Pascal and
the Poisson case: if the push-forward of the factorial moment measures of the
Poisson (or Pascal) process under the map
$\iota_{n}:\mathbb{R}^{n}\to\mathbf{N}$,
$(x_{1},\ldots,x_{n})\mapsto\delta_{x_{1}}+\ldots+\delta_{x_{n}}$ is
reversible and the Markov processes are consistent, then we obtain the
intertwining relations. The proof uses the explicit representation of the
respective infinite-dimensional orthogonal polynomials (the well-known Wiener-
Itô integrals defined as (2.3) and (3.2)).

Thus, we only need to verify the reversible measures for the finite systems.
For correlated Brownian motions, this follows directly from the definition,
while for sticky Brownian motions, we use [BW23].

###  4.1 Consistency

In [CGR21] and [FJRW21], _consistency_ refers to the property that the removal
of a particle uniformly at random commutes with the time-evolution of a
process. This property can be formulated for particle systems with unlabeled
particles. Specifically, if

$\displaystyle\mathbb{E}_{\mu}\left[\int
F(\eta_{t}-\delta_{x})\eta_{t}(\mathrm{d}x)\right]=\int\mathbb{E}_{\mu-\delta_{x}}\left[F(\eta_{t})\right]\mu(\mathrm{d}x),\qquad\mu\in\mathbf{N}_{<\infty}$
(4.1)

holds for all measurable $F:\mathbf{N}_{<\infty}\to[0,\infty)$, then the
system is called consistent. In (4.1), the left-hand side first evolves the
system and then removes a particle uniformly at random. Conversely, the right-
hand side of the equation first removes a particle uniformly at random from
the initial configuration and then evolves the process from the resulting
state.

In this article, we deliberately define consistency differently since (4.1) is
not useful for extending to infinitely many particles. Equation (4.1) lacks a
connection between the dynamics of infinitely many particles and finitely many
particles. More precisely, if one particle is removed from an infinite
configuration, the total number of particles in the configuration remains
infinite, whereas removing a particle from a finite configuration results in a
finite number of remaining particles. However, our objective is to obtain
intertwining relations that reduce the infinite number of particles to a
finite number. According to [FJRW21, Theorem 3.5, Remark 3.6.], our new
definition below is a natural extension of the conventional definition of
consistency, which coincides with the established notation of consistency of
finite particle systems.

Let $(\Omega$, $\mathcal{F}$, $(\eta_{t})_{t\geq 0}$,
$(\mathbb{P}_{\mu})_{\mu\in\mathbf{N}})$ be a Markov family such that
$\eta_{t}$ is proper for each $t\geq 0$.

######  Definition 4.1.

A stochastic process $(\eta_{t})_{t\geq 0}$ is called _consistent_ if

$\displaystyle\mathbb{E}_{\mu}\left[\int
F(\delta_{x_{1}}+\ldots+\delta_{x_{n}})\eta_{t}^{(n)}(\mathrm{d}(x_{1},\ldots,x_{n}))\right]$
$\displaystyle\hskip
40.00006pt=\int\mathbb{E}_{\delta_{x_{1}}+\ldots+\delta_{x_{n}}}\left[F(\eta_{t})\right]\mu^{(n)}(\mathrm{d}(x_{1},\ldots,x_{n})),\qquad\mu\in\mathbf{N}$
(4.2)

holds for all $t\geq 0$, measurable $F:\mathbf{N}_{<\infty}\to[0,\infty)$ and
$n\in\mathbb{N}$.

Interpreting (4.1), the left-hand side describes $n$ particles chosen
uniformly from the evolved state of the process starting at $\mu$. The right-
hand side, on the other hand, describes $n$ particles chosen uniformly from
the initial state $\mu$ and then evolved under the process.

######  Remark 4.2.

We only define consistency for processes in which each $\eta_{t}$, $t\geq 0$,
is proper. This assumption is motivated by a technical subtlety: in general,
the mapping $\mu\mapsto\int
F(\delta_{x_{1}}+\ldots+\delta_{x_{n}})\mu^{(n)}(\mathrm{d}(x_{1},\ldots,x_{n}))$
for $n\geq 2$, even for measurable $F$, is not measurable. By requiring each
$\eta_{t}$ to be proper, we ensure the measurability of $\omega\mapsto\int
F(\delta_{x_{1}}+\ldots+\delta_{x_{n}})\eta_{t}(\omega)^{(n)}(\mathrm{d}(x_{1},\ldots,x_{n}))$.

Our definition can also be interpreted as self-intertwining property (see
[FJRW21, Equation (3.6)]) using the _$K$ -transform_ introduced by Lenard
[Len73], [Len75]. See also [KKO+09] for $K$-transform intertwining relations
in the context of _free Kawasaki dynamics_.

We say that a process $(\eta_{t})_{t\geq 0}$ is _conservative_ if it conserves
the number of particles, meaning that $\eta_{t}(\mathbb{R})=\mu(\mathbb{R})$
holds $\mathbb{P}^{\mu}$-almost surely for all $\mu\in\mathbf{N}$ and $t\geq
0$.

######  Proposition 4.3.

Both the unlabeled correlated Brownian motions with pairwise correlation $a$
and the unlabeled sticky Brownian motions with stickiness $\theta$ are
consistent and conservative.

The proofs of both Lemma 2.2 and Proposition 4.3 rely solely on the strong
consistency (picked up in Remark 2.1) of the family of underlying $n$-particle
evolutions. This demonstrates that from any strongly consistent family, a
consistent, unlabeled process can be constructed. Therefore, strong
consistency is aptly named, as it is indeed a stronger property compared to
consistency. For models of finitely many configurations, see also [FJRW21,
Section 3.1].

###  4.2 Main Result: a Broader Perspective

The definition of consistency enables us to state the following theorem, which
reveals the abstract framework after Theorem 2.4 and Theorem 3.5. Let
$(\Omega$, $\mathcal{F}$, $(\eta_{t})_{t\geq 0}$,
$(\mathbb{P}_{\mu})_{\mu\in\mathbf{N}})$ be a Markov family such that
$\eta_{t}$ is proper for each $t\geq 0$. Recall the mapping
$\iota_{n}:\mathbb{R}^{n}\to\mathbf{N}$,
$(x_{1},\ldots,x_{n})\mapsto\delta_{x_{1}}+\ldots+\delta_{x_{n}}$. Note that
each symmetric function $f_{n}:\mathbb{R}^{n}\to\mathbb{R}$ can be identified
with a function $F:\mathbf{N}_{n}\to\mathbb{R}$ through the relationship
$f_{n}=F\circ\iota_{n}$. Suppose $(\eta_{t})_{t\geq 0}$ is a conservative
process. Then, for $n\in\mathbb{N}$, the $n$-particle semigroup $P_{t}^{[n]}$
can be recovered as follows:

$\displaystyle
P_{t}^{[n]}f_{n}(x):=\mathbb{E}_{\iota_{n}(x)}\left[F(\eta_{t})\right],$

where $x\in\mathbb{R}^{n}$, and $f_{n}:\mathbb{R}^{n}\to\mathbb{R}$ is a
bounded (or non-negative), symmetric and measurable function. Define the
operator $P_{t}^{[0]}$ as the identity operator on $\mathbb{R}$.

######  Theorem 4.4.

Assume that $(\eta_{t})_{t\geq 0}$ is consistent and conservative.

  1. (i)

Let $\zeta$ be a proper Poisson process with intensity measure $\lambda$.
Suppose that the push-forward measure of the Lebesgue measure on
$\mathbb{R}^{n}$ under the mapping $\iota_{n}$ is reversible for
$(\eta_{t})_{t\geq 0}$ for each $n\in\mathbb{N}$. Then, the intertwining
relation
$\mathbb{E}_{\zeta}\left[I_{n}f_{n}(\eta_{t})\right]=I_{n}P_{t}^{[n]}f_{n}(\zeta)$
holds almost surely and for all $t\geq 0$ and $f_{n}\in
L^{2}_{\mathrm{sym}}(\lambda^{\otimes n})$, $n\in\mathbb{N}_{0}$.

  2. (ii)

Let $\zeta$ be a proper Pascal process with parameters $\alpha$ and $0<p<1$.
Suppose that the push-forward measure of $\lambda_{n}$ defined as (3.5) under
the mapping $\iota_{n}$ is reversible for $(\eta_{t})_{t\geq 0}$ for each
$n\in\mathbb{N}$. Then, the intertwining relation
$\mathbb{E}_{\zeta}\left[\mathcal{M}^{p,\alpha}_{n}f_{n}(\eta_{t})\right]=\mathcal{M}^{p,\alpha}_{n}P_{t}^{[n]}f_{n}(\zeta)$
holds almost surely for all $t\geq 0$ and $f_{n}\in
L^{2}_{\mathrm{sym}}(\lambda_{n})$, $n\in\mathbb{N}_{0}$.

We present a proof for Theorem 4.4 in Section 5.2 below.

###### Proof of Theorem 2.4 and Theorem 3.5.

Proposition 4.3 shows that both the correlated and sticky Brownian motions, in
their unlabeled version, are consistent and conservative. By using Proposition
2.5 (or Proposition 3.6), we apply Theorem 4.4 to obtain the intertwining
relations for symmetric functions for both processes.

If $f_{n}$ is a non-symmetric, square-integrable function, we apply Theorem
4.4 to its symmetrization $\widetilde{f_{n}}$, and then use

$\displaystyle P_{t}^{[n]}{\widetilde{f_{n}}}=\widetilde{P_{t}^{[n]}f_{n}},$

which follows from strong consistency, as defined in (2.3), together with
$I_{n}\widetilde{f_{n}}=I_{n}f_{n}$ (or
$\mathcal{M}^{p,\alpha}_{n}\widetilde{f_{n}}=\mathcal{M}^{p,\alpha}_{n}f_{n}$).
∎

The proof of Corollary 2.6 (or Corollary 3.7) also works in the abstract
framework. Recall that $\pi_{\lambda}$ denotes the distribution of the Poisson
process with intensity measure $\lambda$, and $\rho_{p,\alpha}$ denotes the
distribution of the Pascal process with parameters $p$ and $\alpha$.

######  Corollary 4.5.

Under the assumptions of Theorem 4.4, if we assume the condition stated in
(i), then $\pi_{\lambda}$ is reversible for $(\eta_{t})_{t\geq 0}$, whereas if
we assume the condition stated in (ii), then $\rho_{p,\alpha}$ is reversible
for $(\eta_{t})_{t\geq 0}$.

######  Remark 4.6.

Theorem 4.4 and Corollary 4.5 are formulated here for Markov processes with
particles on the real line. However, both results hold true in a much more
general setting. For instance, if $(E,\mathcal{E})$ is a _Borel space_ (see
[LP17]), instead of the Lebesgue measure, an arbitrary $\sigma$-finite measure
can be chosen for both the Poisson and Pascal case.

######  Remark 4.7.

The condition of reversibility in Theorem 4.4 can be relaxed by demanding only
the condition (5.3) below. Moreover, if this equation holds pointwise, rather
than almost everywhere, we obtain the intertwining relation (2.12) pointwise
as well. However, to ensure the existence of integrals, it may be necessary to
reduce the set of configurations. We can consider this idea for systems of
_independent particles_ on a Borel space: if the one-particle dynamics admits
only an invariant, $\sigma$-finite measure, then (5.3) follows pointwise.

By being more rigorous, we obtain the following statement. Consider a Markov
semigroup $(p_{t})_{t\geq 0}$ on a Borel space $(E,\mathcal{E})$ with an
invariant $\sigma$-finite measure $\lambda$. Let $(\Omega$, $\mathcal{F}$,
$(\eta_{t})_{t\geq 0}$, $(\mathbb{P}_{\mu})_{\mu\in\mathbf{N}})$ be the Markov
family of unlabeled particles, where each particle evolves independently
according to the single particle dynamics $(p_{t})_{t\geq 0}$. This family can
be constructed using the arguments presented in Lemma 2.2.

Let $t\geq 0$ be fixed, and assume that $\mu$ is locally finite, meaning that
$\mu(A)<\infty$ for all $A\in\mathcal{E}$ with $\lambda(A)<\infty$, as well as
the measure $\mu p_{t}$, defined by $\mu p_{t}(B):=\int
p_{t}\mathbbm{1}_{B}\>\mathrm{d}\mu$. We also consider a measurable and
bounded function $f_{n}:E^{n}\to\mathbb{R}$, $n\in\mathbb{N}$, such that
$\lambda^{\otimes n}[f_{n}\neq 0]<\infty$. Under these conditions, we have the
following intertwining relation:

$\displaystyle\mathbb{E}_{\mu}\left[I_{n}f_{n}(\eta_{t})\right]=I_{n}P_{t}^{[n]}f_{n}(\mu)$

where $I_{n}$ denotes the multiple Wiener-Itô integral of degree $n$,
constructed with the invariant measure $\lambda$.

##  5 Proofs

###  5.1 Unlabeled Dynamics and Consistency

First, we prove Lemma 2.2 and Proposition 4.3. In the following proof, we use
a construction of unlabeled dynamics that combines two principles. Firstly, we
combine the dynamics of different numbers of particles, choosing which
dynamics to follow based on the initial configuration’s particle number.
Secondly, we map the labeled notation $x=(x_{k})_{k=1}^{n}\in\mathbb{R}^{n}$
to $\iota_{n}(x)=\sum_{k=1}^{n}\delta_{x_{k}}$. These arguments are standard:
the first one follows easily, and the second can be interpreted as a Markov
mapping theorem. However, since this construction is essential for the
intertwining relations, we present it in detail for the reader’s benefit.

###### Proof of Lemma 2.2.

For each $n\in\mathbb{N}\cup{\left\\{\infty\right\\}}$, let
$(\Omega^{n},\mathcal{F}^{n},(Z_{t}^{n})_{t\geq 0}$,
$(\mathbb{P}_{x}^{n})_{x\in\mathbb{R}^{n}})$ be the Markov family that
describes dynamics of $n$ particles. The family consists of a measurable space
$(\Omega^{n},\mathcal{F}^{n})$, measurable mappings
$Z_{t}^{n}:\Omega\to\mathbb{R}^{n}$, $t\geq 0$, and probability measures
$\mathbb{P}_{x}^{n}$ on $(\Omega^{n},\mathcal{F}^{n})$ with corresponding
expected values denoted by $\mathbb{E}_{x}^{n}$. The Markov property is
satisfied with respect to the natural filtration
$\mathcal{F}_{t}^{n}:=\sigma(Z_{s}^{n}:0\leq s\leq t)$.

  * •

We define
$\Omega:={\left\\{0\right\\}}\cup\bigcup_{n\in\mathbb{N}\cup{\left\\{\infty\right\\}}}{\left\\{n\right\\}}\times\Omega_{n}$,
and equip it with the $\sigma$-algebra $\mathcal{F}$ generated by the sets
${\left\\{0\right\\}}$ and ${\left\\{n\right\\}}\times A_{n}$,
$A_{n}\in\mathcal{F}^{n}$, $n\in\mathbb{N}_{0}\cup{\left\\{\infty\right\\}}$

  * •

Put $\eta_{t}(n,\omega^{n}):=\iota_{n}(Z_{t}^{n}(\omega^{n}))$ for
$n\in\mathbb{N}$, $\omega^{n}\in\Omega^{n}$ and $\eta_{t}(0):=0$.

  * •

For every $\mu\in\mathbf{N}$, we choose a fixed $z(\mu)\in\mathbb{R}^{n}$ such
that $\iota_{n}(z(\mu))=\mu$, where $n=\mu(E)$. We then define
$\mathbb{P}_{\mu}$ to be the push-forward measure of $\mathbb{P}_{z(\mu)}^{n}$
under the mapping $\omega^{n}\mapsto(n,\omega^{n})$.

The distribution $\mathbb{P}_{\mu}$ relies on how the components of $z(\mu)$
are permuted, but according to the strong consistency property (2.3), the
distribution of $(\eta_{t})_{t\geq 0}$ under $\mathbb{P}_{\mu}$ remains
unchanged regardless of the choice of permutation. In particular, this
distribution is equal to the distribution of $(\iota_{n}(Z_{t}^{n}))_{t\geq
0}$ under $\mathbb{P}_{x}$, for all $x\in\mathbb{R}^{n}$ that satisfy
$\iota_{n}(x)=\mu$.

We now show that for $\mathcal{F}_{t}=\sigma(\eta_{s}:0\leq s\leq t)$ the
Markov property

$\displaystyle\mathbb{P}_{\mu}[\eta_{t+s}\in
A\mid\mathcal{F}_{s}]=\mathbb{P}_{\eta_{s}}\left[\eta_{t}\in
A\right]\qquad\mathbb{P}_{\mu}\text{-almost surely for all
}\mu\in\mathbf{N},A\in\mathcal{N},t,s\geq 0$

holds by applying the Markov property of the process $(Z_{t}^{n})_{t\geq 0}$.
Indeed, fix $A\in\mathcal{N}$, and $s,t\geq 0$,
$\mu=\sum_{k=1}^{n}\delta_{x_{k}}\in\mathbf{N}$, $x=(x_{k})_{k=1}^{n}$,
$n\in\mathbb{N}\cup{\left\\{\infty\right\\}}$ and $B\in\mathcal{F}_{s}$. We
use the abbreviations $\iota=\iota_{n}$, $Z_{t}=Z_{t}^{n}$,
$\mathbb{E}_{x}=\mathbb{E}_{x}^{n}$. Then, using the definition of
$\mathcal{F}_{s}$, we find a measurable set
$C\subset{\left\\{h:h:[0,t]\to\mathbf{N}\right\\}}$ such that
$\mathbbm{1}_{B}=\mathbbm{1}_{C}((\eta_{u})_{0\leq u\leq s})$. Therefore,

$\displaystyle\mathbb{E}_{\mu}\left[\mathbbm{1}_{A}(\eta_{t+s})\mathbbm{1}_{B}\right]$
$\displaystyle=\mathbb{E}_{x}\left[\mathbbm{1}_{A}(\iota(Z_{t+s}))\mathbbm{1}_{C}{\left({\left(\iota(Z_{u})\right)}_{0\leq
u\leq s}\right)}\right]$
$\displaystyle=\mathbb{E}_{x}\left[\mathbb{E}_{Z_{s}}\left[\mathbbm{1}_{A}(\iota(Z_{t}))\right]\mathbbm{1}_{C}{\left({\left(\iota(Z_{u})\right)}_{0\leq
u\leq
s}\right)}\right]=\mathbb{E}_{\mu}\left[\mathbb{E}_{\eta_{s}}\left[\mathbbm{1}_{A}(\eta_{t})\right]\mathbbm{1}_{B}\right]$

holds since $\mathbbm{1}_{C}{\left({\left(\iota_{n}(Z_{u})\right)}_{0\leq
u\leq s}\right)}$ is $\mathcal{F}_{s}^{n}$-measurable. The remaining
properties can be easily obtained. ∎

The proof of Proposition 4.3 relies solely on strong consistency (see Remark
2.1) and follows by linearity and by the definition of the factorial measure.

###### Proof of Proposition 4.3.

By definition, $(\eta_{t})_{t\geq 0}$ is conservative, so only consistency
requires a proof. Let $n\in\mathbb{N}\cup{\left\\{\infty\right\\}}$ be fixed
and consider $\mu=\sum_{k=1}^{n}\delta_{x_{k}}\in\mathbf{N}$, where
$x=(x_{k})_{k=1}^{n}$. Let $(\Omega^{n},\mathcal{F}^{n},(Z_{t}^{n})_{t\geq
0}$, $(\mathbb{P}_{x}^{n})_{x\in\mathbb{R}^{n}})$,
$Z_{t}^{n}=(Z_{k,t}^{n})_{k=1}^{n},t\geq 0$ denote the Markov family
associated with the $n$-particle dynamics as described in the proof of Lemma
2.2. By applying (2.7) twice, and using strong consistency and linearity, we
obtain consistency:

$\displaystyle\mathbb{E}_{\mu}\left[\int
F(\delta_{y_{1}}+\ldots+\delta_{y_{l}})\eta_{t}^{(l)}(\mathrm{d}(y_{1},\ldots,y_{l}))\right]$
$\displaystyle=\sum_{\begin{subarray}{c}i_{1},\ldots,i_{l}=1\\\ \text{pairwise
different}\end{subarray}}^{n}\mathbb{E}_{x}^{n}\left[F(\delta_{Z_{i_{1},t}^{n}}+\ldots,\delta_{Z_{i_{l},t}^{n}})\right]$
$\displaystyle=\sum_{\begin{subarray}{c}i_{1},\ldots,i_{l}=1\\\ \text{pairwise
different}\end{subarray}}^{n}\mathbb{E}_{\delta_{x_{i_{1}}}+\ldots+\delta_{x_{i_{l}}}}\left[F(\eta_{t})\right]$
$\displaystyle=\int\mathbb{E}_{\delta_{y_{1}}+\ldots+\delta_{y_{l}}}\left[F(\eta_{t})\right]\mu^{(l)}(\mathrm{d}(y_{1},\ldots,y_{l})).\qed$

###  5.2 Intertwining Relations

To begin, we present a proof for part (i) of Theorem 4.4. Recall the mapping
$\iota_{n}:\mathbb{R}^{n}\to\mathbf{N}$,
$(x_{1},\ldots,x_{n})\mapsto\delta_{x_{1}}+\ldots+\delta_{x_{n}}$. Note that

$\displaystyle{\left(\widetilde{u\otimes\mathbbm{1}}\right)}(x_{1},\ldots,x_{n+1})$
$\displaystyle=\frac{1}{(n+1)!}\int
u\>\mathrm{d}{\left(\delta_{x_{1}}+\ldots+\delta_{x_{n+1}}\right)}^{(n)}$
(5.1)

for $x_{1},\ldots,x_{n+1}\in\mathbb{R}$ and measurable
$u:\mathbb{R}^{n}\to\mathbb{R}$ using the notation
$\widetilde{u\otimes\mathbbm{1}}$ for the symmetrization of the function
$(u\otimes\mathbbm{1})(x_{1},\ldots,x_{n+1}):=u(x_{1},\ldots,x_{n})$. With
this notation, the consistency property for a finite number of particles can
be expressed in terms of $P_{t}^{[n]}$ by

$\displaystyle
P_{t}^{[n+1]}{\left(\widetilde{\varphi\otimes\mathbbm{1}}\right)}(x_{1},\ldots,x_{n+1})$
$\displaystyle=\frac{1}{(n+1)!}\int
P_{t}^{[n]}\varphi\>\mathrm{d}{\left(\delta_{x_{1}}+\ldots+\delta_{x_{n+1}}\right)}^{(n)}$
$\displaystyle={\left(\widetilde{(P_{t}^{[n]}\varphi)\otimes\mathbbm{1}}\right)}(x_{1},\ldots,x_{n+1})$
(5.2)

for bounded, symmetric $\varphi:\mathbb{R}^{n}\to\mathbb{R}$. To obtain this
expression, we use (5.1) for both $\varphi$ and $P_{t}^{[n]}\varphi$, and
apply the consistency condition (4.1).

The crucial steps in the following proof are as follows: first, we establish
the intertwining relation for functions in the smaller space $\mathcal{C}_{n}$
using the explicit formulas for the orthogonal polynomials (2.3) and (3.2).
Next, we extend this relation to all functions in $L^{2}(\lambda^{\otimes n})$
using an approximation argument. To accomplish this, we employ the moment
problem to demonstrate that $\pi_{\lambda}$ is invariant.

###### Proof of Theorem 4.4 (i).

We claim the following equation: for all $t\geq 0$, $l\in\mathbb{N}_{0}$ and
measurable $F:\mathbf{N}_{l+1}\to[0,\infty)$,

$\displaystyle\int\mathbb{E}_{\delta_{z_{1}}+\ldots+\delta_{z_{l}}+\delta_{y}}\left[F(\eta_{t})\right]\lambda(\mathrm{d}y)=\int\mathbb{E}_{\delta_{z_{1}}+\ldots+\delta_{z_{l}}}\left[F(\eta_{t}+\delta_{y})\right]\lambda(\mathrm{d}y)$
(5.3)

holds for $z=(z_{1},\ldots,z_{l})\in\mathbb{R}^{l}$ $\lambda^{\otimes
l}$-almost everywhere. The case where $l=0$ reads as follows:
$\int\mathbb{E}_{\delta_{y}}\left[F(\eta_{t})\right]\lambda(\mathrm{d}y)=\int
F(\delta_{y})\lambda(\mathrm{d}y)$.

To prove (5.3), we multiply both the right-hand side and the left-hand side of
the equation by an arbitrary measurable function
$\varphi:\mathbb{R}^{l}\to[0,\infty)$ and integrate with respect to
$\lambda^{\otimes l}$. Since both the right-hand side and the left-hand side
of (5.3) are symmetric in $(z_{1},\ldots,z_{l})$, it is sufficient to
integrate with symmetric functions $\varphi$. Let $t\geq 0$, $l\in\mathbb{N}$.
Using reversibility, we obtain

$\displaystyle\int\varphi(x_{1},\ldots,x_{l})\int\mathbb{E}_{\delta_{x_{1}}+\ldots+\delta_{x_{l}}+\delta_{y}}\left[F(\eta_{t})\right]\lambda(\mathrm{d}y)\lambda^{\otimes
l}(\mathrm{d}(x_{1},\ldots,x_{l}))$
$\displaystyle\int\widetilde{\varphi\otimes\mathbbm{1}}(x_{1},\ldots,x_{l},y)\mathbb{E}_{\delta_{x_{1}}+\ldots+\delta_{x_{l}}+\delta_{y}}\left[F(\eta_{t})\right]\lambda^{\otimes(l+1)}(\mathrm{d}(x_{1},\ldots,x_{l},y))$
$\displaystyle=\int
P_{t}^{[l+1]}{\left(\widetilde{\varphi\otimes\mathbbm{1}}\right)}(x_{1},\ldots,x_{l},y)F(\delta_{x_{1}}+\ldots+\delta_{x_{l}}+\delta_{y})\lambda^{\otimes(l+1)}(\mathrm{d}(x_{1},\ldots,x_{l},y)).$
(5.4)

Applying (5.2) and using reversibility once again, (5.2) can be transformed
into

$\displaystyle\int{\left(\widetilde{(P_{t}^{[l]}\varphi)\otimes\mathbbm{1}}\right)}(x_{1},\ldots,x_{l},y)F(\delta_{x_{1}}+\ldots+\delta_{x_{l}}+\delta_{y})\lambda^{\otimes(l+1)}(\mathrm{d}(x_{1},\ldots,x_{l},y))$
$\displaystyle\hskip 40.00006pt=\int
P_{t}^{[l]}\varphi(x_{1},\ldots,x_{l})\int
F(\delta_{x_{1}}+\ldots+\delta_{x_{l}}+\delta_{y})\lambda(\mathrm{d}y)\lambda^{\otimes
l}(\mathrm{d}(x_{1},\ldots,x_{l}))$ $\displaystyle\hskip
40.00006pt=\int\varphi(x_{1},\ldots,x_{l})\int\mathbb{E}_{\delta_{x_{1}}+\ldots+\delta_{x_{l}}}\left[F(\eta_{t}+\delta_{y})\right]\lambda(\mathrm{d}y)\lambda^{\otimes
l}(\mathrm{d}(x_{1},\ldots,x_{l})).$

which implies (5.3).

Let $\zeta$ be proper Poisson process with intensity measure $\lambda$.
Equation (5.3) enables us to prove the intertwining relation (2.12). Let
$f_{n}\in\mathcal{C}_{n}$ be symmetric and select
$F:\mathbf{N}_{n}\to\mathbb{R}$ such that $f_{n}=F\circ\iota$. Consequently,
by using (2.3) and consistency, we arrive at the following

$\displaystyle\mathbb{E}_{\zeta}\left[I_{n}f_{n}(\eta_{t})\right]$
$\displaystyle\hskip
40.00006pt=\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}\mathbb{E}_{\zeta}\bigg{[}\iint
f_{n}(x_{1},\ldots,x_{n})\lambda^{\otimes(n-k)}(\mathrm{d}(x_{k+1},\ldots,x_{n}))\eta_{t}^{(k)}(\mathrm{d}(x_{1},\ldots,x_{k}))\bigg{]}$
$\displaystyle\hskip
40.00006pt=\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}\iint\mathbb{E}_{\delta_{x_{1}}+\ldots+\delta_{x_{k}}}\left[F(\eta_{t}+\delta_{x_{k+1}}+\ldots+\delta_{x_{n}})\right]$
$\displaystyle\hskip
160.00024pt\lambda^{\otimes(n-k)}(\mathrm{d}(x_{k+1},\ldots,x_{n}))\zeta^{(k)}(\mathrm{d}(x_{1},\ldots,x_{k})).$

Using (5.3) repeatedly for $n-k$, we obtain

$\displaystyle\int\mathbb{E}_{\delta_{x_{1}}+\ldots+\delta_{x_{k}}}\left[F(\eta_{t}+\delta_{x_{k+1}}+\ldots+\delta_{x_{n}})\right]\lambda^{\otimes(n-k)}(\mathrm{d}(x_{k+1},\ldots,x_{n}))$
$\displaystyle\hskip
40.00006pt=\int\mathbb{E}_{\delta_{x_{1}}+\ldots+\delta_{x_{n}}}\left[F(\eta_{t})\right]\lambda^{\otimes(n-k)}(\mathrm{d}(y_{x+1},\ldots,x_{n}))$
$\displaystyle\hskip 40.00006pt=\int
P_{t}^{[n]}f_{n}(x_{1},\ldots,x_{n})\lambda^{\otimes(n-k)}(\mathrm{d}(x_{k+1},\ldots,x_{n}))$

for $\lambda^{\otimes k}$-almost all $(y_{1},\ldots,y_{k})$. Since
$\lambda^{\otimes k}$ is the $k$-th factorial moment measure of the Poisson
process with intensity measure $\lambda$, integrating with respect to the
$k$-th factorial measure of $\zeta$ is well-defined, resulting in

$\displaystyle\mathbb{E}_{\zeta}\left[I_{n}f_{n}(\eta_{t})\right]$
$\displaystyle\hskip 40.00006pt=\sum_{k=0}^{n}\binom{n}{k}(-1)^{n-k}\iint
P_{t}^{[n]}f_{n}(x_{1},\ldots,x_{n})\lambda^{\otimes(n-k)}(\mathrm{d}(x_{k+1},\ldots,x_{n}))\zeta^{\otimes
k}(\mathrm{d}(x_{1},\ldots,x_{k}))$ $\displaystyle\hskip
40.00006pt=I_{n}P_{t}^{[n]}f_{n}(\zeta)$

almost surely.

As a next step, we extend (2.12) to include $f_{n}\in
L^{2}_{\mathrm{sym}}(\lambda^{\otimes n})$. First, we claim that
$\pi_{\lambda}$ is invariant for $(\eta_{t})_{t\geq 0}$, i.e.,
$\mathbb{E}\left[\mathbb{E}_{\zeta}\left[\eta_{t}\in
A\right]\right]=\pi_{\lambda}(A)$ for each $A\in\mathcal{N}$. To show this, we
use the fact that the moment problem for the Poisson process is uniquely
solvable (as shown e.g. in [LP17, Proposition 4.12]). Therefore, it is enough
to check that the factorial moment measures of $\zeta$, given by
$\lambda^{\otimes n}$, and $\eta_{t}$ starting at $\zeta$ coincide. Since the
push-forward measure of $\lambda^{\otimes n}$ under $\iota_{n}$ is invariant
for $(\eta_{t})_{t\geq 0}$, we use consistency to obtain

$\displaystyle\mathbb{E}\left[\mathbb{E}_{\zeta}\left[(\eta_{t}(B))_{n}\right]\right]=\mathbb{E}\left[\int
P_{t}^{[n]}\mathbbm{1}_{B^{n}}\>\mathrm{d}\zeta^{(n)}\right]=\int
P_{t}^{[n]}\mathbbm{1}_{B^{n}}\>\mathrm{d}\lambda^{\otimes
n}=\int(\mu(B))_{n}\pi_{\lambda}(\mathrm{d}\mu)$

for bounded, measurable $B\subset\mathbb{R}$ and $t\geq 0$ which implies that
$\pi_{\lambda}$ is indeed invariant for $(\eta_{t})_{t\geq 0}$.

Thus, the expected value $\mathbb{E}_{\zeta}\left[F(\eta_{t})\right]$ is well-
defined for equivalence classes under $\pi_{\lambda}$. Moreover, we have

$\displaystyle\left\lVert\mathbb{E}_{\zeta}\left[F(\eta_{t})\right]\right\rVert_{L^{2}}\leq\left\lVert
F\right\rVert_{L^{2}(\pi_{\lambda})},\qquad F\in L^{2}(\pi_{\lambda})$ (5.5)

where $L^{2}:=L^{2}(\Omega,\mathcal{F},\mathbb{P})$ and
$(\Omega,\mathcal{F},\mathbb{P})$ denotes the probability space on which
$\zeta$ is defined. There exists a sequence $f_{n}^{k}\in\mathcal{C}_{n}$,
since $\mathcal{C}_{n}$ is dense in $L^{2}(\lambda^{\otimes n})$, such that
$f_{n}^{k}\to f_{n}^{k}$ as $k\to\infty$. Since $f_{n}$ is symmetric, we can
choose each $f_{n}^{k}$ to be symmetric as well. Using (5.5) and the
orthogonality relation (2.10), we obtain

$\displaystyle\left\lVert\mathbb{E}_{\zeta}\left[I_{n}f_{n}(\eta_{t})\right]-I_{n}P_{t}^{[n]}f_{n}(\zeta)\right\rVert_{L^{2}}$
$\displaystyle\hskip
40.00006pt\leq\left\lVert\mathbb{E}_{\zeta}\left[I_{n}f_{n}(\eta_{t})-I_{n}f_{n}^{k}(\eta_{t})\right]\right\rVert_{L^{2}}+\left\lVert
I_{n}P_{t}^{[n]}f_{n}(\zeta)-I_{n}P_{t}^{[n]}f_{n}^{k}(\zeta)\right\rVert_{L^{2}}$
$\displaystyle\hskip 40.00006pt\leq 2\sqrt{n}\left\lVert
f_{n}-f_{n}^{k}\right\rVert_{L^{2}(\lambda^{\otimes n})}\to 0,\qquad
k\to\infty,$

and therefore,
$\mathbb{E}_{\zeta}\left[I_{n}f_{n}(\eta_{t})\right]=I_{n}P_{t}^{[n]}f_{n}(\zeta)$
almost surely. ∎

Recall $(a)^{(k)}:=a(a+1)\cdots(a+k-1)$, $(a)^{(0)}:=1$, the _rising
factorial_ , and $(a)_{k}:=a(a-1)\cdots(a-k+1)$, $(a)_{0}:=1$, the _falling
factorial_.

###### Proof of Theorem 4.4 (ii).

The proof is analogous to the one for the Poisson case up to minor changes:
Equation (5.3) has an analogous form for the Pascal case, given by

$\displaystyle\int\mathbb{E}_{\delta_{z_{1}}+\ldots+\delta_{z_{l}}+\delta_{y}}\left[F(\eta_{t})\right]\kappa_{l+1,l}(z_{1},\ldots,z_{l},\mathrm{d}y)=\int\mathbb{E}_{\delta_{z_{1}}+\ldots+\delta_{z_{l}}}\left[F(\eta_{t}+\delta_{y})\right]\kappa_{l+1,l}(z_{1},\ldots,z_{l},\mathrm{d}y)$

for $t\geq 0$, $l\in\mathbb{N}_{0}$, measurable
$F:\mathbf{N}_{l+1}\to[0,\infty)$ and
$z=(z_{1},\ldots,z_{l})\in\mathbb{R}^{l}$ $\lambda_{l}$-almost everywhere.

Additionally, Proposition 3.4 establishes that the $n$-th factorial moment
measure of the Pascal process equals to
${\left(\frac{p}{1-p}\right)}^{n}\lambda_{n}$. We prove the unique solvability
of the moment problem using the criterion presented in [LP17, Proposition
4.12]. Fix a bounded, measurable set $B\subset\mathbb{R}$. Since
$\frac{1}{2^{n}n!}\alpha(B)^{(n)}$ as $n\to\infty$, there exists a $D\geq 1$
such that $\frac{1}{2^{n}n!}\alpha(B)^{(n)}\leq D$ for all $n\in\mathbb{N}$.
Therefore, we estimate the factorial moment measure as follows:

$\displaystyle{\left(\frac{p}{1-p}\right)}^{n}\lambda_{n}(B^{n})$
$\displaystyle={\left(\frac{p}{1-p}\right)}^{n}\alpha(B)^{(n)}\leq{\left(2D\frac{p}{1-p}\right)}^{n}n!.\qed$

###### Proof of Proposition 3.4.

Let $\zeta$ be a Pascal process with parameters $p$ and $\alpha$ and fix
$A=A_{1}^{d_{1}}\times\cdots\times A_{N}^{d_{N}}$ where
$A_{1},\ldots,A_{N}\subset\mathbb{R}$ are disjoint measurable sets and
$d_{1}+\ldots+d_{N}=n$. Firstly, note that (see e.g. [FJRW21, Equation (3.4)])

$\displaystyle\int\mathbbm{1}_{A}\>\mathrm{d}\mu^{(n)}=(\mu(A_{1}))_{d_{1}}\cdots(\mu(A_{N}))_{d_{N}}.$
(5.6)

Secondly, Lemma 5.2 below implies

$\displaystyle\int\mathbbm{1}_{A}\>\mathrm{d}\lambda_{n}=(\alpha(A_{1}))^{(d_{1})}\cdots(\alpha(A_{N}))^{(d_{N})}.$
(5.7)

By combining (5.6) and (5.7), and using the fact that the $k$-th factorial
moment of the negative Binomial distribution with parameters $p$ and $a$ is
${\left(\frac{p}{1-p}\right)}(a)^{(k)}$ we obtain
$\mathbb{E}\left[\int\mathbbm{1}_{A}\>\mathrm{d}\mu^{(n)}\right]={\left(\frac{p}{1-p}\right)}^{n}\lambda_{n}(A)$.
This equation can be extended to all measurable subsets of $\mathbb{R}^{n}$
using standard measure-theoretic arguments. ∎

###  5.3 Infinite-Dimensional Meixner Polynomials: an Explicit Formula

In this section, we prove Proposition 3.2. Our strategy for the proof is as
follows: on the one hand, as mentioned in Remark 3.3,
$\mathcal{M}^{p,\alpha}_{n}f_{n}$ defined by (3.2) for specific $f_{n}$
reduces to a product of univariate Meixner polynomials. We prove this property
in Proposition 5.3 below, for which we state two preliminary lemmas. On the
other hand, it is well-known (see [FJRW21, Proposition 5.3], see also [Lyt03b,
Lemma 3.1]) that the infinite-dimensional orthogonal polynomials defined by
the right-hand side of (3.3) have the same product structure for this $f_{n}$.
The equality then is obtained through standard measure-theoretical arguments.

Fix a partition of measurable sets of $B_{1},\ldots,B_{N}$ of $\mathbb{R}$,
$n<m$, $z_{1},\ldots,z_{n}\in\mathbb{R}$ and put
$c_{k}:={\left(\delta_{z_{1}}+\ldots+\delta_{z_{n}}\right)}(B_{k})$.

######  Lemma 5.1.

Let $i_{n+1},\ldots,i_{m}\in{\left\\{1,\ldots,N\right\\}}$ and put
$e_{k}:=\sum_{l=n+1}^{m}\mathbbm{1}_{{\left\\{i_{l}=k\right\\}}}$ for
$k\in{\left\\{1,\ldots,N\right\\}}$. Then,

$\displaystyle\int\mathbbm{1}_{B_{i_{n+1}}\times\ldots\times
B_{i_{m}}}(y_{n+1},\ldots,y_{m})\>\kappa_{m,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m}))=\prod_{k=1}^{N}{\left(\alpha(B_{k})+c_{k}\right)}^{(e_{k})}$

holds true.

Thereby, we put $\infty^{(k)}:=\infty$ for $k\geq 1$ and $\infty^{(0)}:=0$.

###### Proof.

We prove the equation by induction over $m$. For $m=n+1$ the statement is a
direct consequence of the definition of $\kappa_{n+1,n}$. Assume that the
statement is true for some fixed $m>n$. Let

$\displaystyle\sum_{l=n+1}^{m+1}\mathbbm{1}_{{\left\\{i_{l}=k\right\\}}}=e_{k}+\mathbbm{1}_{{\left\\{k=i_{m+1}\right\\}}}$

for an arbitrary $i_{m+1}\in{\left\\{1,\ldots,N\right\\}}$. Then, by (3.2),

$\displaystyle\int\mathbbm{1}_{B_{i_{n+1}}\times\ldots
B_{i_{m+1}}}(y_{n+1},\ldots,y_{m+1})\kappa_{m+1,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m+1}))$
$\displaystyle\hskip 40.00006pt=\int\mathbbm{1}_{B_{i_{n+1}}\times\ldots\times
B_{i_{m}}}(y_{n+1},\ldots,y_{m})\kappa_{m+1,m}(z_{1},\ldots,z_{n},y_{n+1},\ldots,y_{m},B_{i_{m+1}})$
$\displaystyle\hskip
210.00032pt\kappa_{m,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m}))$
$\displaystyle\hskip
40.00006pt=(\alpha(B_{i_{m+1}})+c_{i_{m+1}}+e_{i_{m+1}})\prod_{k=1}^{N}{\left(\alpha(B_{k})+c_{k}\right)}^{(e_{k})}=\prod_{k=1}^{N}{\left(\alpha(B_{k})+c_{k}\right)}^{(e_{k}+\mathbbm{1}_{{\left\\{k=i_{m+1}\right\\}}})}.\qed$

######  Lemma 5.2.

Let $d_{1},\ldots,d_{N}\in\mathbb{N}_{0}$ with $d_{1}+\ldots+d_{N}=m$. Then,

$\displaystyle\int\widetilde{\mathbbm{1}}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(z_{1},\ldots,z_{n},y_{n+1},\ldots,y_{m})\>\kappa_{m,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m}))$
$\displaystyle\hskip
40.00006pt=\frac{1}{(m)_{n}}\prod_{k=1}^{N}(d_{k})_{c_{k}}{\left(\alpha(B_{k})+c_{k}\right)}^{(d_{k}-c_{k})}$
(5.8)

holds where $\widetilde{\mathbbm{1}}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}$ denotes the symmetrization of
$\mathbbm{1}_{B_{1}^{d_{1}}\times\cdots\times B_{N}^{d_{N}}}$.

In case there exists a $k$ for which $d_{k}<c_{k}$, $(d_{k})_{c_{k}}$ becomes
zero, leading to the right-hand side of (5.2) being zero. Moreover, if there
exists a $k$ for which $d_{k}>c_{k}$ and $\alpha(B_{k})=\infty$, the right-
hand side of (5.2) is equal to infinity.

###### Proof.

We decompose the the integral into

$\displaystyle\int\widetilde{\mathbbm{1}}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(z_{1},\ldots,z_{n},y_{n+1},\ldots,y_{m})\kappa_{m,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m}))$
$\displaystyle\hskip
20.00003pt=\sum_{i_{n+1},\ldots,i_{m}\in{\left\\{1,\ldots,N\right\\}}}\int\mathbbm{1}_{B_{i_{n+1}}\times\ldots\times
B_{i_{m}}}(y_{n+1},\ldots,y_{m})\widetilde{\mathbbm{1}}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(z_{1},\ldots,z_{n},y_{n+1},\ldots,y_{m})$ $\displaystyle\hskip
160.00024pt\kappa_{m,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m})).$
(5.9)

If $x_{1},\ldots,x_{m}\in\mathbb{R}$ are given such that
$(\delta_{x_{1}}+\ldots+\delta_{x_{m}})(B_{k})=d_{k}$ for all
$k\in{\left\\{1,\ldots,N\right\\}}$, then the symmetrization
$\widetilde{\mathbbm{1}}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(x_{1},\ldots,x_{m})$ is equal to $\frac{d_{1}!\cdots
d_{N}!}{m!}$. Otherwise, it is equal to zero. Thus,
$\widetilde{\mathbbm{1}}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(z_{1},\ldots,z_{n},y_{n+1},\ldots,y_{m})>0$ is equivalent to
$\sum_{l=n+1}^{m}\mathbbm{1}_{{\left\\{i_{l}=k\right\\}}}+c_{k}=d_{k}$. By
applying Lemma 5.1 to (5.3), we obtain:

$\displaystyle\frac{d_{1}!\cdots
d_{N}!}{m!}\sum_{\begin{subarray}{c}i_{n+1},\ldots,i_{m}\in{\left\\{1,\ldots,N\right\\}}\\\
\sum_{l=n+1}^{m}\mathbbm{1}_{{\left\\{i_{l}=k\right\\}}}+c_{k}=d_{k}\end{subarray}}\int\mathbbm{1}_{B_{i_{n+1}}\times\ldots\times
B_{i_{m}}}(y_{n+1},\ldots,y_{m})\kappa_{m,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m}))$
$\displaystyle=\frac{d_{1}!\cdots
d_{N}!}{m!}\sum_{\begin{subarray}{c}i_{n+1},\ldots,i_{m}\in{\left\\{1,\ldots,N\right\\}}\\\
\sum_{l=n+1}^{m}\mathbbm{1}_{{\left\\{i_{l}=k\right\\}}}+c_{k}=d_{k}\end{subarray}}\prod_{k=1}^{N}{\left(\alpha(B_{k})+c_{k}\right)}^{(d_{k}-c_{k})}.$

Finally, using the identity

$\displaystyle\sum_{\begin{subarray}{c}i_{n+1},\ldots,i_{m}\in{\left\\{1,\ldots,N\right\\}}\\\
\sum_{l=n+1}^{m}\mathbbm{1}_{{\left\\{i_{l}=k\right\\}}}+c_{k}=d_{k}\end{subarray}}1=(m-n)!\prod_{k=1}^{N}\mathbbm{1}_{{\left\\{d_{k}\geq
c_{k}\right\\}}}\frac{1}{(d_{k}-c_{k})!},$

we conclude the proof. ∎

######  Lemma 5.3.

Let $d_{1},\ldots,d_{N}\in\mathbb{N}_{0}$ with $d_{1}+\ldots+d_{N}=m$ and
$\mu\in\mathbf{N}$. Assume $\mu(B_{k})<\infty$ and $0<\alpha(B_{k})<\infty$
for all $k$ with $d_{k}>0$. Then, the equation

$\displaystyle\mathcal{M}^{p,\alpha}_{m}\mathbbm{1}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(\mu)=\prod_{k=1}^{N}\mathscr{M}^{p,\alpha(A_{k})}_{d_{k}}{\left(\mu(B_{k})\right)}$

is satisfied.

###### Proof.

Let $b_{k}:=\mu(B_{k})\in\mathbb{N}_{0}$. By applying Lemma 5.1 and the
definition of the factorial measure $\mu^{(n)}$, we obtain

$\displaystyle\iint\widetilde{\mathbbm{1}}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(z_{1},\ldots,z_{n},y_{n+1},\ldots,y_{m})\>\kappa_{m,n}(z_{1},\ldots,z_{n},\mathrm{d}(y_{n+1},\ldots,y_{m}))\mu^{(n)}(\mathrm{d}(z_{1},\ldots,z_{n}))$
$\displaystyle\hskip
40.00006pt=\frac{n!}{(m)_{n}}\sum_{\begin{subarray}{c}c_{1},\ldots,c_{N}\in\mathbb{N}_{0}\\\
c_{1}+\ldots+c_{N}=n\end{subarray}}\prod_{k=1}^{N}\frac{(b_{k})_{c_{k}}}{c_{k}!}(d_{k})_{c_{k}}{\left(\alpha(B_{k})+c_{k}\right)}^{(d_{k}-c_{k})}.$

It follows that

$\displaystyle\mathcal{M}^{p,\alpha}_{m}\mathbbm{1}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}(\mu)$
$\displaystyle=\prod_{k=1}^{N}\sum_{c_{k}=d_{k}}^{\infty}{\left(1-\frac{1}{p}\right)}^{c_{k}-d_{k}}\frac{(b_{k})_{c_{k}}}{c_{k}!}(d_{k})_{c_{k}}{\left(\alpha(B_{k})+c_{k}\right)}^{(d_{k}-c_{k})}$
$\displaystyle=\prod_{k=1}^{N}\mathscr{M}^{p,\alpha(B_{k})}_{d_{k}}(\mu(B_{k})).\qed$

We define $\hat{\mathcal{M}}^{p,\alpha}_{n}f_{n}$ as the expression on the
right-hand side of (3.3). Recall that the monic, univariate Meixner
polynomials are given by (3.4).

###### Proof of Proposition 3.2.

Fix $m\in\mathbb{N}$ and a bounded set $B\subset\mathbb{R}$. We consider the
function

$\displaystyle f_{m}=\mathbbm{1}_{B_{1}^{d_{1}}\times\cdots\times
B_{N}^{d_{N}}}$ (5.10)

where $B_{1},\ldots,B_{N}$ are disjoint and measurable subsets of $B$,
$d_{1}+\ldots+d_{N}=m$, and $d_{1},\ldots,d_{N}\in\mathbb{N}$. We claim that
(3.3) holds true for the function $f_{m}$.

Indeed, if $0<\alpha(B_{k})$ for all $k$, then applying Lemma 5.3 and [FJRW21,
Proposition 5.3] (see also [Lyt03b, Lemma 3.1]) yields

$\displaystyle\hat{\mathcal{M}}^{p,\alpha}_{m}f_{m}(\mu)$
$\displaystyle=\prod_{k=1}^{N}\mathscr{M}_{d_{k}}^{p,\alpha(A_{k})}{\left(\mu(B_{k})\right)}=\mathcal{M}^{p,\alpha}_{m}f_{m}(\mu)$

for $\rho_{p,\alpha}$-almost all $\mu\in\mathbf{N}$.

On the other hand, if there exists a $k$ such that $\alpha(B_{k})=0$, then
$f_{m}=\mathbbm{1}_{B_{1}^{d_{1}}\times\cdots\times B_{N}^{d_{N}}}=0$
$\lambda_{m}$-almost everywhere. Therefore, using (3.2), Proposition 3.4 and
(3.6), we obtain

$\displaystyle\int{\left|\mathcal{M}^{p,\alpha}_{m}f_{m}\right|}\>\mathrm{d}\rho_{p,\alpha}\leq\sum_{k=0}^{m}\binom{m}{k}{\left(\frac{p}{1-p}\right)}^{m}\int\widetilde{f_{m}}\>\mathrm{d}\lambda_{m}=0,$

which implies $\mathcal{M}^{p,\alpha}_{m}f_{m}=0$ $\rho_{p,\alpha}$-almost
surely. Furthermore, $\hat{\mathcal{M}}^{p,\alpha}$ is well-defined on
$L^{2}(\lambda_{n})$, as shown by the orthogonality relation (3.7) (proved by
[FJRW21] or [Lyt03a]) for $\hat{\mathcal{M}}^{p,\alpha}$. Thus,
$\hat{\mathcal{M}}^{p,\alpha}f_{n}=0=\mathcal{M}^{p,\alpha}_{m}f_{m}$
$\rho_{p,\alpha}$-almost surely.

Using the _functional monotone class theorem_ (see e.g. [Bog07, Theorem
2.12.9.]) we obtain that (3.3) holds true for $f_{m}\in\mathcal{C}_{m}$ as
well. More precisely, let

$\displaystyle\mathcal{H}$
$\displaystyle={\left\\{g_{m}:B\to\mathbb{R}:f\text{ measurable, bounded,
}\mathcal{M}^{p,\alpha}_{m}\hat{g}_{m}=\hat{\mathcal{M}}^{p,\alpha}_{m}\hat{g}_{m}\right\\}}$

where $\hat{g}_{m}$ is defined to be equal to $g_{m}$ on $B$ and equal to zero
on $\mathbb{R}\setminus B$. Then, $\mathcal{H}$ contains the constant
functions and is closed with respect the formation of uniform and monotone
limits. Furthermore, the set

$\displaystyle\mathcal{H}_{0}$
$\displaystyle={\left\\{\mathbbm{1}_{C_{1}\times\cdots\times
C_{m}}:C_{1},\ldots,C_{m}\subset B\text{ measurable}\right\\}}$

is closed under forming products and is contained in $\mathcal{H}$, since
$\widetilde{\mathbbm{1}}_{C_{1}\times\cdots\times C_{m}}$ can be written as a
linear combination of symmetrizations of functions of the type (5.10).
Therefore, by the functional monotone class theorem, $\mathcal{H}$ equals the
set of all bounded, measurable functions on $B$. ∎

###  5.4 Reversible Measures for the $n$-Particle Dynamics

###### Proof of Proposition 2.5.

To show that $\lambda^{\otimes n}$ is reversible for the $n$-particle dynamics
where $\lambda$ denotes the Lebesgue measure, we use the following
straightforward computation. Let $n\in\mathbb{N}$ and $t>0$ be fixed. We
define $Y:=(Y_{k})_{k=1}^{n}$, where $Y_{k}:=\sqrt{1-a}B_{k,t}+\sqrt{a}B_{t}$
for $k\in{\left\\{1,\ldots,n\right\\}}$. By using the construction of
correlated Brownian motions given in (2.1), we obtain $X_{k,t}=Y_{k}+x_{k}$.
Then, using Fubini’s theorem, substitution, and the fact that $-Y$ is equal in
distribution to $Y$, we obtain

$\displaystyle\int\mathbb{E}\left[f_{n}(Y+x)\right]g_{n}(x)\lambda^{\otimes
n}(\mathrm{d}x)$
$\displaystyle=\mathbb{E}\left[\int_{\mathbb{R}^{n}}f_{n}(-Y+x)g_{n}(x)\lambda^{\otimes
n}(\mathrm{d}x)\right]$ $\displaystyle=\mathbb{E}\left[\int
f_{n}(x)g_{n}(x+Y)\lambda^{\otimes n}(\mathrm{d}x)\right]$ $\displaystyle=\int
f_{n}(x)\mathbb{E}\left[g_{n}(x+Y)\right]\lambda^{\otimes n}(\mathrm{d}x)$

for all $f_{n}$, $g_{n}\in\mathcal{C}_{n}$. ∎

As a next step, we prove Proposition 3.6. A reversible measure for a system of
$n$ ordered sticky Brownian motions is known, as established in [BW23, Theorem
4.17]. Starting from this reversible measure, we obtain that its
symmetrization is equal to the measure $\lambda_{n}$ up to a constant, which
is the factorial moment measure of the Pascal process up to a constant.

Let $n\in\mathbb{N}$ be fixed. We use the notation $\Sigma_{n}$ for the set of
_partitions_ of the set ${\left\\{1,\ldots,n\right\\}}$ and we define
$\Pi_{n}$ to be the set of _ordered partitions_ of
${\left\\{1,\ldots,n\right\\}}$, where

$\displaystyle\Pi_{n}:={\left\\{(a_{1},\ldots,a_{k}):a_{1},\ldots,a_{k}\in\mathbb{N},k\in\mathbb{N},a_{1}+\ldots+a_{k}=n\right\\}}.$

For each $\pi=(a_{1},\ldots,a_{k})\in\Pi_{n}$ and each function
$f_{n}:\mathbb{R}^{n}\to\mathbb{R}$, we define the function
$(f_{n})_{\pi}:\mathbb{R}^{k}\to\mathbb{R}$ by

$\displaystyle(x_{1},\ldots,x_{k})\mapsto
f_{n}(\underbrace{x_{1},\ldots,x_{1}}_{a_{1}\text{
times}},\ldots,\underbrace{x_{k},\ldots,x_{k}}_{a_{k}\text{ times}}).$

We define the measure $\lambda_{\pi}^{\geq}$ on $\mathbb{R}^{n}$ by

$\displaystyle\int
f_{n}\>\mathrm{d}\lambda_{\pi}^{\geq}:=\int\mathbbm{1}_{{\left\\{x_{1}\geq\ldots\geq
x_{k}\right\\}}}(f_{n})_{\pi}(x_{1},\ldots x_{k})\lambda^{\otimes
k}(\mathrm{d}(x_{1},\ldots,x_{k}))$

and put

$\displaystyle
m_{\theta}^{(n)}:=\sum_{\pi\in\Pi_{n}}\theta^{{\left|\pi\right|}-n}{\left(\prod_{A\in\pi}\right)}\frac{1}{{\left|A\right|}}\lambda_{\pi}^{\geq}.$

We consider the mapping
$\varphi:\mathbb{R}^{n}\to{\left\\{(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:x_{1}\geq\ldots\geq
x_{n}\right\\}}$, which orders the components of a vector in descending order.
Let $(\Omega^{n},\mathcal{F}^{n},(Z_{t}^{n})_{t\geq 0}$,
$(\mathbb{P}_{x}^{n})_{x\in\mathbb{R}^{n}})$ denote the Markov family
associated with the $n$-particle dynamics as described in the proof of Lemma
2.2, abbreviate $Z_{t}=Z_{t}^{n}$, $\mathbb{E}_{x}=\mathbb{E}_{x}^{n}$ and put
$Y_{t}:=\varphi(Z_{t})$.

We define the _symmetrization_ of $m_{\theta}^{(n)}$ by
$\widetilde{m}_{\theta}^{(n)}:=\frac{1}{n!}\sum_{s\in\mathfrak{S}_{n}}{\left(T_{s}\right)}_{\\#}m_{\theta}^{(n)}$
where ${\left(T_{s}\right)}_{\\#}m_{\theta}^{(n)}$ denotes the push-forward of
$m_{\theta}^{(n)}$ under the mapping
$T_{s}:(x_{1},\ldots,x_{n})\mapsto(x_{s(1)},\ldots,x_{s(n)})$.
$\mathfrak{S}_{n}$ denotes the set of permutations on
${\left\\{1,\ldots,n\right\\}}$.

###### Proof of Proposition 3.6.

To prove that the forward measure of the symmetrization
$\widetilde{m}_{\theta}^{(n)}$ under the mapping
$(x_{1},\ldots,x_{n})\mapsto\delta_{x_{1}}+\ldots+\delta_{x_{n}}$ is
reversible for unlabeled sticky Brownian motions $(\eta_{t})_{t\geq 0}$, it
suffices to verify

$\displaystyle\int(P_{t}^{[n]}f_{n})g_{n}\>\mathrm{d}\lambda_{n}=\int(P_{t}^{[n]}g_{n})f_{n}\>\mathrm{d}\lambda_{n}$

for symmetric $f_{n},g_{n}\in\mathcal{C}_{n}$. Indeed, using
$f_{n}\circ\varphi=f_{n}$ and
$\varphi_{\\#}\widetilde{m}_{\theta}^{(n)}=m_{\theta}^{(n)}$, we obtain

$\displaystyle\int
f_{n}(x)\mathbb{E}_{x}\left[g_{n}(Z_{t})\right]\widetilde{m}_{\theta}^{(n)}(\mathrm{d}x)$
$\displaystyle=\int
f_{n}(\varphi(x))\mathbb{E}_{x}\left[g_{n}(Z_{t})\right]\widetilde{m}_{\theta}^{(n)}(\mathrm{d}x)$
$\displaystyle=\int
f_{n}(\varphi(x))\mathbb{E}_{\varphi(x)}\left[g_{n}(Y_{t})\right]\widetilde{m}_{\theta}^{(n)}(\mathrm{d}x)$
$\displaystyle=\int
f_{n}(y)\mathbb{E}_{y}\left[g_{n}(Y_{t})\right]m_{\theta}^{(n)}(\mathrm{d}y).$
(5.11)

Thereby, in the second equation, we use the strong consistency property (2.3).
To be more precise, let $x\in\mathbb{R}^{n}$ be fixed and let
$s\in\mathfrak{S}_{n}$ be such that $T_{s}(x)=\varphi(x)$. By applying (2.3)
and using the fact that $g_{n}\circ T_{s}=g_{n}$, we obtain

$\displaystyle\mathbb{E}_{x}\left[g_{n}(Z_{t})\right]=\mathbb{E}_{x}\left[g_{n}(T_{s}(Z_{t}))\right]=\mathbb{E}_{T_{s}(x)}\left[g_{n}(Z_{t})\right]=\mathbb{E}_{\varphi(x)}\left[g_{n}(Y_{t})\right].$

Brockington and Warren demonstrated in [BW23, Theorem 4.17] that
$m_{\theta}^{(n)}$ is reversible for $n$ ordered uniform sticky Brownian
motions with stickiness $\theta$. As a result, (5.4) is symmetric in $f_{n}$
and $g_{n}$.

Hence, to complete the proof, it only remains to show that the symmetrization
of $m_{\theta}^{(n)}$ coincides with $\lambda_{n}$ up to a constant.
Specifically, we claim that
$\widetilde{m}_{\theta}^{(n)}=\frac{1}{\theta^{n}n!}\lambda_{n}$. To recall,
we construct $\lambda_{n}$ using the measure $\alpha:=\theta\lambda$ through
(3.5). Note that both $\widetilde{m}_{\theta}^{(n)}$ and $\lambda_{n}$ are
invariant under permutation of variables. Hence, it suffices to prove that
$\int f_{n}\>\mathrm{d}\widetilde{m}_{\theta}^{(n)}=\int
f_{n}\>\mathrm{d}\lambda_{n}$ holds for all symmetric functions
$f_{n}\in\mathcal{C}_{n}$.

By using the fact that the Lebesgue measure is diffuse, meaning that
$\lambda({\left\\{x\right\\}})=0$ for all $x\in\mathbb{R}$, we obtain for all
$\pi=(a_{1},\ldots,a_{k})\in\Pi_{n}$

$\displaystyle\int(f_{n})_{\pi}\>\mathrm{d}\lambda^{\otimes k}$
$\displaystyle=\sum_{s\in\mathfrak{S}_{k}}\int\mathbbm{1}_{{\left\\{x_{s(1)}\geq\ldots\geq
x_{s(k)}\right\\}}}(f_{n})_{\pi}(x_{1},\ldots,x_{k})\lambda^{\otimes
k}(\mathrm{d}(x_{1},\ldots,x_{k}))$
$\displaystyle=\sum_{s\in\mathfrak{S}_{k}}\int\mathbbm{1}_{{\left\\{x_{1}\geq\ldots\geq
x_{k}\right\\}}}(f_{n})_{\pi}(x_{s^{-1}(1)},\ldots,x_{s^{-1}(k)})\lambda^{\otimes
k}(\mathrm{d}(x_{1},\ldots,x_{k}))$
$\displaystyle=\sum_{s\in\mathfrak{S}_{k}}\int
f_{n}\>\mathrm{d}\lambda_{{\left(a_{s(1)},\ldots,s_{s(k)}\right)}}^{\geq},$

as $(f_{n})_{\pi}$ is symmetric.

We observe that every permutation $(a_{s(1)},\ldots,a_{s(k)})$ of an ordered
partition is itself an ordered partition. Using this fact along with (3.5), we
get

$\displaystyle\int f_{n}\>\mathrm{d}\widetilde{m}_{\theta}^{(n)}$
$\displaystyle=\int
f_{n}\>\mathrm{d}m_{\theta}^{(n)}=\sum_{\pi=(a_{1},\ldots,a_{k})\in\Pi_{n}}\theta^{k-n}\frac{1}{k!}\frac{1}{a_{1}\cdots
a_{k}}\int(f_{n})_{\pi}\>\mathrm{d}\lambda^{\otimes k}$
$\displaystyle=\frac{1}{n!}\sum_{\sigma={\left\\{B_{1},\ldots,B_{k}\right\\}}\in\Sigma_{n}}\theta^{k-n}({\left|B_{1}\right|}-1)!\cdots({\left|B_{k}\right|}-1)!\int(f_{n})_{\sigma}\lambda^{\otimes
k}=\frac{1}{\theta^{n}n!}\int f_{n}\>\mathrm{d}\lambda_{n}.$

We can switch from the summation over ordered partitions $\Pi_{n}$ to the
summation over partitions $\Sigma_{n}$, since the equality

$\displaystyle{\left|{\left\\{B_{1},\ldots,B_{k}\right\\}}\in\Sigma_{n}:{\left\\{{\left|B_{1}\right|},\ldots,{\left|B_{k}\right|}\right\\}}={\left\\{l_{1},\ldots,l_{k}\right\\}}\right|}$
$\displaystyle\hskip 40.00006pt=\frac{n!}{k!}\frac{1}{l_{1}!\cdots
l_{k}!}{\left|(a_{1},\ldots,a_{k})\in\Pi_{n}:{\left\\{a_{1},\ldots,a_{k}\right\\}}={\left\\{l_{1},\ldots,l_{k}\right\\}}\right|},$

holds for all $l_{1},\ldots,l_{k}\in\mathbb{N}$, $k\in\mathbb{N}$ with
$l_{1}+\ldots+l_{k}=n$. ∎

#### Acknowledgments

I thank Sabine Jansen, Frank Redig, Simone Floreani and Jan Philipp Neumann
for helpful discussions. This work was supported under Germany’s Excellence
Strategy - EXC-2111 - 390814868.

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