Intertwinings for Continuum Particle Systems: an Algebraic Approach

We develop the algebraic approach to duality, more precisely to intertwinings, within the context of particle systems in general spaces, focusing on the $\mathfrak{su}(1,1)$ 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, Giardin\`a, Groenevelt, and Redig (2019).


Introduction
In recent years the algebraic approach to duality has been developed in the context of interacting particle systems (see, e.g., [42] 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 have the same semigroup, 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., [28]), population genetics models (see, e.g., [13]), (stochastic) partial differential equations (see, e.g., [34]).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., [10,11,24,25]).So far, the algebraic approach to duality has been developed only in the case of particles hopping on discrete spaces.The basic idea of the algebraic approach is as follows: Write the generator of the Markov process as a sum of single edge-generators describing the dynamics of particles among the edges of the underlying graph.Then, identify an underlying Lie algebra such that the single edge-generators are elements of the universal enveloping algebra in a given representation.One can then exploit the commutation relations of the Lie algebra to find symmetries of the generator.This procedure was applied by Carinci, Franceschini, Giardinà, Groenevelt and Redig in the context of selfduality functions for independent random walkers, the symmetric exclusion process and the symmetric inclusion process.For the latter process, a self-duality relation in terms of Meixner polynomials was recovered, which first appeared in [18].
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., [26], 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., [30]).
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(ξ, θ), indexed by parameters ξ ∈ C and θ ∈ R, in a suitable L 2 space of finite counting measures (Section 2.3).The reference measure ρ p,α 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 [37,Section 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 [30] and to representations of the algebra of the square of white noise, the current algebra of sl(2, R), and the finite difference algebra [2,6,41].
We prove that the unitaries U(ξ, θ) belong to the symmetry group of consistent Markov processes that admit the law of the Pascal process as a reversible measure (Theorem 2.3).Put differently, each of these unitaries is a self-intertwiner for the semigroup: U(ξ, θ)P t = P t U(ξ, θ).Consistency roughly means that random removal of a particle and time evolution commute [9].Theorem 2.3 applies in particular to the generalized symmetric inclusion process and thereby generalizes to the continuum item 1 (i) in [7,Theorem 3.1].
For a concrete choice of the parameters ξ and θ, the unitary U(ξ, θ) maps functions supported on n-particle configurations to generalized Meixner polynomials of degree n (Theorem 2.4), up to proportionality constants.This result generalizes item 1 (ii) in [7,Theorem 3.1].As a byproduct to Theorems 2.3 and 2.4, we obtain a new algebraic proof of a self-intertwining relation proven by F. Redig and us in [15], see Corollary 2.5.Our results are complemented by some considerations on how to rewrite infinitesimal generators with our raising, lowering and neutral operators (Section 2.6).
We treat only systems with finitely many particles.However, 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 [28,Section III.4].Accordingly our unitary self-intertwining 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 α(E) = ∞, and the unitary operator is the isomorphism from chaos decompositions for the Pascal point process [29,Corollary 5.3].Such an intertwining relation is proven for sticky Brownian motions in [44,Theorem 3.5].
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 [37]; 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 2.3 and 2.4 are proven in Sections 4 and 5, respectively.Finally, Appendix A gathers some known formulas on univariate Meixner polynomials.

Setting and main results
In the following, Cartesian products are always equipped with product σ-algebras.For finite interacting particle systems on lattices, the configuration space is 0 with a space of finite counting measures: Let (E, E) be a Borel space, for example, E = R d or more generally a Polish space with its Borel σ-algebra.Let N <∞ be the space of finite counting measures on E. Every element η ∈ N <∞ is either zero or a finite sum η = δ x 1 + • • • + δ xn of Dirac measures, where x 1 , . . ., x n ∈ E, n ∈ N. The space N <∞ is equipped with the σ-algebra N <∞ generated by the counting variables η → η(B), B ∈ E. For background on point processes and counting measures, we refer the reader to Last and Penrose [27].

Symmetric inclusion process
We are interested in Markov processes with state space N <∞ .As a guiding example we take the process with formal generator where α is a finite measure on E and c : In [15] we called this process generalized inclusion process.When E is a finite set, we may identify finite counting measures η with vectors n = (n x ) x∈Λ ∈ N Λ 0 , integrals over E turn into sums, and the generator turns into where e x (y) = δ x,y and α y = α({y}).This is the generator of the inhomogeneous symmetric inclusion process (see, e.g., [17, equation (2.2)]), which first appeared in the homogeneous case as a dual process to a model for energy and momentum transport, see [20] 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 [21,33] when E is finite, measure-valued Moran model (see [12,Section 2.6] or [14,Section 5.4]) when E is uncountable.The Moran model describes a population in which individuals have genetic types x ∈ 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 (2.1) (with c(x, y) ≡ 1) coincides with the n-particle Moran model [12, equation (2.5.2)] with so-called mutation operator Aφ(x) = (φ(y) − φ(x))α(dx).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 η ∈ N <∞ but rather with the empirical measure η η(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 [40] and references therein).

Representation of the su(1, 1) current algebra
Let D ⊂ H be the set of bounded measurable functions F for which there exists a cutoff m F ∈ N such that F is supported in {η : η(E) ≤ m F }. We define operators k ± (φ), k 0 (φ) : D → D indexed by bounded measurable functions φ : E → C as follows: for η ∈ N <∞ .The raising operator k + (φ) and the lowering operator k − (φ) map functions supported in {η : η(E) = n} to functions supported in {η : η(E) = n + 1} and {η : η(E) = n − 1} respectively.The indicator 1 {0} that there is no particle at all (vacuum) is annihilated by all lowering operators: k − (φ)1 {0} = 0. Our operators are a representation of the su(1, 1) current algebra, i.e., they represent the matrix-valued maps The following commutation relations can be checked by an explicit computation: where [T, S] := T S − ST .Indeed, it can be readily verified that Subtracting these equations yields the commutation relation The other relations follow similarly.The factor √ p included in the definitions of k ± (φ) is irrelevant for the commutation relations but it matters for the adjointness relation ⟨f, k + (φ)g⟩ = ⟨k − (φ)f, g⟩ proven in Lemma 3.2.We leave a thorough analysis of the current algebra generated by the operators k ± (φ), k 0 (φ) to another article [16] and focus on the operators associated with the constant function φ = 1, equal to 1 on all of E. In Lemma 4.2 below, we check that for every ξ ∈ C, the operator ) is, by definition, the unitary exp(iA).For θ ∈ R, the unitary exp 2iθk 0 (1) is defined as a multiplication operator, it multiplies a function f (η) with exp(iθ(α(E) + 2η(E))).In this way we obtain a family of unitary operators For the definition of the unitary (2.4), it is essential that α has finite total mass and that each configuration η has finitely many particles.

Symmetries of consistent particle processes
Let (P t ) t≥0 be the semigroup of a Markov process with state space N <∞ .For a measurable f : Definition 2.2.The semigroup (P t ) t≥0 is consistent if P t Af = AP t f for all measurable f : N <∞ → R + and all t ≥ 0.
Notice that, up to questions of domains, A = √ p k + (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 A by the total mass η(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 2.3.Let (P t ) t≥0 be the semigroup of a Markov process with state space N <∞ .Let p ∈ (0, 1) and α be a finite measure on E. Assume that the process is consistent, and admits the Pascal law ρ p,α as a reversible measure.Then P t commutes with all unitaries U(ξ, θ) from (2.4): The generalized symmetric inclusion process is a consistent Markov process and admits the Pascal law ρ p,α as a reversible measure (as proved in [15, Theorem 5.2.]) and, thus, Theorem 2.3 applies.

Intertwining with Meixner polynomials
Let P n be the L 2 -closure of the space of polynomials of degree at most n in the counting variables η(B).Thus P n is the closed linear hull of monomials η → η(B 1 ) ) be given by where f (η) := f n dη ⊗n .The orthogonalized version of is a product of univariate Meixner polynomials with leading coefficient 1: Our second main theorem generalizes item 1 (ii) in Theorem 3.1 in Carinci et al. [7].For f : N <∞ → C we define a sequence of functions (f n ) n∈N by 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 artanh. .Then for all n ∈ N and all bounded measurable f : for all measurable bounded f : N <∞ → C and x 1 , . . ., x n ∈ E.
Corollary 2.5.Under the assumptions of Theorem 2.3: For all t ≥ 0, all n ∈ N, and every bounded measurable f n : E n → C, Proof .Apply Theorem 2.3 to U(ξ, θ) for ξ = artanh √ p and θ = 0, combine with Theorem 2.4, and observe that n-dependent constants may be dropped because the process is conservative (see Proposition 4.1 below).■ Furthermore, we note that, in addition to unitary intertwinings, relation [15, equation (3.6)] in terms of the K-transform can also be derived using algebraic techniques: The K-transform can be expressed with the operator k + (1).More precisely, it is equal to exp This relation serves as the continuum counterpart to [9, Lemma 4.2].

Algebraic expression for the generator
On the lattice the generator (2.2) of the symmetric inclusion process can be expressed directly in terms of raising and lowering operators associated with lattice sites [19]: Suppose that α y = α does not depend on y.Let e x ∈ N E 0 be the vector given by e x (y) = δ x,y .Set 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 [19, Section 6.2].
In the continuum, equation (2.5) is problematic as we cannot define operators k # x -in the lingo of quantum field theory, x → k # x only make sense as operator-valued distributions [43, Section 3.7].However, using the operators k # (φ) introduced above, equation (2.5) generalizes nicely, when c(x, y) is of product form c(x, y) = 2φ(x)φ(y).In that case This expression makes perfect sense in the continuum setup as well.Moreover, it can readily be checked that (2.6) holds if L is the formal generator (2.1) of the generalized symmetric inclusion process as well.A similar rewrite is possible if c(x, y) is a linear combination of symmetrized products φ 1 (x)φ 2 (y) + φ 2 (x)φ 1 (y).More generally, say on E = R d with homogeneous measures α(dx) = αdx, one may hope to give meaning to expressions like by first defining it as (2.6) 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 µ(k)a(k) † a(k)dk in quantum field theory and quantum many-body theory (see, e.g., the non-numbered proposition preceding Theorem X.45 in Reed and Simon [37] or Talagrand [43,Section 3.7]).

Papangelou kernel for the Pascal point process
If X is a negative binomial variable with parameters p ∈ (0, 1) and a > 0, then for all n ∈ N 0 , accordingly for all f : N 0 → R + .The following proposition gives the analogous property for Pascal point processes.For future reference we state and prove the proposition for σ-finite measures α, allowing for configurations with infinitely many particles.Thus, let N be the space of s-finite counting measures on E. As E is a Borel space, N consists of the measures η on E that are finite or countable sums of Dirac measures.The space N is equipped with the σ-algebra N generated by the counting variables B → η(B), B ∈ E.
A Pascal point process with parameters p and α is a random variable η : (Ω, F, P) → (N, N ) that satisfies the properties listed in the definition of ρ p,α : Counting variables for disjoint regions B i are independent, and each counting variable η(B) has a negative binomial distribution with parameters p and α(B).For sets with infinite mass α(B) = ∞, the counting variable η(B) is almost surely infinite.Pascal point processes have Laplace functional 1 − e −jf (x) p j j α(dx) for measurable f : E → R + .
Proposition 3.1.Fix p ∈ (0, 1) and a σ-finite measure α on E. Let η be a Pascal point process with parameters p and α.Then for all measurable F : E × N → R + .
The proposition says that the kernel E × N → R + , (x, B) → κ(x, B) = p(α(B) + η(B)) is a Papangelou kernel for the Pascal point process.For a general discussion on Papangelou kernels, we refer to [31] or [35, Let φ : E × N(E) → R + be a measurable function.We apply the Mecke equation for Π to G (x, n), i δ (x i ,n i ) := nφ x, i n i δ x i and get Using (3.1) for φ(x, µ) = F (x, µ + δ x ), we obtain We apply (3.1) to φ(x, µ) = F (x, µ) and eliminate the sum over n ≥ 2 using (3.2).This gives Equivalently, and the proof is complete as ξ and η are equal in distribution.■ Turning back to the law ρ p,α on N <∞ for finite measures α, we obtain for all measurable F : E × N <∞ → R + , hence also all bounded measurable functions F (here we use that the expected total number of particles is finite when α has finite total mass).
Lemma 3.2.For all bounded measurable φ : E → C and all f, g ∈ D, Proof .Let us denote integration with respect to ρ p,α as E[f (η)] = f dρ p,α .We apply equation (3.3) to F (x, η) = φ(x)f (η)g(η − δ x ) and obtain Before we turn to the proof of Theorem 2.3, 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≥0 be Markov semigroup on (N <∞ , N <∞ ) that is consistent, i.e., P t Af = AP t f for all t ≥ 0 and all measurable f : N <∞ → R + .Then, for all k ∈ N, A straightforward computation shows that A k 1 is a descending factorial power of the total number of particles, for all η ∈ N <∞ .Let (η t ) t≥0 be a Markov process with semigroup (P t ) t≥0 and deterministic initial condition η 0 = µ ∈ N <∞ , defined on some probability space Ω, F , P µ .Equations (4.1) and (4.2) show that under P µ , at all times t ≥ 0, the total number of particles η t (E) has the same factorial moments as η 0 (E).The factorial moments of η 0 (E) = µ(E) (hence η t (E)) vanish for k ≥ µ(E) + 1, therefore the moment problem is uniquely solvable and we conclude η t (E) = η 0 (E) = µ(E), P µ -almost surely, and the process is conservative.■ Remember that the set D ⊂ H consists of the bounded measurable functions f supported in {η : η(E) ≤ n f } for some n f < ∞.Lemma 4.2.For every ξ ∈ C, the operator 1 i ξk + (1) − ξk − (1) with domain D is closable and its closure is self-adjoint.
Put differently, the operator is essentially self-adjoint on D and the latter is a core for the self-adjoint closure [36,Section VIII.2].
Proof .The operator A = 1 i ξk + (1) − ξk − (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 [37,Theorem X.41 (a)].The domain D is dense in H and satisfies ⟨f, Ag⟩ = ⟨Af, g⟩ for all f, g ∈ D by Lemma 3.2.Thus, A with domain D(A) = D is symmetric.Symmetric operators are always closable.To show that the closure is self-adjoint, we check that every vector f ∈ D is analytic for A and conclude with Nelson's analytic vector theorem [37,Theorem X.39].As a reminder, Nelson's analytic vector theorem states that a symmetric operator on a Hilbert space whose domain contains a total set of analytic vectors is essential self-adjoint.A vector f is analytic for A if there exists ε > 0 such that for all η ∈ N <∞ .Set β := α(E), we obtain a common upper bound to k ± (1)f : We expand the power k + (1) − k − (1) n f and bound each term in the expansion by repeated use of the previous inequality.This gives The indicator on the right has norm which is finite for ε < 1/s.Thus, every vector f ∈ D is analytic and A is essentially self-adjoint on D. ■ Proof of Theorem 2.3.Let t ≥ 0. As ρ p,α is reversible, P t is a well-defined, self-adjoint and bounded operator on L 2 (N <∞ , N <∞ , ρ p,α ).The operator exp 2iθk 0 (1) acts as multiplication with exp(iθ(α(E) + 2η(E))), it commutes with P t because (P t ) t≥0 preserves the total number of particles (see Proposition 4.1 below).
For exp ξk + (1) − ξk − (1) , the short informal reasoning is simple: P t is self-adjoint and commutes with k + (1), therefore it also commutes with the adjoint k − (1) and also with the difference ξk + (1) − ξk − (1) and the exponential exp ξk + (1) − ξk − (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 D. Arguing as in the proof of Lemma 3.2, we obtain ⟨k − (1)f, P t g⟩ = ⟨f, AP t g⟩.Therefore, ⟨k − (1)f, P t g⟩ = ⟨f, AP t g⟩ = ⟨f, P t Ag⟩ = P t f, k + (1)g for all non-negative f, g ∈ D. Taking complex conjugates and switching f and g, we get k + (1)f, P t g = ⟨P t f, k − (1)g⟩ and then, by linear combination, for all f, g ∈ D. By Lemma 4.2, the operator −i ξk + (1) − ξk − (1) with domain D is closable and its closure A is self-adjoint.Taking limits in the above equation, we obtain ⟨Af, P t g⟩ = ⟨P t f, Ag⟩ for all f , g in the domain of A. In particular, for fixed g in the domain of A, and all f ∈ D(A), we have ⟨Af, P t g⟩ = ⟨f, P t Ag⟩, hence P t g is in the domain of A * and A * P t g = P t Ag.
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 [38, Proposition 5.26] and then, by spectral calculus, P t exp(iA) = exp(iA)P t .■ 5 Unitary operator vs. Meixner polynomials.Proof of Theorem 2.4 Here we prove Theorem 2.4.Our proof is similar in spirit to the proof with generating functions from Carinci et al. [7], see in particular Proposition 5.4 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 → C with sup |z| < 1, we define the exponential state The exponential state owes its name to the relation following analogous arguments as in [15, equation (5.7)].For ξ ∈ C, consider the matrix The matrix is in SU(1, 1), the group of 2 × 2 matrices of the form Proof .To lighten notation we write down the proof for real-valued ξ only, the general case is similar.Thus let , and (iii) ∂ t f t = iAf t .The key equations are (5.1),(5.3), (5.4), and (5.5).Items (i)-(iii) imply f t = exp(itA)f 0 for all t ∈ R. The proposition follows upon specializing to t = ξ.
(i) The map t → z t is differentiable, pointwise for all x and actually in supremum norm The multiplier C(t) is differentiable with derivative To prove that the map t → E zt is differentiable, we express For the empty configuration η = 0, we note that both E z t+h (0) and E zt (0) are equal to 1 hence the difference vanishes.For single-particle configurations η = δ x 1 , we have E z t+h (δ x 1 ) − E zt (δ x 1 ) = v t,h (x 1 ).For n ≥ 2, we have We define a function Q t,h : (this is precisely the second term in (5.2)).Let us check that ∥Q t,h ∥ = O h 2 as h → 0. At fixed number of particles n ≥ 2, Q t,h is bounded by Therefore, the L 2 -norm satisfies It follows that t → E zt is norm-differentiable with derivative for n ≥ 1 and ∂ t E zt (0) = 1.The map t → f t is norm-differentiable with derivative (ii) We show that every exponential vector E z , ∥z∥ ∞ < 1, is in the domain of the closure of It follows that k + (1) N n=0 ψ n converges in norm.Turning to k − (1), we note and ∞ n=1 ∥k − (1)ψ n ∥ 2 < ∞ as well.Therefore, the partial sums S N = N n=0 ψ n are in D, they converge in norm to E z , and k + (1) − k − (1) S N converges in norm to n k + (1) − k − (1) ψ n .It follows that E z is in the domain of the closure of k + (1) − k − (1) and The function f t = C(t)E zt is in the domain of the closure too.
(iii) Comparing the previous equation with (5.4), (5.3), and (5.1), we get The proof is complete.■ Remark 5.2.The proof can be concluded using an explicit formula given in [30,Proposition 3.1] for generating functionals G φ : N <∞ → C defined by for measurable φ : E → C with sup |φ| < 1: Applying Proposition 5.1 to exponential vectors E φ √ p with φ an arbitrary bounded measurable function, we find that (1 − p) −α(E)/2 exp ξk + (1) − ξk − (1) maps the function f : N <∞ → C given by to G φ , from which we deduce that Theorem 2.4 holds true when f n = φ ⊗n , much in the same way we prove Proposition 5.4 below.The general statement follows with polarization formulas and (multi)linearity, and density arguments-notice that (f a symmetric multilinear mapping and as such, uniquely determined by its value for 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 [30, Proposition 3.1] which uses the machinery of Jacobi fields and distribution theory).
We start by applying Proposition 5.1 to exponential vectors associated with indicators. .
Proof .The function f s is the exponential state associated with z(x) = s √ p1 B (x).The transformed z is and the associated exponential state is .
The multiplier is In the last line we have used (cosh ξ) 2 = 1/ 1 − (tanh ξ) 2 .We apply Proposition 5.1 and obtain the corollary.■ We remind that the univariate monic Meixner polynomials are denoted by M n (x; α, p), see Appendix A below.
Proof .Let f (0) := 1 {0} .For s ∈ (−1, 1), with the notation of Corollary 5.3, The series converges in norm as the f (n) 's are orthogonal and ∞ n=0 |s| 2n f (n) 2 < ∞.As U is unitary, hence continuous, we may exchange summation and the application of U which gives and the series on the right converges in norm.On the other hand, in the expression for U f s from Corollary 5.3, we recognize the generating function of Meixner polynomials with leading coefficient (1 − p) n : see equation (A.2).The right side is not only absolutely convergent, pointwise for each η ∈ N <∞ , but also absolutely convergent in the Hilbert space H. Indeed as η → η(B), under ρ p,α is a negative binomial variable with parameters α(B) and p (see equation (2.3)), the orthogonality relations for univariate Meixner polynomials recalled in Appendix A tell us that the summands are orthogonal and The sum over n ∈ N 0 of the right side is 1 − ps 2 α(B) and in particular, finite.Therefore, the series on the right side in (5.7) converges in norm.Thus, we have two convergent series expansions for the analytic function s → U f s from (−1, 1) to H.As the expansion coefficients are uniquely determined, the coefficients of s n in equations (5.6) and (5.7) must be the same.The proposition follows. ■ The symmetric tensor product of f 1 , . . ., f n : E → C is the function from E n to C given by Above, S n denotes the set of permutations of the numbers {1, . . ., n}.Consider disjoint sets B 1 , . . . ,B ℓ , integers n 1 , . . . ,n ℓ ∈ N adding up to N, and The associated function f is supported on configurations that have exactly n i particles in B i , for each i = 1, . . ., ℓ, and n particles in total.On the relevant configurations there is an additional combinatorial contribution.One finds 1 {η(B i )=n i } . (5.8) Set B 0 := E \ ℓ i=1 B i , n 0 = 0, and f i (η) = 1 {η(B i )=n i } , i = 0, . . ., ℓ. (5.9) The next lemma says that U maps the product of f i 's to products of U f i 's.
The map t → G t from R → H is norm-differentiable, G t is in the domain of the closure L of k + (1) − k − (1), and d dt G t (η) = LG t .It follows that G t = V (t)G 0 .We apply the statement to g j (n) = 1 {n=n j } and t = ξ and deduce (U j (ξ)g j )(η(B j )). (5.10) Finally, we notice that each term on the right side is given by a univariate Meixner polynomial as (U j (ξ)g j )(m) = 1 n j !(1 − p) n j +α(B j )/2 M n j (m; α(B j ), p). with ℓ ∈ N, n 1 , . . ., n ℓ ∈ N 0 , and B 1 , . . ., B ℓ ∈ E (not necessarily disjoint).These sets form a generating π-system of the σ-algebra N n on N n = {η : η(E) = n} that is the trace of N <∞ on N n .We conclude with a monotone class argument.Let V be the set of bounded functions f : N ∞ → C supported in N n for which U f = n! −1 (1 − p) n+α(E)/2 I n (f n ).Clearly V is a vector space.We check that it is closed under monotone pointwise limits of uniformly bounded sequences: Suppose f (1) ≤ f (2) ≤ . . .and f (m) → f with C = sup n f (m) ∞ < ∞ and f (m) ∈ V for all m.Then, on the one hand f (m) → f also in L 2 -norm and U f (m) − U f → 0 because U is unitary.On the other hand

Proposition 5 . 4 .
Fix B ∈ E. The unitary U maps the functions f
Section 1.2.2].Proof .Pascal point processes are compound Poisson; we deduce the proposition from the Mecke formula for Poisson point processes.Let Π = i δ (X i ,N i ) be a Poisson point process on E × N 0 with intensity measure β = α ⊗ δ n .Then ξ = i N i δ X i is a Pascal point process with law ρ p,α .The Mecke equation for Π [27, Theorem 4.1] and measurable n 1 {0} , compare Meyer [32, Section IV.1].The state has norm By definition of z t , the set {z t (x) : x ∈ E} is contained in the image of the diskD 0 := {ζ ∈ C : |ζ| ≤ ∥z 0 ∥ ∞ } under the Möbius transform ζ → (ζ + tanh t)/(1 + ζ tanh t).It is known that the Möbius transform maps the open unit disk to itself, and it maps compact sets to compact sets because it is continuous.Furthermore, D 0 is compact and contained in the open unit disk (because ∥z 0 ∥ ∞ < 1).Therefore, the Möbius transform maps D 0 to a compact subset K t of the open unit disk.In particular, K t stays away from the boundary of the unit disk and we deduce ∥z t ∥ ∞ < 1, and then κ < ∞.The differentiability of t → z t in supremum norm yields ∥v t,h