#  Intertwinings for continuum particle systems:
an algebraic approach

Simone Floreani  Institute for applied mathematics, University of Bonn, Bonn,
Germany<EMAIL_ADDRESS>,  Sabine Jansen  Mathematisches Institut,
Ludwig-Maximilians-Universität, 80333 München; Munich Center for Quantum
Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany.
<EMAIL_ADDRESS>and  Stefan Wagner  Mathematisches Institut, Ludwig-
Maximilians-Universität, 80333 München; Munich Center for Quantum Science and
Technology (MCQST), Schellingstr. 4, 80799 München, Germany.
<EMAIL_ADDRESS>
(Date: 15 November 2023)

###### Abstract.

We develop the algebraic approach to duality, more precisely to intertwinings,
within the context of particle systems in general spaces, focusing on the
$su(1,1)$ current algebra. We introduce raising, lowering, and neutral
operators indexed by test functions and we use them to construct unitary
operators, which act as self-intertwiners for some Markov processes having the
Pascal process’s law as a reversible measure. We show that such unitaries
relate to generalized Meixner polynomials. Our primary results are continuum
counterparts of results in the discrete setting obtained by Carinci,
Franceschini, Giardinà, Groenevelt, and Redig (2019).

_Keywords:_ algebraic approach to stochastic duality; intertwining; inclusion
process; Lie algebra $su(1,1)$; orthogonal polynomials.

_MSC2020:_ 60J25; 60K35; 82C22; 22E60

##  1\. Introduction

In recent years the algebraic approach to duality has been developed in the
context of interacting particle systems (see, e.g., [41] for a review).
Stochastic duality is a tool that connects two Markov processes via an
observable of both processes (the duality function): The relation then tells
that the expected evolution of such observable with respect to one Markovian
dynamics is equal to the expected evolution with respect to a second one, for
any time and initial conditions. When the two processes are the same one, we
speak of self-duality. Duality becomes relevant when one of the two processes
is easier to be studied and the duality function is a meaningful observable
for the other process. In the particular case of self-duality, the
simplification may arise by a simpler initial condition in the dual process.

Stochastic duality has been used in various contexts, such as interacting
particle systems (see, e.g., [27]), population genetics models (see, e.g.
[14]), (stochastic) partial differential equations (see, e.g., [33]). However,
finding duality relations is not easy. The algebraic approach provides a
structured way to find duality functions; for several processes, duality
relations have been found only via the algebraic approach (see, e.g., [10, 11,
23, 24]). So far, the algebraic approach to duality has been developed only in
the case of particles hopping on discrete spaces. In fact the very notion of
self-duality in the continuum is not trivial and only recently a
generalization for particles on a Polish space, where the concept of self-
duality is replaced by the one of self-intertwining, has been developed by the
authors together with F. Redig. In [15], we formulated self-intertwining
relations with respect to infinite-dimensional orthogonal and falling
factorial polynomials. In particular, we used the modern language of point
processes, see, e.g., [25], identifying the particle system with an evolving
counting measure and intriguing relations were found with the literature of
chaos decomposition and extended Fock spaces in infinite dimensions (see,
e.g., [29]).

In the present article we extend the algebraic approach to the continuum, in
the context of the $su(1,1)$ algebra. To that aim we first introduce raising,
lowering and neutral operators indexed by test functions rather than (lattice)
sites and use them to define a family of unitary operators $U(\xi,\theta)$,
indexed by parameters $\xi\in\mathbb{C}$ and $\theta\in\mathbb{R}$, in a
suitable $L^{2}$ space of finite counting measures (Section 2.3). The
reference measure $\rho_{p,\alpha}$ is the law of a _Pascal point process_ or
_negative binomial process_ (Section 2.2). It is the continuum counterpart to
the product of negative binomial laws, one for each lattice site, for
interacting particle systems on a lattice. Families of operators indexed by
functions are standard, e.g., for the canonical commutation relations for
bosons in quantum many-body mechanics or quantum field theory [35, Chapter
X.7]; they also appear in connection with current algebras and quantum
probability [1, 3]. We leave a thorough analysis of the algebraic setting to a
companion article [16] but mention already that our raising and lowering
operators are closely related to operators studied for infinite-dimensional
orthogonal polynomials [29] and to representations of the algebra of the
square of white noise, the current algebra of $sl(2,\mathbb{R})$, and the
finite difference algebra [2, 6, 40].

We prove that the unitaries $U(\xi,\theta)$ belong to the symmetry group of
consistent Markov processes that admit the law of the Pascal process as a
reversible measure (Theorem 2.1). Put differently, each of these unitaries is
a self-intertwiner for the semigroup: $U(\xi,\theta)P_{t}=P_{t}U(\xi,\theta)$.
_Consistency_ roughly means that random removal of a particle and time
evolution commute [9]. We treat only systems with finitely many particles.
Theorem 2.1 applies in particular to the generalized symmetric inclusion
process and thereby generalizes to the continuum Item 1(i) in Theorem 3.1 by
Carinci, Franceschini, Giardinà, Groenevelt and Redig [7].

For a concrete choice of the parameters $\xi$ and $\theta$, the unitary
$U(\xi,\theta)$ maps functions supported on $n$-particle configurations to
generalized Meixner polynomials of degree $n$ (Theorem 2.2), up to
proportionality constants. This result generalizes Item 1(ii) in [7, Theorem
3.1]. As a by-product to Theorems 2.1 and 2.2, we obtain a new—algebraic—proof
of a self-intertwining relation proven by F. Redig and us in [15], see
Corollary 2.3. Our results are complemented by some considerations on how to
rewrite infinitesimal generators with our raising, lowering and neutral
operators (Section 2.6).

We should emphasize that our method also applies to consistent Markov
processes that have a Poisson law as a reversible measure instead of a Pascal
law. The relevant algebra is the Heisenberg algebra, the raising, lowering and
neutral operators are replaced with creation, annihilation and number
operators, and Charlier polynomials take the place of Meixner polynomials. The
outcome is a generalization of Item 3 in Theorem 3.1 from Carinci et al. [7].
Creation and annihilation operators in the continuum are well known in the
context of many-body quantum mechanics [35]; this is why we focus our
presentation on $su(1,1)$.

In contrast, our method does not apply to the $SU(2)$ symmetry and Krawtchouk
polynomials relevant for exclusion processes [7]. We use a reference measure
on configurations with respect to which raising and lowering operators are
dual, and that reference measure should be infinitely divisible. Poisson and
negative binomial laws are infinitely divisible and thus have natural Lévy
processes or fields as continuum counterparts. Bernoulli and binomial laws are
not infinitely divisible and it is not clear what the associated continuum
random field should be.

The article is organized as follows. We introduce the space of finite counting
measures, the Pascal law, and the lowering, raising and neutral operators and
then we present our main results. All this is done in Section 2. In Section 3
we give a Papangelou kernel for the Pascal point process and apply it to prove
that raising and lowering operators are adjoint to each other. Theorems 2.1
and 2.2 are proven in Sections 4 and 5 respectively. Finally Appendix A
gathers some known formulas on univariate Meixner polynomials.

##  2\. Setting and main results

In the following Cartesian products are always equipped with product
$\sigma$-algebras. For finite interacting particle systems on lattices, the
configuration space is $\mathbb{N}_{0}^{\Lambda}$ with
$\Lambda\subset\mathbb{Z}^{d}$. We replace $\mathbb{N}_{0}^{\Lambda}$ with a
space of finite counting measures: Let $(E,\mathcal{E})$ be a Borel space, for
example, $E=\mathbb{R}^{d}$ or more generally a Polish space with its Borel
$\sigma$-algebra. Let $\mathbf{N}_{<\infty}$ be the space of finite counting
measures on $E$. Every element $\eta\in\mathbf{N}_{<\infty}$ is either zero or
a finite sum $\eta=\delta_{x_{1}}+\cdots+\delta_{x_{n}}$ of Dirac measures,
where $x_{1},\ldots,x_{n}\in E$, $n\in\mathbb{N}$. The space
$\mathbf{N}_{<\infty}$ is equipped with the $\sigma$-algebra
$\mathcal{N}_{<\infty}$ generated by the counting variables
$\eta\mapsto\eta(B)$, $B\in\mathcal{E}$. For background on point processes and
counting measures, we refer the reader to Last and Penrose [26].

###  2.1. Symmetric inclusion process

We are interested in Markov processes with state space $\mathbf{N}_{<\infty}$.
As a guiding example we take the process with formal generator

$Lf(\eta)=\iint\bigl{(}f(\eta-\delta_{x}+\delta_{y})-f(\eta)\bigr{)}c(x,y)\eta(\mathrm{d}x)\bigl{(}\alpha+\eta\bigr{)}(\mathrm{d}y)$
(1)

where $\alpha$ is a finite measure on $E$ and $c:E\times E\to\mathbb{R}_{+}$
is a bounded measurable function with $c(x,y)=c(y,x)$ on $E^{2}$. In [15] we
called this process _generalized inclusion process_. When $E$ is a finite set,
we may identify finite counting measures $\eta$ with vectors
$\boldsymbol{n}=(n_{x})_{x\in\Lambda}\in\mathbb{N}_{0}^{\Lambda}$, integrals
over $E$ turn into sums, and the generator turns into

$Lf(\boldsymbol{n})=\sum_{x,y\in
E}\bigl{(}f(\boldsymbol{n}-\boldsymbol{e}_{x}+\boldsymbol{e}_{y})-f(\boldsymbol{n})\bigr{)}c(x,y)n_{x}(\alpha_{y}+n_{y}),$
(2)

where $\boldsymbol{e}_{x}(y)=\delta_{x,y}$ and $\alpha_{y}=\alpha(\\{y\\})$.
This is the generator of the inhomogeneous _symmetric inclusion process_ (see,
e.g., [17, Eq. (2.2)]), which first appeared in the homogeneous case as a dual
process to a model for energy and momentum transport, see [19] and references
therein.

When $c(x,y)$ is the constant function with value $1$, the process corresponds
to the Moran process from mathematical population genetics—classical Moran
model [20, 32] when $E$ is finite, measure-valued Moran model (see [12,
Section 2.6] or [13, Section 5.4]) when $E$ is uncountable. The Moran model
describes a population in which individuals have genetic types $x\in E$ that
may mutate or evolve by sampling replacement (death of an individual followed
by replacement with the offspring of another individual).

For the homogeneous symmetric inclusion process with finite state space /
Moran model with finite type space, the correspondence was already noticed in
[8, Section 5]. For uncountable state spaces, we notice that the $n$-particle
dynamics of the process generated by (1) (with $c(x,y)\equiv 1$) coincides
with the $n$-particle Moran model [12, Eq. (2.5.2)] with so-called _mutation
operator_ $A\varphi(x)=\int(\varphi(y)-\varphi(x))\alpha(\mathrm{d}x)$. Our
process for counting measures is similar to the measure-valued Moran model.
The only difference is in bookkeeping: In mathematical population genetics,
the population is often modeled not with $\eta\in\mathbf{N}_{<\infty}$ but
rather with the empirical measure $\frac{\eta}{\eta(E)}$. This is helpful for
scaling limits in which the population size goes to infinity, notably Fleming-
Viot limits, which are important in population genetics (see [39] and
references therein).

###  2.2. Pascal point process

The symmetric inclusion process with generator (1) has a family of reversible
measures $\rho_{p,\alpha}$ indexed by $p\in(0,1)$ [15, Theorem 5.2] whose
definition we recall here. For $a\in\mathbb{R}$ and $n\in\mathbb{N}$, the
rising factorial (also known as Pochhammer symbol) is

$(a)_{0}=1,\quad(a)_{n}=a(a+1)\cdots(a+n-1).$

Let $p\in(0,1)$ and $\alpha$ be a finite measure on $E$. Let $\rho_{p,\alpha}$
be the law of the Pascal point process or negative binomial process (see,
e.g., [38, Section 2.7.], [22] and [15, Section 5.2]). That is,
$\rho_{p,\alpha}$ is the uniquely defined probability measure on
$\mathbf{N}_{<\infty}$ such that:

  * •

For all $B\in\mathcal{E}$, and $n\in\mathbb{N}_{0}$

$\rho_{p,\alpha}\bigl{(}\\{\eta:\eta(B)=n\\}\bigr{)}=(1-p)^{\alpha(B)}\frac{p^{n}}{n!}(\alpha(B))_{n}.$
(3)

  * •

For all $n\geq 2$ and all disjoint $B_{1},\ldots,B_{n}\in\mathcal{E}$, the
variables $\eta\mapsto\eta(B_{i})$, $i=1,\ldots,n$, are independent.

We work in the Hilbert space
$\mathcal{H}=L^{2}(\mathbf{N}_{<\infty},\mathcal{N}_{<\infty},\rho_{p,\alpha})$
of complex-valued square-integrable functions. The scalar product is

$\langle f,g\rangle=\int\overline{f}\,g\,\mathrm{d}\rho_{p,\alpha}.$

###  2.3. Representation of the $su(1,1)$ current algebra

Let $\mathcal{D}\subset\mathcal{H}$ be the set of bounded measurable functions
$F$ for which there exists a cutoff $m_{F}\in\mathbb{N}$ such that $F$ is
supported in $\\{\eta:\eta(E)\leq m_{F}\\}$. We define operators
$k^{\pm}(\varphi),k^{0}(\varphi):\mathcal{D}\to\mathcal{D}$ indexed by bounded
measurable functions $\varphi:E\to\mathbb{C}$ as follows:

$\displaystyle k^{+}(\varphi)F(\eta)$
$\displaystyle=\frac{1}{\sqrt{p}}\int\varphi(x)F(\eta-\delta_{x})\eta(\mathrm{d}x),$
$\displaystyle k^{-}(\varphi)F(\eta)$
$\displaystyle=\sqrt{p}\int\overline{\varphi(x)}F(\eta+\delta_{x})\bigl{(}\alpha+\eta)\bigr{(}\mathrm{d}x),$
$\displaystyle k^{0}(\varphi)F(\eta)$
$\displaystyle=F(\eta)\Bigl{(}\int\varphi\mathrm{d}\eta+\frac{1}{2}\int\varphi\mathrm{d}\alpha\Bigr{)}.$

for $\eta\in\mathbf{N}_{<\infty}$. The raising operator $k^{+}(\varphi)$ and
the lowering operator $k^{-}(\varphi)$ map functions supported in
$\\{\eta:\eta(E)=n\\}$ to functions supported in $\\{\eta:\eta(E)=n+1\\}$ and
$\\{\eta:\eta(E)=n-1\\}$ respectively. The indicator $\mathbbm{1}_{\\{0\\}}$
that there is no particle at all (_vacuum_) is annihilated by all lowering
operators: $k^{-}(\varphi)\mathbbm{1}_{\\{0\\}}=0$.

The following commutation relations can be checked by an explicit computation:

$\displaystyle[k^{-}(\varphi),k^{+}(\theta)]=2k^{0}(\overline{\varphi}\theta),$
$\displaystyle\quad[k^{0}(\theta),k^{+}(\varphi)]=k^{+}(\varphi\theta),\quad[k^{0}(\theta),k^{-}(\varphi)]=-k^{-}(\overline{\varphi}\theta),$

where $[T,S]:=TS-ST$. The factor $\sqrt{p}$ included in the definitions of
$k^{\pm}(\varphi)$ is irrelevant for the commutation relations but it matters
for the adjointness relation $\langle f,k^{+}(\varphi)g\rangle=\langle
k^{-}(\overline{\varphi})f,g\rangle$ proven in Lemma 3.2.

We leave a thorough analysis of the current algebra generated by the operators
$k^{\pm}(\varphi),k^{0}(\varphi)$ for another article (see [16]) and focus on
the operators associated with the constant function $\varphi=\mathbbm{1}$,
equal to $1$ on all of $E$. In Lemma 4.2 below we check that for every
$\xi\in\mathbb{C}$, the operator $\frac{1}{\mathrm{i}}\bigl{(}\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1}))$ with domain
$\mathcal{D}$ is closable with self-adjoint closure $A$. The operator
$\exp(\xi k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1}))$ is, by
definition, the unitary $\exp(\mathrm{i}A)$. For $\theta\in\mathbb{R}$, the
unitary $\exp(2\mathrm{i}\theta k^{0}(\mathbbm{1}))$ is defined as a
multiplication operator, it multiplies a function $f(\eta)$ with
$\exp(\mathrm{i}\theta(\alpha(E)+2\eta(E)))$. In this way we obtain a family
of unitary operators

$U(\xi,\theta)=\exp\Bigl{(}\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1})\Bigr{)}\exp\bigl{(}2\mathrm{i}\theta
k^{0}(\mathbbm{1})\bigr{)},\quad\xi\in\mathbb{C},\theta\in\mathbb{R}.$ (4)

For the definition of the unitary (4) it is essential that $\alpha$ has finite
total mass and that each configuration $\eta$ has finitely many particles.

###  2.4. Symmetries of consistent particle processes

Let $(P_{t})_{t\geq 0}$ be the semigroup of a Markov process with state space
$\mathbf{N}_{<\infty}$. For a measurable
$f:\mathbf{N}_{<\infty}\to\mathbb{R}_{+}$ let
$\mathcal{A}f:\mathbf{N}_{<\infty}\to\mathbb{R}_{+}$ be the function given by

$\mathcal{A}f(\eta)=\int f(\eta-\delta_{x})\eta(\mathrm{d}x).$

The semigroup is _consistent_ if $P_{t}\mathcal{A}f=\mathcal{A}P_{t}f$ for all
measurable $f:\mathbf{N}_{<\infty}\to\mathbb{R}_{+}$. Notice that, up to
questions of domains, $\mathcal{A}=\sqrt{p}\,k^{+}(\mathbbm{1})$. In
Proposition 4.1 below we show that if a process is consistent then it is
_conservative_ , i.e., the total number of particles is constant in time. In
view of that, we could normalize $\mathcal{A}$ by the total mass $\eta(E)$
(which is constant in time) and read consistency as the property that the
action of removing a particle uniformly at random commutes with the dynamics
[9, 15].

The following theorem generalizes item 1(i) in Theorem 3.1 in Carinci et al.
[7].

######  Theorem 2.1.

Let $(P_{t})_{t\geq 0}$ be the semigroup of a Markov process with state space
$\mathbf{N}_{<\infty}$. Let $p\in(0,1)$ and $\alpha$ be a finite measure on
$E$. Assume that the process is consistent, and admits the Pascal law
$\rho_{p,\alpha}$ as a reversible measure. Then $P_{t}$ commutes with all
unitaries $U(\xi,\theta)$ from (4):

$P_{t}U(\xi,\theta)f(\eta)=U(\xi,\theta)P_{t}f(\eta)$

for all $t\geq 0$, $f\in
L^{2}(\mathbf{N}_{<\infty},\mathcal{N}_{<\infty},\rho_{p,\alpha})$ and
$\rho_{p,\alpha}$-almost all $\eta\in\mathbf{N}_{<\infty}$.

The generalized symmetric inclusion process is a consistent Markov process and
admits the Pascal law $\rho_{p,\alpha}$ as a reversible measure (as proved in
[15, Theorem 5.2.]) and, thus, Theorem 2.1 applies.

###  2.5. Intertwining with Meixner polynomials

Let $\mathcal{P}_{n}$ be the the $L^{2}$-closure of the space of polynomials
of degree at most $n$ in the counting variables $\eta(B)$. Thus
$\mathcal{P}_{n}$ is the closed linear hull of monomials
$\eta\mapsto\eta(B_{1})^{n_{1}}\cdots\eta(B_{\ell})^{n_{\ell}}$ with
$n_{1}+\cdots+n_{\ell}\leq n$ and $B_{1},\ldots,B_{\ell}\in\mathcal{E}$. For
$f_{n}:E^{n}\to\mathbb{C}$ a bounded measurable function, let $I_{n}(f_{n})\in
L^{2}(\mathbf{N}_{<\infty},\mathcal{N}_{<\infty},\rho_{p,\alpha})$ be given by
$\eta\mapsto\int f_{n}\mathrm{d}\eta^{\otimes n}$ minus the orthogonal
projection of the latter onto $\mathcal{P}_{n-1}$. The orthogonalized version
of

$\eta(B_{1})^{n_{1}}\cdots\eta(B_{\ell})^{n_{\ell}}$

with disjoint $B_{1},\ldots,B_{\ell}$ and $n_{1}+\cdots+n_{\ell}=n$, is a
product of univariate Meixner polynomials with leading coefficient $1$:

$I_{n}\bigl{(}\mathbbm{1}_{B_{1}}^{\otimes
n_{1}}\otimes\cdots\otimes\mathbbm{1}_{B_{\ell}}^{\otimes
n_{\ell}}\bigr{)}(\eta)=\prod_{i=1}^{\ell}\mathcal{M}_{n_{i}}\bigl{(}\eta(B_{i});\alpha(B_{i}),p\bigr{)}$
(5)

for $\rho_{p,\alpha}$-almost all $\eta$. See, e.g., Lytvynov [29, Lemma 3.1]
or [15, Prop. 5.3]. The definition of univariate Meixner polynomials is
recalled in Appendix A. The letter $I_{n}$ reflects an analogy with Poisson-
Charlier polynomials and multiple stochastic integrals with respect to
compensated Poisson random measures [26, Chapter 12].

Our second main theorem generalizes item 1(ii) in Theorem 3.1 in Carinci et
al. [7]. For $f:\mathbf{N}_{<\infty}\to\mathbb{C}$ we define a sequence of
functions $(f_{n})_{n\in\mathbb{N}}$ by

$f_{n}(x_{1},\ldots,x_{n})=f(\delta_{x_{1}}+\cdots+\delta_{x_{n}}\bigr{)}.$

Each function $f_{n}$ is _symmetric_ , i.e., its value stays the same when the
variables are permuted. The inverse of the hyperbolic tangent is denoted
$\mathrm{artanh}$.

######  Theorem 2.2.

Let $\xi=\mathrm{artanh}\,\sqrt{p}$ and $U=\exp(\xi k^{+}(\mathbbm{1})-\xi
k^{-}(\mathbbm{1}))$. Then for all $n\in\mathbb{N}$ and all bounded measurable
$f:\mathbf{N}_{<\infty}\to\mathbb{C}$ supported in $\\{\eta:\eta(E)=n\\}$, we
have

$Uf=(1-p)^{\alpha(E)/2}\,\frac{1}{n!}\,(1-p)^{n}I_{n}(f_{n}).$

Furthermore, $U\mathbbm{1}_{\\{0\\}}=(1-p)^{\alpha(E)/2}\,\mathbbm{1}$.

Combining Theorem 2.1 and Theorem 2.2, we obtain a new proof of an
intertwining relation proven by other means in [15], see Theorems 3.15 and 5.2
therein. Let $(p_{t}^{(n)})_{t\geq 0}$ be a semigroup on symmetric bounded and
measurable functions from $E^{n}$ to $\mathbb{C}$ that satisfies

$(P_{t}f)(\delta_{x_{1}}+\cdots+\delta_{x_{n}})=(p_{t}^{(n)}f_{n})(x_{1},\ldots,x_{n}).$

for all measurable bounded $f:\mathbf{N}_{<\infty}\to\mathbb{C}$ and
$x_{1},\ldots,x_{n}\in E$.

######  Corollary 2.3.

Under the assumptions of Theorem 2.1: For all $t\geq 0$, all $n\in\mathbb{N}$,
and every bounded measurable $f_{n}:E^{n}\to\mathbb{C}$,

$P_{t}I_{n}(f_{n})=I_{n}(p_{t}^{(n)}f_{n}).$

###### Proof.

Apply Theorem 2.1 to $U(\xi,\theta)$ for $\xi=\mathrm{artanh}\,\sqrt{p}$ and
$\theta=0$, combine with Theorem 2.2, and observe that $n$-dependent constants
may be dropped because the process is conservative (see Proposition 4.1
below). ∎

Let us briefly remark on Markov processes with infinitely many particles.
Usually the duality is between on the one hand a process with possibly
infinitely many particles and on the other hand the same process but with
finitely many particles [27, Chapter III.4]. Accordingly our unitary self-
intertwining unitary $U$ is replaced with a unitary operator from a Hilbert
space for _finitely_ many particles onto a Hilbert space for infinitely many
particles. The first Hilbert space may be chosen as an extended Fock space,
the second Hilbert space is still an $L^{2}$ space with respect to a Pascal
law with infinite mass $\alpha(E)=\infty$, and the unitary operator is the
isomorphism from chaos decompositions for the Pascal point process [28,
Corollary 5.3]. Such an intertwining relation is proven for sticky Brownian
motions in [43, Theorem 3.5].

###  2.6. Algebraic expression for the generator

On the lattice the generator (2) of the symmetric inclusion process can be
expressed directly in terms of raising and lowering operators associated with
lattice sites [18]: Suppose that $\alpha_{y}=\alpha$ does not depend on $y$.
Let $\boldsymbol{e}_{x}\in\mathbb{N}_{0}^{E}$ be the vector given by
$\boldsymbol{e}_{x}(y)=\delta_{x,y}$. Set

$\displaystyle
k_{x}^{+}f(\boldsymbol{n})=\frac{1}{\sqrt{p}}\,n_{x}f(\boldsymbol{n}-\boldsymbol{e}_{x}),\quad
k_{x}^{-}f(\boldsymbol{n})=\sqrt{p}\,(\alpha+n_{x})f(\boldsymbol{n}+\boldsymbol{e}_{x}),$
$\displaystyle
k_{x}^{0}f(\boldsymbol{n})=\Bigl{(}\frac{\alpha}{2}+n_{x}\Bigr{)}f(\boldsymbol{n}).$

(Our operators $k^{\\#}(\varphi)$ correspond to
$k^{\\#}(\varphi)=\sum_{x}\varphi(x)k_{x}^{\\#}$ for $\\#=+,0$ and
$k^{-}(\varphi)=\sum_{x}\overline{\varphi(x)}k_{x}^{-}$.) Then

$\displaystyle L$ $\displaystyle=\sum_{\begin{subarray}{c}x,y\in E\\\ x\neq
y\end{subarray}}c(x,y)\Bigl{(}k_{x}^{+}k_{y}^{-}+k_{x}^{-}k_{y}^{+}-2k_{x}^{0}k_{y}^{0}+\frac{\alpha^{2}}{2}I\Bigr{)}$
$\displaystyle=\sum_{x,y\in
E}c(x,y)\Bigl{(}k_{x}^{+}k_{y}^{-}+k_{x}^{-}k_{y}^{+}-2k_{x}^{0}k_{y}^{0}+\left(\frac{\alpha^{2}}{2}-\alpha\mathbbm{1}_{\\{x=y\\}}\right)I\Bigr{)}$
(6)

where $I$ is the identity operator. The rewrite is a key ingredient to the
algebraic approach for duality: Dual processes such as the Brownian energy
process have the same algebraic expression but for a different representation
of the $su(1,1)$ algebra [18, Section 6.2].

In the continuum Eq. (2.6) is problematic as we cannot define operators
$k_{x}^{\\#}$—in the lingo of quantum field theory, $x\mapsto k_{x}^{\\#}$
only make sense as operator-valued _distributions_ [42, Chapter 3.7]. However,
using the operators $k^{\\#}(\varphi)$ introduced above, Eq. (2.6) generalizes
nicely, when $c(x,y)$ is of product form $c(x,y)=2\varphi(x)\varphi(y)$. In
that case

$L=k^{+}(\varphi)k^{-}(\varphi)+k^{-}(\varphi)k^{+}(\varphi)-2k^{0}(\varphi)^{2}+\Bigl{(}\frac{1}{2}\Bigl{(}\int\varphi\mathrm{d}\alpha\Bigr{)}^{2}-\int\varphi^{2}\mathrm{d}\alpha\Bigr{)}I.$
(7)

This expression makes perfect sense in the continuum setup as well. Moreover,
it can readily be checked that (7) holds if $L$ is the formal generator (1) of
the generalized symmetric inclusion process as well. A similar rewrite is
possible if $c(x,y)$ is a linear combination of symmetrized products
$\varphi_{1}(x)\varphi_{2}(y)+\varphi_{2}(x)\varphi_{1}(y)$. More generally,
say on $E=\mathbb{R}^{d}$ with homogeneous measures
$\alpha(\mathrm{d}x)=\alpha\mathrm{d}x$, one may hope to give meaning to
expressions like

$L=\int
c(x,y)\Bigl{(}k_{x}^{+}k_{y}^{-}+k_{y}^{+}k_{x}^{-}-2k_{x}^{0}k_{y}^{0}+\frac{\alpha^{2}}{2}-\alpha\mathbbm{1}_{\\{x=y\\}}\Bigr{)}\,\mathrm{d}x\,\mathrm{d}y$

by first defining it as (7) and its siblings when $c(x,y)$ is of product form
or a linear combination of symmetrized products, and second by approximating
sufficiently nice functions $c(x,y)$ by linear combinations of symmetrized
products. We leave as an open problem to carry out this program, to clarify
its usefulness for Markov processes, and to figure out its relation to
Hamilton operators $\int\mu(k)a(k)^{\dagger}a(k)\,\mathrm{d}k$ in quantum
field theory and quantum many-body theory (see, e.g., the non-numbered
proposition preceding Theorem X.45 in Reed and Simon [35] or Talagrand [42,
Chapter 3.7]).

##  3\. Papangelou kernel for the Pascal point process

If $X$ is a negative binomial variable with parameters $p\in(0,1)$ and $a>0$,
then

$(n+1)\mathbb{P}(X=n+1)=(1-p)^{a}\,\frac{(a)_{n+1}}{n!}p^{n+1}=p(a+n)\mathbb{P}(X=n),$

for all $n\in\mathbb{N}_{0}$, accordingly

$\mathbb{E}\bigl{[}Xf(X)\bigr{]}=\mathbb{E}\bigl{[}p(a+X)f(X+1)\bigr{]}$

for all $f:\mathbb{N}_{0}\to\mathbb{R}_{+}$. The following proposition gives
the analogous property for Pascal point processes. For future reference we
state and prove the proposition for $\sigma$-finite measures $\alpha$,
allowing for configurations with infinitely many particles. Thus, let
$\mathbf{N}$ be the space of $s$-finite counting measures on $E$. As $E$ is a
Borel space, $\mathbf{N}$ consists of the measures $\eta$ on $E$ that are
finite or countable sums of Dirac measures. The space $\mathbf{N}$ is equipped
with the $\sigma$-algebra $\mathcal{N}$ generated by the counting variables
$B\mapsto\eta(B)$, $B\in\mathcal{E}$.

A _Pascal point process_ with parameters $p$ and $\alpha$ is a random variable
$\eta:(\Omega,\mathcal{F},\mathbb{P})\to(\mathbf{N},\mathcal{N})$ that
satisfies the properties listed in the definition of $\rho_{p,\alpha}$:
Counting variables for disjoint regions $B_{i}$ are independent, and each
counting variable $\eta(B)$ has a negative binomial distribution with
parameters $p$ and $\alpha(B)$. For sets with infinite mass
$\alpha(B)=\infty$, the counting variable $\eta(B)$ is almost surely infinite.
Pascal point processes have Laplace functional

$\displaystyle\mathbb{E}\bigl{[}\mathrm{e}^{-\int f\mathrm{d}\eta}\bigr{]}$
$\displaystyle=\exp\Bigl{(}-\int\log\Bigl{(}\frac{1-p\,\mathrm{e}^{-f(x)}}{1-p}\Bigr{)}\alpha(\mathrm{d}x)\Bigr{)}$
$\displaystyle=\exp\Bigl{(}-\sum_{j=1}^{\infty}\int\bigl{(}1-\mathrm{e}^{-jf(x)}\bigr{)}\frac{p^{j}}{j}\alpha(\mathrm{d}x)\Bigr{)}.$

for measurable $f:E\to\mathbb{R}_{+}$.

######  Proposition 3.1.

Fix $p\in(0,1)$ and a $\sigma$-finite measure $\alpha$ on $E$. Let $\eta$ be a
Pascal point process with parameters $p$ and $\alpha$. Then

$\mathbb{E}\Bigl{[}\int
F(x,\eta)\eta(\mathrm{d}x)\Bigr{]}=\mathbb{E}\Bigl{[}\int
F(x,\eta+\delta_{x})p(\alpha+\eta)(\mathrm{d}x)\Bigr{]}$

for all measurable $F:E\times\mathbf{N}\to\mathbb{R}_{+}$.

The proposition says that the kernel $E\times\mathcal{N}\to\mathbb{R}_{+}$,
$(x,B)\mapsto\kappa(x,B)=p(\alpha(B)+\eta(B))$ is a _Papangelou kernel_ for
the Pascal point process. For a general discussion on Papangelou kernels, we
refer to [30] or [34, Section 1.2.2].

###### Proof.

Pascal point processes are compound Poisson; we deduce the proposition from
the Mecke formula for Poisson point processes. Let
$\Pi=\sum_{i}\delta_{(X_{i},N_{i})}$ be a Poisson point process on
$E\times\mathbb{N}_{0}$ with intensity measure
$\beta=\alpha\otimes(\sum_{n=1}^{\infty}\frac{p^{n}}{n}\delta_{n})$. Then
$\xi=\sum_{i}N_{i}\delta_{X_{i}}$ is a Pascal point process with law
$\rho_{p,\alpha}$. The Mecke equation for $\Pi$ [26, Theorem 4.1] and
measurable
$G:(E\times\mathbb{N}_{0})\times\mathbf{N}(E\times\mathbb{N}_{0})\to\mathbb{R}_{+}$
gives

$\mathbb{E}\Bigl{[}\sum_{i}G\bigl{(}(X_{i},N_{i}),\Pi\bigr{)}\Bigr{]}=\sum_{n=1}^{\infty}\frac{p^{n}}{n}\int\mathbb{E}\Bigl{[}G\bigl{(}(x,n),\delta_{(x,n)}+\Pi\bigr{)}\Bigr{]}\alpha(\mathrm{d}x).$

Let $\varphi:E\times\mathbf{N}(E)\to\mathbb{R}_{+}$ be a measurable function.
We apply the Mecke equation for $\Pi$ to
$G((x,n),\sum_{i}\delta_{(x_{i},n_{i})}):=n\varphi(x,\sum_{i}n_{i}\delta_{x_{i}})$
and get

$\mathbb{E}\Bigl{[}\sum_{i}N_{i}\varphi\bigl{(}X_{i},\xi\bigr{)}\Bigr{]}=\sum_{n=1}^{\infty}p^{n}\int\mathbb{E}\Bigl{[}\varphi(x,\xi+n\delta_{x})\Bigr{]}\alpha(\mathrm{d}x).$
(8)

Using (8) for $\varphi(x,\mu)=F(x,\mu+\delta_{x})$, we obtain

$\displaystyle
p\,\mathbb{E}\Bigl{[}\sum_{i}N_{i}F\bigl{(}X_{i},\xi+\delta_{X_{i}}\bigr{)}\Bigr{]}$
$\displaystyle=p\sum_{n=1}^{\infty}p^{n}\int\mathbb{E}\Bigl{[}F(x,\xi+(n+1)\delta_{x})\Bigr{]}\alpha(\mathrm{d}x)$
$\displaystyle=\sum_{n=2}^{\infty}p^{n}\int\mathbb{E}\Bigl{[}F(x,\xi+n\delta_{x})\Bigr{]}\alpha(\mathrm{d}x).$
(9)

We apply (8) to $\varphi(x,\mu)=F(x,\mu)$ and eliminate the sum over $n\geq 2$
using (9). This gives

$\mathbb{E}\Bigl{[}\sum_{i}N_{i}\varphi\bigl{(}X_{i},\xi\bigr{)}\Bigr{]}=p\int\mathbb{E}\Bigl{[}F(x,\xi+\delta_{x})\Bigr{]}\alpha(\mathrm{d}x)+p\,\mathbb{E}\Bigl{[}\sum_{i}N_{i}F\bigl{(}X_{i},\xi+\delta_{X_{i}}\bigr{)}\Bigr{]}.$

Equivalently,

$\mathbb{E}\Bigl{[}\int
F(x,\xi)\xi(\mathrm{d}x)\Bigr{]}=p\,\mathbb{E}\Bigl{[}\int
F(x,\xi+\delta_{x})(\alpha+\xi)(\mathrm{d}x)\Bigr{]}$

and the proof is complete as $\xi$ and $\eta$ are equal in distribution. ∎

Turning back to the law $\rho_{p,\alpha}$ on $\mathbf{N}_{<\infty}$ for finite
measures $\alpha$, we obtain

$\int\Bigl{(}\int
F(x,\eta)\eta(\mathrm{d}x)\Bigr{)}\rho_{p,\alpha}(\mathrm{d}x)=\int\Bigl{(}\int
F(x,\eta+\delta_{x})p(\alpha+\eta)(\mathrm{d}x)\Bigr{)}\rho_{p,\alpha}(\mathrm{d}x)$
(10)

for all measurable $F:E\times\mathbf{N}_{<\infty}\to\mathbb{R}_{+}$, hence
also all bounded measurable functions $F$ (here we use that the expected total
number of particles is finite when $\alpha$ has finite total mass).

######  Lemma 3.2.

For all bounded measurable $\varphi:E\to\mathbb{C}$ and all
$f,g\in\mathcal{D}$,

$\langle f,k^{+}(\varphi)g\rangle=\langle
k^{-}(\overline{\varphi})f,g\rangle.$

###### Proof.

Let us denote integration with respect to $\rho_{p,\alpha}$ as
$\mathbb{E}[f(\eta)]=\int f\mathrm{d}\rho_{p,\alpha}$. We apply Eq. (10) to
$F(x,\eta)=\varphi(x)\overline{f(\eta+\delta_{x})}g(\eta)$ and obtain

$\displaystyle\langle k^{-}(\overline{\varphi})f,g\rangle$
$\displaystyle=\frac{1}{\sqrt{p}}\mathbb{E}\Bigl{[}\int\varphi(x)\overline{f(\eta+\delta_{x})}g(\eta)\eta(\mathrm{d}x)\Bigr{]}$
$\displaystyle=\mathbb{E}\Bigl{[}\int\varphi(x)\overline{f(\eta)}g(\eta-\delta_{x})\sqrt{p}\,(\alpha+\eta)(\mathrm{d}x)\Bigr{]}$
$\displaystyle=\langle k^{+}(\varphi)f,g\rangle.\qed$

##  4\. Symmetries. Proof of Theorem 2.1

Before we turn to the proof of Theorem 2.1, we shows that consistency implies
conservativity, which is of interest in its own.

######  Proposition 4.1.

Every consistent Markov semigroup is conservative.

###### Proof.

Let $(P_{t})_{t\geq 0}$ be Markov semigroup on
$(\mathbf{N}_{<\infty},\mathcal{N}_{<\infty})$ that is consistent, i.e.,
$P_{t}\mathcal{A}f=\mathcal{A}P_{t}f$ for all $t\geq 0$ and all measurable
$f:\mathbf{N}_{<\infty}\to\mathbb{R}_{+}$. Then, for all $k\in\mathbb{N}$,

$P_{t}\mathcal{A}^{k}\mathbbm{1}=\mathcal{A}^{k}P_{t}\mathbbm{1}=\mathcal{A}^{k}\mathbbm{1}.$
(11)

A straightforward computation shows that $\mathcal{A}^{k}\mathbbm{1}$ is a
descending factorial power of the total number of particles,

$\bigl{(}\mathcal{A}^{k}\mathbbm{1}\bigr{)}(\eta)=\eta(E)\bigl{(}\eta(E)-1\bigr{)}\cdots\bigl{(}\eta(E)-k+1\bigr{)},$
(12)

for all $\eta\in\mathbf{N}_{<\infty}$. Let $(\eta_{t})_{t\geq 0}$ be a Markov
process with semigroup $(P_{t})_{t\geq 0}$ and deterministic initial condition
$\eta_{0}=\mu\in\mathbf{N}_{<\infty}$, defined on some probability space
$(\Omega,\mathscr{F},\mathbb{P}_{\mu})$. Eqs. (11) and (12) show that under
$\mathbb{P}_{\mu}$, at all times $t\geq 0$, the total number of particles
$\eta_{t}(E)$ has the same factorial moments as $\eta_{0}(E)$. The factorial
moments of $\eta_{0}(E)=\mu(E)$ (hence $\eta_{t}(E)$) vanish for
$k\geq\mu(E)+1$, therefore the moment problem is uniquely solvable and we
conclude $\eta_{t}(E)=\eta_{0}(E)=\mu(E)$, $\mathbb{P}_{\mu}$-almost surely,
and the process is conservative. ∎

Remember that the set $\mathcal{D}\subset\mathcal{H}$ consists of the bounded
measurable functions $f$ supported in $\\{\eta:\eta(E)\leq n_{f}\\}$ for some
$n_{f}<\infty$.

######  Lemma 4.2.

For every $\xi\in\mathbb{C}$, the operator $\frac{1}{\mathrm{i}}\bigl{(}\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1}))$ with domain
$\mathcal{D}$ is closable and its closure is self-adjoint.

Put differently, the operator is _essentially self-adjoint_ on $\mathcal{D}$
and the latter is a _core_ for the self-adjoint closure [36, Chapter VIII.2].

###### Proof.

The operator $A=\frac{1}{\mathrm{i}}(\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1}))$ is similar to the Segal
field operators for free quantum fields, we adapt the proof of essential self-
adjointness for the latter from Reed and Simon [35, Theorem X.41(a)]. The
domain $\mathcal{D}$ is dense in $\mathcal{H}$ and satisfies $\langle
f,Ag\rangle=\langle Af,g\rangle$ for all $f,g\in\mathcal{D}$ by Lemma 3.2.
Thus, $A$ with domain $\mathcal{D}(A)=\mathcal{D}$ is symmetric. Symmetric
operators are always closable. To show that the closure is self-adjoint, we
check that every vector $f\in\mathcal{D}$ is _analytic_ for $A$ and conclude
with Nelson’s analytic vector theorem [35, Theorem X.39]. Remember that a
vector $f$ is analytic for $A$ if there exists $\varepsilon>0$ such that
$\sum_{n=0}^{\infty}n!^{-1}\varepsilon^{n}||A^{n}\varphi||<\infty$. Let
$f\in\mathcal{D}$ and $m=m_{f}\in\mathbb{N}$ be such that
$|f(\eta)|\leq||f||_{\infty}\mathbbm{1}_{\\{\eta(E)\leq m_{f}\\}}$ on
$\mathbf{N}_{<\infty}$. Then

$\displaystyle\bigl{|}k^{+}(\mathbbm{1})f(\eta)\bigr{|}$
$\displaystyle\leq\frac{1}{\sqrt{p}}||f||_{\infty}(m_{f}+1)\mathbbm{1}_{\\{\eta(E)\leq
m_{f}+1\\}},$ $\displaystyle\bigl{|}k^{-}(\mathbbm{1})f(\eta)\bigr{|}$
$\displaystyle\leq\sqrt{p}\,||f||_{\infty}\bigl{(}\alpha(E)+m_{f}-1\bigr{)}\mathbbm{1}_{\\{\eta(E)\leq
m_{f}-1\\}}$

for all $\eta\in\mathbf{N}_{<\infty}$. Set $\beta:=\alpha(E)$, we obtain a
common upper bound to $k^{\pm}(\mathbbm{1})f$:

$\bigl{|}k^{\pm}(\mathbbm{1})f(\eta)\bigr{|}\leq\frac{1}{\sqrt{p}}||f||_{\infty}(\beta+m_{f}+1)\mathbbm{1}_{\\{\eta(E)\leq
m_{f}\\}}.$

We expand the power $(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))^{n}f$ and bound
each term in the expansion by repeated use of the previous inequality. This
gives

$\bigl{|}(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1})\bigr{)}^{n}f(\eta)\bigr{|}\leq\Bigl{(}\frac{2||f||_{\infty}}{\sqrt{p}}\Bigr{)}^{n}(\beta+m_{f}+1)_{n}\mathbbm{1}_{\\{\eta(E)\leq
m_{f}+n\\}}.$

The indicator on the right has norm $\leq 1$. Set
$s=2|\xi|\,||f||_{\infty}/\sqrt{p}$, we obtain

$\sum_{n=0}^{\infty}\frac{\varepsilon^{n}}{n!}||(\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1}))^{n}f||\leq\sum_{n=0}^{\infty}\frac{(s\varepsilon)^{n}}{n!}\bigl{(}\beta+m_{f}+1)_{n}$

which is finite for $\varepsilon<1/s$. Thus, every vector $f\in\mathcal{D}$ is
analytic and $A$ is essentially self-adjoint on $\mathcal{D}$. ∎

###### Proof of Theorem 2.1.

Let $t\geq 0$. As $\rho_{p,\alpha}$ is reversible, $P_{t}$ is a well-defined,
self-adjoint and bounded operator on
$L^{2}(\mathbf{N}_{<\infty},\mathcal{N}_{<\infty},\rho_{p,\alpha})$. The
operator $\exp(2\mathrm{i}\theta k^{0}(\mathbbm{1}))$ acts as multiplication
with $\exp(\mathrm{i}\theta(\alpha(E)+2\eta(E))$, it commutes with $P_{t}$
because $(P_{t})_{t\geq 0}$ preserves the total number of particles (see
Proposition 4.1 below).

For $\exp(\xi k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1}))$, the short
informal reasoning is simple: $P_{t}$ is self-adjoint and commutes with
$k^{+}(\mathbbm{1})$, therefore it also commutes with the adjoint
$k^{-}(\mathbbm{1})$ and also with the difference $\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1})$ and the exponential
$\exp(\xi k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1}))$. The precise
reasoning is slightly longer because the operators are unbounded and we have
to consider domains carefully. Let $f,g$ be non-negative functions in
$\mathcal{D}$. Arguing as in the proof of Lemma 3.2, we obtain $\langle
k^{-}(\mathbbm{1})f,P_{t}g\rangle=\langle f,\mathcal{A}P_{t}g\rangle$; .
Therefore

$\langle k^{-}(\mathbbm{1})f,P_{t}g\rangle=\langle
f,\mathcal{A}P_{t}g\rangle=\langle f,P_{t}\mathcal{A}g\rangle=\langle
P_{t}f,k^{+}(\mathbbm{1})g\rangle$

for all non-negative $f,g\in\mathcal{D}$. Taking complex conjugates and
switching $f$ and $g$, we get $\langle
k^{+}(\mathbbm{1})f,P_{t}g\rangle=\langle P_{t}f,k^{-}(\mathbbm{1})g\rangle$
and then, by linear combination,

$\Bigl{\langle}\frac{1}{\mathrm{i}}\bigl{(}\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1})\bigr{)}f,P_{t}g\Bigr{\rangle}=\Bigl{\langle}P_{t}f,\frac{1}{\mathrm{i}}\bigl{(}\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1})\bigr{)}g\Bigr{\rangle}$

for all $f,g\in\mathcal{D}$. By Lemma 4.2, the operator $-\mathrm{i}(\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1})$ with domain $\mathcal{D}$
is closable and its closure $A$ is self-adjoint. Taking limits in the above
equation, we obtain

$\langle Af,P_{t}g\rangle=\langle P_{t}f,Ag\rangle$

for all $f,g$ in the domain of $A$. In particular, for fixed $g$ in the domain
of $\mathcal{A}$, and all $f\in\mathcal{D}(A)$, we have $\langle
Af,P_{t}g\rangle=\langle f,P_{t}Ag\rangle$, hence $P_{t}g$ is in the domain of
$A^{*}$ and $A^{*}P_{t}g=P_{t}Af$.

As $A$ is self-adjoint, we conclude that $P_{t}$ leaves the domain of $A$
invariant and $P_{t}A=AP_{t}$ on the domain of $A$. It follows that $P_{t}$
commutes with all spectral projections of $A$ [37, Proposition 5.26] and then,
by spectral calculus, $P_{t}\exp(\mathrm{i}A)=\exp(\mathrm{i}A)P_{t}$. ∎

##  5\. Unitary operator vs. Meixner polynomials. Proof of Theorem 2.2

Here we prove Theorem 2.2. Our proof is similar in spirit to the proof with
generating functions from Carinci et al. [7], see in particular Proposition
5.3 below. In addition, we point out a connection with the representation of
the group $SU(1,1)$ with the Möbius transform exploited by Bargmann [4,
Section 9].

For measurable $z:E\to\mathbb{C}$ with $\sup|z|<1$, we define the _exponential
state_

$\mathcal{E}_{z}(0)=1,\quad\mathcal{E}_{z}\bigl{(}\delta_{x_{1}}+\cdots+\delta_{x_{n}}\bigr{)}=\frac{1}{\sqrt{p}^{n}}\,z(x_{1})\cdots
z(x_{n}).$

The exponential state owes its name to the relation
$\mathcal{E}_{z}=\exp(k^{+}(z))\mathbbm{1}_{\\{0\\}}$, compare Meyer [31,
Chapter IV.1]. The state has norm

$||\mathcal{E}_{z}||^{2}=(1-p)^{\alpha(E)}\exp\Bigl{(}-\int\log(1-|z|^{2})\mathrm{d}\alpha\Bigr{)}.$

For $\xi\in\mathbb{C}$, consider the matrix

$A(\xi)=\exp\Biggl{(}\begin{pmatrix}0&\mathrm{i}\xi\\\
-\mathrm{i}\overline{\xi}&0\end{pmatrix}\Biggr{)}=\begin{pmatrix}\cosh|\xi|&\mathrm{i}\frac{\xi}{|\xi|}\sinh|\xi|\\\
-\mathrm{i}\frac{\overline{\xi}}{|\xi|}\sinh|\xi|&\cosh|\xi|\end{pmatrix}.$

The matrix is in $SU(1,1)$, the group of $2\times 2$ matrices of the form

$A=\begin{pmatrix}a&b\\\ \overline{b}&\overline{a}\end{pmatrix}$

with $a,b\in\mathbb{C}$ and $|a|^{2}-|b|^{2}=1$. It is known that $SU(1,1)$
acts on the open complex unit disk $\\{z:|z|<1\\}$ by the Möbius transform

$\phi_{A}(z)=\frac{az+b}{\overline{b}z+\overline{a}}.$

For a function $z$ from $E$ to the open unit disk, we define a new function
$z_{\xi}$ by

$z_{\xi}(x)=\frac{1}{\mathrm{i}}\phi_{A(\xi)}\bigl{(}\mathrm{i}z(x)\bigr{)}=\frac{z(x)+\frac{\xi}{|\xi|}\tanh|\xi|}{1+z(x)\frac{\overline{\xi}}{|\xi|}\tanh|\xi|}$

and a scalar $C(\xi)\in\mathbb{C}$ by

$C(\xi)=\exp\Biggl{(}-\int\log\Bigl{(}\cosh|\xi|+z(y)\frac{\overline{\xi}}{\xi}\sinh|\xi|\Bigr{)}\alpha(\mathrm{d}y)\Biggr{)}.$

The logarithm is the principal branch of the complex logarithm on
$\mathbb{C}\setminus(-\infty,0]$, i.e., $\log z=\log|z|+\mathrm{i}\theta$ with
$\theta\in(-\pi,\pi)$ an argument of $z$. More concretely, in view of
$|z(y)|<1$ and $\tanh|\xi|<1$,

$\log\Bigl{(}\cosh|\xi|+z(y)\frac{\overline{\xi}}{\xi}\sinh|\xi|\Bigr{)}=\log(\cosh|\xi|)+\log\Bigl{(}1+z(y)\frac{\overline{\xi}}{\xi}\tanh|\xi|\Bigr{)}$

with $\log(1+u)=\sum_{n=1}^{\infty}(-1)^{n-1}u^{n}/n$.

The scalar $C(\xi)$ is similar to multipliers introduced by Bargmann when he
constructed group actions $SU(1,1)$ on spaces of functions from the complex
unit disk to $\mathbb{C}$ [4, Section 9].

######  Proposition 5.1.

Let $\xi\in\mathbb{C}$ and $z:\mathbb{E}\to\mathbb{C}$ be measurable with
$\sup|z|<1$. Then

$\exp\Bigl{(}\xi
k^{+}(\mathbbm{1})-\overline{\xi}k^{-}(\mathbbm{1})\Bigr{)}\mathcal{E}_{z}=C(\xi)\mathcal{E}_{z_{\xi}}.$

###### Proof.

To lighten notation we write down the proof for real-valued $\xi$ only, the
general case is similar. Thus let $\xi=t\in\mathbb{R}$. Set
$f_{t}:=C(t)\mathcal{E}_{z_{t}}$. We show that (i) $t\mapsto f_{t}$ is norm
differentiable, (ii) $f_{t}$ is in the domain of the closure $A$ of
$\frac{1}{\mathrm{i}}(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))$, and (iii)
$\partial_{t}f_{t}=\mathrm{i}Af_{t}$. The key equations are (13), (14), (15),
and (16). Items (i)–(iii) imply $f_{t}=\exp(\mathrm{i}tA)f_{0}$ for all
$t\in\mathbb{R}$. The proposition follows upon specializing to $t=\xi$.

(i) The map $t\mapsto z_{t}$ is differentiable,

$\partial_{t}z_{t}(x)=\frac{(1-z_{0}(x))^{2}(1-\tanh(t))^{2}}{(1+z_{0}(x)\tanh
t)^{2}}=1-z_{t}(x)^{2}$

pointwise for all $x$ and actually in supremum norm

$\lim_{h\to 0}||\frac{1}{t}(z_{t+h}-z_{t})-(1-z_{t})^{2}||_{\infty}=0.$

The multiplier $C(t)$ is differentiable with derivative

$\displaystyle C^{\prime}(t)$
$\displaystyle=-C(t)\frac{\mathrm{d}}{\mathrm{d}t}\int\Bigl{(}\log(\cosh
t)+\log\bigl{(}1+z_{0}(y)\tanh t\bigr{)}\Bigr{)}\alpha(\mathrm{d}y)$ (13)
$\displaystyle=-C(t)\int z_{t}(y)\alpha(\mathrm{d}y).$

For $t\mapsto\mathcal{E}_{z_{t}}$: Given $t>0$, let $h_{0},\delta>0$ such that
$||z_{t+h}||_{\infty}\leq 1-\delta$ for all $h\in[-h_{0},h_{0}]$. Such a $t$
exists because $t\mapsto z_{t}$ is continuous in supremum norm and because the
Möbius transform is continuous and maps the open unit disk to the open unit
disk, hence compact subsets $\\{\zeta:|\zeta|\leq||z_{0}||_{\infty}\\}$ to
compact subsets, bounded away from the open disk’s boundary. For $n\geq 2$,

$\bigl{(}\mathcal{E}_{z_{t+h}}-\mathcal{E}_{z_{t}}\bigr{)}(\delta_{x_{1}}+\cdots+\delta_{x_{n}})\\\
=\frac{1}{{\sqrt{p}}^{n}}\sum_{i=1}^{n}v(x_{i})\prod_{j\neq
i}z_{t}(x_{j})+\frac{1}{{\sqrt{p}}^{n}}\sum_{\begin{subarray}{c}I\subset[n]:\\\
|I|\geq 2\end{subarray}}\prod_{i\in I}v(x_{i})\prod_{j\notin I}z_{t}(x_{j}).$

The second term on the right is bounded in supremum norm by
$p^{-n/2}\binom{n}{2}||v||_{\infty}^{2}||z_{t}||_{\infty}^{n-2}$. The
$L^{2}$-norm of the corresponding function of
$\eta=\delta_{x_{1}}+\cdots+\delta_{x_{n}}$ is bounded by
$\sqrt{\kappa}||v||_{\infty}$ where

$\displaystyle\kappa=\mathbb{E}\Bigl{[}\eta(E)^{4}p^{-\eta(E)}||z_{t}||_{\infty}^{\eta(E)-2}\Bigr{]}=(1-p)^{\alpha(E)}\sum_{n=2}^{\infty}\frac{1}{n!}(\alpha(E))_{n}n^{4}||z_{t}||_{\infty}^{2n-4}<\infty.$

In view of $||v_{h}||_{\infty}=O(h)$ we get a bound of the order of
$\sqrt{\kappa}||v_{h}||_{\infty}^{2}=O(h^{2})$. A similar bound based on
$v_{h}=h(1-z_{t})^{2}+O(h^{2})$ shows that the linear (in $v$) term is

$h\,\frac{1}{{\sqrt{p}}^{n}}\sum_{i=1}^{n}(1-z_{t}(x_{i}))^{2}\prod_{j\neq
i}z_{t}(x_{j})+O(h^{2}).$

It follows that $t\mapsto\mathcal{E}_{z_{t}}$ is norm-differentiable with
derivative

$\partial_{t}\mathcal{E}_{z_{t}}\bigl{(}\delta_{x_{1}}+\cdots+\delta_{x_{n}}\bigr{)}=\frac{1}{\sqrt{p}}\sum_{i=1}^{n}(1-z_{t}(x_{i})^{2})\prod_{j\neq
i}\frac{z_{t}(x_{j})}{\sqrt{p}}$ (14)

for $n\geq 1$ and $\partial_{t}\mathcal{E}_{z_{t}}(0)=1$. The map $t\mapsto
f_{t}$ is norm-differentiable with derivative

$\partial_{t}f_{t}=C^{\prime}(t)\mathcal{E}_{z_{t}}+C(t)\partial_{t}\mathcal{E}_{z_{t}}.$
(15)

(ii) We show that every exponential vector $\mathcal{E}_{z}$,
$||z||_{\infty}<1$, is in the domain of the closure of
$k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1})$. Set $\psi_{0}=\mathbbm{1}_{\\{0\\}}$
and
$\psi_{n}(\delta_{x_{1}}+\cdots+\delta_{x_{m}})=\delta_{n,m}\prod_{i=1}^{m}z(x_{i})/\sqrt{p}$
so that $\mathcal{E}_{z}=\sum_{n=0}^{\infty}\psi_{n}$. Every $\psi_{n}$ is in
$\mathcal{D}$. Furthermore,

$k^{+}(\mathbbm{1})\psi_{n}(\delta_{x_{1}}+\cdots+\delta_{x_{m}})=\delta_{m,n+1}\frac{1}{\sqrt{p}}\sum_{i=1}^{m}\prod_{j\neq
i}\frac{z(x_{j})}{\sqrt{p}}$

and

$\sum_{n=0}^{\infty}||k^{+}(\mathbbm{1})\psi_{n}||^{2}\leq\mathbb{E}\Bigl{[}\mathbbm{1}_{\\{\eta(E)\geq
1\\}}p^{-\eta(E)}\eta(E)^{2}||z||_{\infty}^{2(\eta(E)-1)}\Bigr{]}<\infty.$

It follows that $k^{+}(\mathbbm{1})\sum_{n=0}^{N}\psi_{n}$ converges in norm.
Turning to $k^{-}(\mathbbm{1})$, we note

$\displaystyle
k^{-}(\mathbbm{1})\psi_{n}(\delta_{x_{1}}+\cdots+\delta_{x_{m}})$
$\displaystyle\qquad=\delta_{m,n-1}\Bigl{(}\int
z\mathrm{d}\alpha+\sum_{i=1}^{m}z(x_{i})\Bigr{)}\prod_{j=1}^{m}\frac{z(x_{i})}{\sqrt{p}}$
$\displaystyle\qquad=\delta_{m,n-1}\Bigl{(}\int
z\mathrm{d}\alpha\Bigr{)}\prod_{j=1}^{m}\frac{z(x_{j})}{\sqrt{p}}+\delta_{m,n-1}\frac{1}{\sqrt{p}}\sum_{i=1}^{m}z(x_{i})^{2}\prod_{j\neq
i}\frac{z(x_{j})}{\sqrt{p}}$

and $\sum_{n=1}^{\infty}||k^{-}(\mathbbm{1})\psi_{n}||^{2}<\infty$ as well.
Therefore, the partial sums $S_{N}=\sum_{n=0}^{N}\psi_{n}$ are in
$\mathcal{D}$, they converge in norm to $\mathcal{E}_{z}$, and
$(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))S_{N}$ converges in norm to
$\sum_{n}(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))\psi_{n}$. It follows that
$\mathcal{E}_{z}$ is the domain of the closure of
$(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))$ and

$(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))^{\mathrm{cl}}\mathcal{E}_{z}(\delta_{x_{1}}+\cdots+\delta_{x_{n}})=-\Bigl{(}\int
z\mathrm{d}\alpha\Bigr{)}\mathcal{E}_{z}(\delta_{x_{1}}+\cdots+\delta_{x_{n}})\\\
+\frac{1}{\sqrt{p}}\sum_{i=1}^{n}\bigl{(}1-z(x_{i})^{2}\bigr{)}\prod_{j\neq
i}\frac{z(x_{j})}{\sqrt{p}}.$ (16)

The function $f_{t}=C(t)\mathcal{E}_{z_{t}}$ is in the domain of the closure
too.

(iii) Comparing the previous equation with (15), (14), and (13) we get
$\partial_{t}f_{t}=(k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))^{\mathrm{cl}}f_{t}=\mathrm{i}Af_{t}$.
The proof is complete. ∎

We now sketch the conclusion of the proof using an explicit formula for
generating functionals given in [29, Prop. 3.1]. For measurable
$\varphi:E\to\mathbb{C}$, define
$\mathcal{G}_{\varphi}:\mathbf{N}_{<\infty}\to\mathbb{C}$ with
$\sup|\varphi|<1$ by

$\mathcal{G}_{\varphi}(\eta)=1+\sum_{n=1}^{\infty}\frac{1}{n!}(1-p)^{n}I_{n}(\varphi^{\otimes
n}).$

Then

$\mathcal{G}_{\varphi}(\eta)=\exp\Bigl{(}-\int\log(1+p\varphi)\mathrm{d}\alpha+\int\log\left(\frac{1+\varphi}{1+p\varphi}\right)\,\mathrm{d}\eta\Bigr{)}.$

Applying Prop. 5.1 to exponential vectors $\mathcal{E}_{\varphi\sqrt{p}}$ with
$\varphi$ an arbitrary bounded measurable function, we find that
$(1-p)^{-\alpha(E)/2}$ maps the function $f:\mathbf{N}_{<\infty}\to\mathbb{C}$
given by

$f(0)=1,\quad
f(\delta_{x_{1}}+\cdots+\delta_{x_{n}})=\prod_{i=1}^{n}\varphi(x_{i})$

to $\mathcal{G}_{\varphi}$, from which we deduce that Theorem 2.2 holds true
when $f_{n}=\varphi^{\otimes n}$, much in the same way we prove Prop. 5.3
below. The general statement follows with polarization formulas and
(multi)linearity, and density arguments—notice that
$(f_{1},\ldots,f_{n})\mapsto
I_{n}(f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n})$ is a
symmetric multilinear mapping and as such, uniquely determined by its value
for $f_{1}=\cdots=f_{n}$.

For convenience of the reader, we provide a self-contained proof that does not
require adaption of previously proven statements to our setup (such as [29,
Prop. 3.1] which uses the machinery of Jacobi fields and distribution theory).
We start by applying Prop. 5.1 to exponential vectors associated with
indicators.

######  Corollary 5.2.

Let $\xi>0$ with $\tanh\xi=\sqrt{p}$. Then, for every $B\in\mathcal{E}$ and
$s\in(-1,1)$, the unitary $U=\exp(\xi k^{+}(\mathbbm{1})-\xi
k^{-}(\mathbbm{1}))$ maps the function
$f_{s}:\mathbf{N}_{<\infty}\to\mathbb{R}$ given by

$f_{s}(0)=1,\quad
f_{s}(\delta_{x_{1}}+\cdots+\delta_{x_{n}})=s^{n}\prod_{i=1}^{n}\mathbbm{1}_{B}(x_{i})$

to

$Uf_{s}(\eta)=(1-p)^{\alpha(E)/2}\bigl{(}1+ps\bigr{)}^{-\alpha(B)}\Bigl{(}\frac{1+s}{1+ps}\Bigr{)}^{\eta(B)}.$

###### Proof.

The function $f_{s}$ is the exponential state associated with
$z(x)=s\sqrt{p}\mathbbm{1}_{B}(x)$. The transformed $z$ is

$z_{\xi}(x)=\frac{s\sqrt{p}\mathbbm{1}_{B}(x)+\sqrt{p}}{1+sp\mathbbm{1}_{B}(x)}=\sqrt{p}\,\frac{1+s\mathbbm{1}_{B}(x)}{1+ps\mathbbm{1}_{B}(x)}$

and the associated exponential state is

$\mathcal{E}_{z_{\xi}}(\delta_{x_{1}}+\cdots+\delta_{x_{n}})=\prod_{i=1}^{n}\Bigl{(}\frac{1+s\mathbbm{1}_{B}(x_{i})}{1+ps\mathbbm{1}_{B}(x_{i})}\Bigr{)}^{n}$

or equivalently,

$\mathcal{E}_{z_{\xi}}(\eta)=\Bigl{(}\frac{1+s}{1+ps}\Bigr{)}^{\eta(B)}.$

The multiplier is

$\displaystyle C(\xi)$
$\displaystyle=(\cosh\xi)^{-\alpha(E)}\exp\Bigl{(}-\int\log\bigl{(}1+s\sqrt{p}\mathbbm{1}_{B}\tanh\xi\bigr{)}\mathrm{d}\alpha\Bigr{)}$
$\displaystyle=(1-p)^{\alpha(E)/2}(1+ps)^{-\alpha(B)}.$

In the last line we have used $(\cosh\xi)^{2}=1/(1-(\tanh\xi)^{2})$. We apply
Prop. 5.1 and obtain the corollary. ∎

We remind the reader that the univariate monic Meixner polynomials are denoted
by $\mathcal{M}_{n}(x;\alpha,p)$, see Appendix A below.

######  Proposition 5.3.

Fix $B\in\mathcal{E}$. The unitary $U$ maps the functions
$f^{(n)}:\mathbf{N}_{<\infty}\to\mathbb{R}$, $n\in\mathbb{N}$, given by

$f^{(n)}(\delta_{x_{1}}+\cdots+\delta_{x_{m}})=\begin{cases}\prod_{i=1}^{m}\mathbbm{1}_{B}(x_{i}),&\quad
m=n,\\\ 0,&\quad m\neq n\end{cases}$

to

$Uf^{(n)}(\eta)=(1-p)^{\alpha(E)/2}\times\frac{1}{n!}\,(1-p)^{n}\mathcal{M}_{n}(\eta(B);\alpha(B),p).$

###### Proof.

Let $f^{(0)}:=\mathbbm{1}_{\\{0\\}}$. For $s\in(-1,1)$, with the notation of
Corollary 5.2,

$f_{s}=\sum_{n=0}^{\infty}s^{n}f^{(n)}.$

The series converges in norm as the $f^{(n)}$’s are orthogonal and
$\sum_{n=0}^{\infty}|s|^{2n}\,||f^{(n)}||^{2}<\infty$. As $U$ is unitary,
hence continuous, we may exchange summation and the application of $U$ which
gives

$Uf_{s}=\sum_{n=0}^{\infty}s^{n}Uf^{(n)}$ (17)

and the series on the right converges in norm. On the other hand, in the
expression for $Uf_{s}$ from Corollary 5.2, we recognize the generating
function of Meixner polynomials with leading coefficient $(1-p)^{n}$:

$Uf_{s}=(1-p)^{\alpha(E)/2}\sum_{n=0}^{\infty}\frac{s^{n}}{n!}(1-p)^{n}\mathcal{M}_{n}\bigl{(}\eta(B);\alpha(B),p\bigr{)},$
(18)

see Eq. (25). The right side is not only absolutely convergent, pointwise for
each $\eta\in\mathbf{N}_{<\infty}$, but also absolutely convergent in the
Hilbert space $\mathcal{H}$. Indeed as $\eta\mapsto\eta(B)$, under
$\rho_{p,\alpha}$ is a negative binomial variable with parameters $\alpha(B)$
and $p$ (see Eq. (3)), the orthogonality relations for univariate Meixner
polynomials recalled in Appendix A tell us that the summands are orthogonal
and

$\frac{s^{2n}}{n!^{2}}(1-p)^{2n}\int\bigl{|}\mathcal{M}_{n}\bigl{(}\eta(B);\alpha(B),p\bigr{)}\bigr{|}^{2}\rho_{p,\alpha}(\mathrm{d}\eta)=\frac{1}{n!}\bigl{(}\alpha(B)\bigr{)}_{n}(ps^{2})^{n}.$

The sum over $n\in\mathbb{N}_{0}$ of the right side is $(1-ps^{2})^{\alpha}$
and in particular, finite. Therefore the series on the right side in (18)
converges in norm.

Thus, we have two convergent series expansions for the analytic function
$s\mapsto Uf_{s}$ from $(-1,1)$ to $\mathcal{H}$. As the expansion
coefficients are uniquely determined, the coefficients of $s^{n}$ in Eq. (17)
and (18) must be the same. The proposition follows. ∎

The _symmetric tensor product_ of $f_{1},\ldots,f_{n}:E\to\mathbb{C}$ is the
function from $E^{n}$ to $\mathbb{C}$ given by

$f_{1}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}f_{n}(x_{1},\ldots,x_{n})=\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}_{n}}f_{\sigma(1)}(x_{1})\cdots
f_{\sigma(n)}(x_{n}).$

Above, $\mathfrak{S}_{n}$ denotes the set of permutations of the numbers
${\left\\{1,\ldots,n\right\\}}$. Consider disjoint sets
$B_{1},\ldots,B_{\ell}$, integers $n_{1},\ldots,n_{\ell}\in\mathbb{N}$ adding
up to $\mathbb{N}$, and
$f_{n}=\mathbbm{1}_{B_{1}}^{\otimes_{\mathrm{s}}n_{1}}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}\mathbbm{1}_{B_{\ell}}^{\otimes_{\mathrm{s}}n_{\ell}}$.
The associated function $f$ is supported on configurations that have exactly
$n_{i}$ particles in $B_{i}$, for each $i=1,\ldots,\ell$, and $n$ particles in
total. On the relevant configurations there is an additional combinatorial
contribution. One finds

$f(\eta)=\frac{n_{1}!\cdots
n_{\ell}!}{n!}\,\mathbbm{1}_{\\{\eta(E\setminus\bigcup_{i=1}^{\ell}B_{i})=0\\}}\prod_{i=1}^{\ell}\mathbbm{1}_{\\{\eta(B_{i})=n_{i}\\}}.$
(19)

Set $B_{0}:=E\setminus(\bigcup_{i=1}^{\ell}B_{i})$, $n_{0}=0$, and

$f_{i}(\eta)=\mathbbm{1}_{\\{\eta(B_{i})=n_{i}\\}}\quad i=0,\ldots,\ell.$ (20)

The next lemma says that $U$ maps the product of $f_{i}$’s to products of
$Uf_{i}$’s.

######  Lemma 5.4.

Let $\\{B_{0},\ldots,B_{\ell}\\}$ be a set partition of $E$,
$n_{0},n_{1},\ldots,n_{\ell}\in\mathbb{N}_{0}$, and $f_{0},\ldots,f_{\ell}$ as
in Eq. (20). Then

$\bigl{(}U(f_{0}\cdots
f_{\ell})\bigr{)}(\eta)=\prod_{j=0}^{\ell}\frac{1}{n_{j}!}(1-p)^{n_{j}+\alpha(B_{j})/2}\mathcal{M}_{n_{j}}\bigl{(}\eta(B_{j});\alpha(B_{j}),p\bigr{)}.$

###### Proof.

Let $V(t):=\exp(tk^{+}(\mathbbm{1})-k^{-}(\mathbbm{1}))$. Notice that

$k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1})=\sum_{j=0}^{\ell}\bigl{(}k^{+}(\mathbbm{1}_{B_{j}})-k^{-}(\mathbbm{1}_{B_{j}})\bigr{)}.$

For $j=0,\ldots,\ell$, let $\nu_{j}$ be the negative binomial law on
$\mathbb{N}_{0}$ with parameters $p$ and $\alpha(B_{j})$. The space $d_{0}$ of
sequences with at most finitely many non-zero entries is dense in
$\ell^{2}(\mathbb{N}_{0},\nu_{j})$. Consider the operators
$k_{j}^{\pm}:d_{0}\to\ell^{2}(\mathbb{N}_{0},\nu_{j})$ given by

$k_{j}^{+}g_{j}(n)=\frac{1}{\sqrt{p}}ng(n-1),\quad
k_{i}^{-}g_{i}(n)=\sqrt{p}\bigl{(}\alpha(B_{j})+n\bigr{)}g_{i}(n+1).$

The operator $\frac{1}{\mathrm{i}}(k_{j}^{+}-k_{j}^{-})$ is closable with
self-adjoint closure, let $U_{j}(t)=\exp(t(k_{j}^{+}-k_{j}^{-}))$,
$t\in\mathbb{R}$ be the associated strongly continuous unitary group. For
$g_{0},\ldots,g_{\ell}\in d_{0}$ and $t\in\mathbb{R}$, set

$G_{t}(\eta)=\prod_{j=0}^{\ell}\bigl{(}U_{j}(t)g_{j}\bigr{)}(\eta(B_{j})\bigr{)}.$

The map $t\mapsto G_{t}$ from $\mathbb{R}\to\mathcal{H}$ is norm-
differentiable, $G_{t}$ is in the domain of the closure $L$ of
$k^{+}(\mathbbm{1})-k^{-}(\mathbbm{1})$, and
$\frac{\mathrm{d}}{\mathrm{d}t}G_{t}(\eta)=LG_{t}$. It follows that
$G_{t}=V(t)G_{0}$. We apply the statement to
$g_{j}(n)=\mathbbm{1}_{\\{n=n_{j}\\}}$ and $t=\xi$ and deduce

$U(f_{0}\cdots
f_{\ell})(\eta)=\prod_{j=0}^{\ell}\bigl{(}U_{j}(\xi)g_{j}\bigr{)}(\eta(B_{j})\bigr{)}.$
(21)

Finally, we notice that each term on the right side is given by a univariate
Meixner polynomial as

$\bigl{(}U_{j}(\xi)g_{j}\bigr{)}(m)=\frac{1}{n_{j}!}\,(1-p)^{n_{j}+\alpha(B_{j})/2}\mathcal{M}_{n_{j}}\bigl{(}m;\alpha(B_{j}),p\bigr{)}.$
(22)

This follows from Proposition 5.3 applied to a set $E^{\prime}$ that is a
singleton $\\{x_{0}\\}$, to $B=\\{x_{0}\\}$, and the measure $\alpha^{\prime}$
that gives mass $\alpha(B_{j})$ to $x$. Alternatively, one may deduce the
univariate equation (22) directly from from Theorem 3.1 in Carinci et al. [7].
To conclude, we insert (22) into (21) and obtain the lemma. ∎

###### Proof of Theorem 2.2.

Let $B_{1},\ldots,B_{\ell}\in\mathcal{E}$ disjoint,
$n_{1},\ldots,n_{\ell}\in\mathbb{N}$ with $n_{1}+\cdots+n_{\ell}=n$, and
$f:\mathbf{N}_{<\infty}\to\mathbb{C}$ the function with
$f(\delta_{x_{1}}+\cdots+\delta_{x_{n}})=\mathbbm{1}_{B_{1}}^{\otimes_{\mathrm{s}}n_{1}}\otimes_{\mathrm{s}}\cdots\otimes_{\mathrm{s}}\mathbbm{1}_{B_{\ell}}^{\otimes_{\mathrm{s}}n_{\ell}}(x_{1},\ldots,x_{n})$.
Set $B_{0}=E\setminus(\bigcup_{j=1}^{\ell}B_{j})$ and $n_{0}=0$. Eq. (19) and
Lemma 5.4 yield

$\displaystyle Uf$ $\displaystyle=\frac{n_{1}!\cdots
n_{\ell}!}{n!}\,(1-p)^{\alpha(B_{0})/2}$
$\displaystyle\qquad\qquad\times\prod_{j=1}^{\ell}\frac{1}{n_{j}!}(1-p)^{n_{j}+\alpha(B_{j})/2}\mathcal{M}_{n_{j}}\bigl{(}\eta(B_{j});\alpha(B_{j}),p\bigr{)}$
$\displaystyle=(1-p)^{\alpha(E)/2}\times\frac{1}{n!}(1-p)^{n}\prod_{j=1}^{\ell}\mathcal{M}_{n_{j}}\bigl{(}\eta(B_{j});\alpha(B_{j}),p\bigr{)}$
$\displaystyle=(1-p)^{\alpha(E)/2}\times\frac{1}{n!}(1-p)^{n}I_{n}(f_{n}).$

This proves the theorem for all functions $f$ of the form (19) with disjoint
$B_{1},\ldots,B_{\ell}$. By linearity, the statement extends to all indicators
of sets of the form

$\\{\eta:\eta(E)=n\\}\cap\Biggl{(}\bigcap_{j=1}^{\ell}\\{\eta:\eta(B_{j})=n_{j}\\}\Biggr{)}$

with $\ell\in\mathbb{N}$, $n_{1},\ldots,n_{\ell}\in\mathbb{N}_{0}$, and
$B_{1},\ldots,B_{\ell}\in\mathcal{E}$ (not necessarily disjoint). These sets
form a generating $\pi$-system of the $\sigma$-algebra $\mathcal{N}_{n}$ on
$\mathbf{N}_{n}=\\{\eta:\eta(E)=n\\}$ that is the trace of
$\mathcal{N}_{<\infty}$ on $\mathbf{N}_{n}$.

We conclude with a monotone class argument. Let $\mathcal{V}$ be the set of
bounded functions $f:\mathbf{N}_{\infty}\to\mathbb{C}$ supported in
$\mathbf{N}_{n}$ for which $Uf=n!^{-1}(1-p)^{n+\alpha(E)/2}I_{n}(f_{n})$.
Clearly $\mathcal{V}$ is a vector space. We check that it is closed under
monotone pointwise limits of uniformly bounded sequences: Suppose
$f^{(m)}\uparrow f$ with $C=\sup_{n}||f^{(m)}||_{\infty}<\infty$ and
$f^{(m)}\in\mathcal{V}$ for all $m$. Then, on the one hand $f^{(m)}\to f$ also
in $L^{2}$-norm and $||Uf^{(m)}-Uf||\to 0$ because $U$ is unitary. On the
other hand

$\displaystyle||I_{n}(f^{(m)}-f)||^{2}$
$\displaystyle\leq\int\Bigl{|}\int\bigl{(}f^{(m)}(\boldsymbol{x})-f(\boldsymbol{x})\bigr{)}\eta^{n}(\mathrm{d}\boldsymbol{x})\Bigr{|}^{2}\rho_{p,\alpha}(\mathrm{d}\eta)$

because orthogonal projections onto a subspace decrease the norm. The
integrand $f^{(m)}-f$ in the inner integral on the right side goes to zero
pointwise on $E^{n}$ and is bounded by $2C$ with $\int
2C\mathrm{d}\eta^{n}\leq 2C\eta(E)^{n}<\infty$, therefore by dominated
convergence it goes to zero. All moments of the negative binomial variable
$\eta\mapsto\eta(E)$ are finite, therefore dominated convergence applied to
the outer integral yields $||I_{n}(f^{(m)})-I_{n}(f)||\to 0$ as $m\to\infty$.
Thus, we may pass to the limit on both sides of the equality
$Uf^{(m)}=n!^{-1}(1-p)^{n+\alpha(E)/2}I_{n}(f^{(m)}_{n})$ and we deduce
$f\in\mathcal{V}$.

We have shown that $\mathcal{V}$ is a vector space that contains all
indicators of a $\pi$-system that generates the $\sigma$-algebra and that it
is stable under pointwise limits of uniformly bounded sequences. The
functional monotone class theorem [5, Theorem 2.12.9] implies that it contains
all bounded measurable functions. ∎

As an alternative route to obtain the above results, one could skip
exponential states and Propositions 5.1 and 5.3 and start directly with Lemma
5.4, deducing the crucial equation (22) directly from Theorem 3.1 in [7].

##  Appendix A Univariate Meixner polynomials

We recall a few facts about the Meixner polynomials from Koekoek, Lesky and
Swarttouw [21, Chapter 9]. The Meixner polynomials with parameters $\alpha>0$
and $p\in(0,1)$ are given by

$M_{n}(x)=M_{n}(x;\alpha;p):=\sum_{k=0}^{\infty}\frac{(-x)_{k}(-n)_{k}}{(\alpha)_{k}}\frac{1}{k!}\Bigl{(}1-\frac{1}{p}\Bigr{)}^{k}.$
(23)

where $(a)_{0}=1$ and $(a)_{k}=a(a+1)\cdots(a+k-1)$. They satisfy the symmetry
$M_{n}(x)=M_{x}(n)$ and the orthogonality relation

$\sum_{x=0}^{\infty}M_{n}(x)M_{m}(x)(1-p)^{\alpha}w_{p,\alpha}(x)=\delta_{n,m}\frac{1}{w_{p,\alpha}(n)},\quad
w_{p,\alpha}(n)=\frac{p^{n}}{n!}(\alpha)_{n}.$

The generating function is

$G(t,x)=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}t^{n}M_{n}(x)=\Bigl{(}1-\frac{t}{p}\Bigr{)}^{x}(1-t)^{-\alpha-x}.$

The Meixner polynomials from (23) do not have leading coefficient $1$, instead
they are related to the monic Meixner polynomials
$\mathcal{M}_{n}(x)=\mathcal{M}_{n}(x;\alpha,p)$ by

$M_{n}(x;\alpha,p)=\frac{1}{(\alpha)_{n}}\Bigl{(}1-\frac{1}{p}\Bigr{)}^{n}\mathcal{M}_{n}(x;\alpha,p)$
(24)

from which we get

$\sum_{n=0}^{\infty}\frac{s^{n}}{n!}\,(1-p)^{n}\mathcal{M}_{n}(x)=G(-ps,x)=(1+s)^{x}(1+ps)^{-x-\alpha}.$
(25)

The formula enters the proof of Proposition 5.3.

#### Acknowledgments

S.F. acknowledges financial support from the Engineering and Physical Sciences
Research Council of the United Kingdom through the EPSRC Early Career
Fellowship EP/V027824/1 and from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 –
390685813. S.J. and S.W. were supported under Germany’s excellence strategy
EXC-2111-390814868. S.F. and S.W. thank the Hausdorff Institute for
Mathematics (Bonn) for its hospitality during the Junior Trimester Program
Stochastic modelling in life sciences funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC-2047/1 - 390685813. The authors thank F. Redig for
his insights at the beginning of the project. S.J. and S.W. thank T. Kuna and
E. Lytvynov for helpful discussions.

## References

  * [1] L. Accardi and A. Boukas.  Quantum probability, renormalization and infinite-dimensional $\ast$-Lie algebras.  SIGMA Symmetry Integrability Geom. Methods Appl., 5:Paper 056, 31, 2009.
  * [2] L. Accardi, U. Franz, and M. Skeide.  Renormalized squares of white noise and other non-Gaussian noises as Lévy processes on real Lie algebras.  Comm. Math. Phys., 228(1):123–150, 2002.
  * [3] H. Araki.  Factorizable representation of current algebra. Non commutative extension of the Lévy-Kinchin formula and cohomology of a solvable group with values in a Hilbert space.  Publ. Res. Inst. Math. Sci., 5:361–422, 1969/70.
  * [4] V. Bargmann.  Irreducible unitary representations of the Lorentz group.  Ann. of Math. (2), 48:568–640, 1947.
  * [5] V.I. Bogachev.  Measure theory, volume 1.  Springer, Berlin, 2007.
  * [6] A. Boukas.  An example of a quantum exponential process.  Monatsh. Math., 112(3):209–215, 1991.
  * [7] G. Carinci, C. Franceschini, C. Giardinà, W. Groenevelt, and F. Redig.  Orthogonal dualities of Markov processes and unitary symmetries.  SIGMA Symmetry Integrability Geom. Methods Appl., 15(053):1–27, 2019.
  * [8] G. Carinci, C. Giardinà, C. Giberti, and F. Redig.  Dualities in population genetics: a fresh look with new dualities.  Stochastic Process. Appl., 125(3):941–969, 2015.
  * [9] G. Carinci, C. Giardinà, and F. Redig.  Consistent particle systems and duality.  Electron. J. Probab., 26(125):1–31, 2021.
  * [10] G. Carinci, C. Giardinà, F. Redig, and T. Sasamoto.  A generalized asymmetric exclusion process with $U_{q}(\mathfrak{sl}_{2})$ stochastic duality.  Probab. Theory Related Fields, 166(3):887–933, 2016.
  * [11] G. Carinci, C. Giardinà, F. Redig, and T. Sasamoto.  Asymmetric Stochastic Transport Models with $\mathscr{U}_{q}(\mathfrak{su}(1,1))$ Symmetry.  J. Stat. Phys., 163(2):239–279, 2016.
  * [12] D.A. Dawson.  Measure-valued Markov processes.  In École d’Été de Probabilités de Saint-Flour XXI—1991, volume 1541 of Lecture Notes in Math., pages 1–260. Springer, Berlin, 1993.
  * [13] A. Etheridge.  An Introduction to Superprocesses.  AMS, University Lecture Series, 2000.
  * [14] A. Etheridge.  Course 11 - Evolution in Fluctuating Populations.  In In Mathematical statistical physics, pages 489–545. Elsevier, 2006.
  * [15] S. Floreani, S. Jansen, F. Redig, and S. Wagner.  Intertwining and Duality for Consistent Markov Processes.  Preprint. arXiv:2112.11885, 2021.
  * [16] S. Floreani, S. Jansen, and S. Wagner.  Representations of the $su(1,1)$ current algebra.  In preparation, 2023.
  * [17] S. Floreani, F. Redig, and F. Sau.  Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations.  Ann. Inst. Henri Poincaré Probab. Stat., 58(1):220–247, 2022\.
  * [18] C. Giardinà, J. Kurchan, F. Redig, and K. Vafayi.  Duality and hidden symmetries in interacting particle systems.  J. Stat. Phys., 135(1):25–55, 2009.
  * [19] C. Giardinà, F. Redig, and K. Vafayi.  Correlation inequalities for interacting particle systems with duality.  J. Stat. Phys., 141(2):242–263, 2010.
  * [20] S. Karlin and J. McGregor.  On a genetics model of Moran.  Proc. Cambridge Philos. Soc., 58:299–311, 1962.
  * [21] R. Koekoek, P.A. Lesky, and R.F. Swarttouw.  Hypergeometric Orthogonal Polynomials and their q-Analogues.  Springer Monographs in Mathematics. Springer, Berlin, 2010.
  * [22] T.J. Kozubowski and K. Podgórski.  Distributional properties of the negative binomial Lévy process.  Probab. Math. Statist., 29(1):43–71, 2009.
  * [23] J. Kuan.  Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two.  J. Phys. A, 49(11):115002, 29, 2016.
  * [24] J. Kuan.  An Algebraic Construction of Duality Functions for the Stochastic $\mathcal{U}_{q}(A_{n}^{(1)})$ Vertex Model and Its Degenerations.  Comm. Math. Phys., 359(1):121–187, 2018.
  * [25] G. Last.  Stochastic analysis for Poisson processes.  In G. Peccati and M. Reitzner, editors, Stochastic analysis for Poisson point processes. Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry, volume 7 of Bocconi Springer Ser., pages 1–36. Bocconi Univ. Press, 2016.
  * [26] G. Last and M. Penrose.  Lectures on the Poisson Process.  Institute of Mathematical Statistics Textbooks. Cambridge University Press, 2017.
  * [27] T.M. Liggett.  Interacting particle systems.  Classics in Mathematics. Springer, Berlin, 2 edition, 2005.
  * [28] E. Lytvynov.  Orthogonal decompositions for Lévy processes with an application to the gamma, Pascal, and Meixner processes.  Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6(1):73–102, 2003\.
  * [29] E. Lytvynov.  Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures.  J. Funct. Anal., 200(1):118–149, 2003.
  * [30] K. Matthes, W. Warmuth, and J. Mecke.  Bemerkungen zu einer Arbeit von NGUYEN XUAN XANH und HANS ZESSIN.  Math. Nachr., 88(1):117–127, 1979.
  * [31] P. Meyer.  Quantum Probability for Probabilists, volume 1538 of Lecture Notes in Mathematics.  Springer, Berlin, 2 edition, 1995.
  * [32] P.A.P. Moran.  Random processes in genetics.  Proc. Cambridge Philos. Soc., 54:60–71, 1958.
  * [33] C. Mueller.  Stochastic PDE from the Point of View of Particle Systems and Duality.  In R.C. Dalang, M. Dozzi, F. Flandoli, and F. Russo, editors, Stochastic Analysis: A Series of Lectures, pages 271–295. Springer, Basel, Basel, 2015.
  * [34] M. Rafler.  Gaussian loop- and Pólya processes: a point process approach.  PhD thesis, Universität Potsdam, 2009.
  * [35] M. Reed and B. Simon.  II: Fourier Analysis, Self-Adjointness, volume 2.  Elsevier, 1975.
  * [36] M. Reed and B. Simon.  I: Functional Analysis.  Methods of Modern Mathematical Physics. Elsevier Science, 1981.
  * [37] K. Schmüdgen.  Unbounded self-adjoint operators on Hilbert space, volume 265\.  Springer Science & Business Media, 2012.
  * [38] R.F. Serfozo.  Point processes.  In Stochastic models, volume 2 of Handbooks in operations research and management science, pages 1–93. North-Holland, Amsterdam, 1990\.
  * [39] T. Shiga.  A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes.  J. Math. Kyoto Univ., 30(2):245–279, 1990.
  * [40] P. Śniady.  Quadratic bosonic and free white noises.  Comm. Math. Phys., 211(3):615–628, 2000.
  * [41] A. Sturm, J.M. Swart, and F. Völlering.  The algebraic approach to duality: an introduction.  In Genealogies of Interacting Particle Systems, volume 38 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 81–150. World Sci. Publ., Hackensack, NJ, 2020.
  * [42] M. Talagrand.  What is a quantum field theory?: a first introduction for mathematicians.  Cambridge University Press, Cambridge, 2022.
  * [43] S. Wagner.  Orthogonal Intertwiners for Infinite Particle Systems In The Continuum.  Preprint. arXiv:2305.03367, 2023.