Duality and stationary distributions of the"Immediate Exchange Model"and its generalizations

We prove that the"Immediate Exchange Model"of econophysics has a discrete dual, where the duality functions are those connecting the Brownian Energy Process and the Symmetric Inclusion Process. As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties of a generalized model, where the exchange fraction is $Beta(s,t)$ distributed (instead of uniform), and product measures with $\mbox{Gamma}(s+t)$ marginals are invariant. We prove that the discrete dual has the self-duality property, and prove full SU(1,1) for both the continuous and discrete model.


Introduction
Kinetic wealth exchange models (KWEMs) constitute a popular class of econophysical models in which agents exchange their wealth according to some stochastic rules, always preserving the total amount of wealth in the economy. The aim is to understand some important properties of the dynamics of wealth distribution, such as wealth concentration, stationary distributions and time dependent correlation functions. For a recent review about KWEMs, we refer to [3]. The apparently economically strong assumption of wealth conservation -which also rules out the possibility of (endogenous) growth -is justifiable by choosing the appropriate time scale (or time unit) for the economy. An interesting feature of KWEMs is their similarity with another family of models, known as KMP [1]. Introduced in [7], KMP models are microscopic models of heat conduction and are meant to provide a microscopic foundation of the Fourier law; in those models the exchanged quantity represents energy. As shown in [1], duality is a powerful tool to study the properties of KMP models. Thanks to duality it is possible to investigate invariant measures, ergodic results, and important macroscopic properties such as hydrodynamic limits, the propagation of local equilibrium, and the local equilibrium of boundary-driven non-equilibrium states. In [4], the authors show that duality can also be fruitfully applied to kinetic wealth exchange models, obtaining relevant information about the stationary distributions of a model with saving propensities.
In this paper we aim to extend the use of duality techniques in the field of KWEMs, by focusing our attention on a recent model, the so-called "Immediate Exchange Model". The model has been first proposed in [5], where it is studied via simulations, and it has been later analytically explored in [6]. In that model, upon exchange, each agent gives a fraction of his/her wealth to the other. In [6] it is proved that, if this fraction is a uniformly distributed random variable with support [0, 1], then the exchange process is characterized by an invariant measure, which can be expressed as the product of Gamma (2) distributions. It is now worth noticing that an invariant measure which is a product of Gammas also occur in the redistribution models presented in [1]. In these models duality is characterized by duality polynomials that are naturally associated with the Gamma distribution and it is shown that these polynomials are also the duality functions linking a discrete particle system, the symmetric inclusion process SIP (k), with a diffusion process, the Brownian energy process BEP (k). It is therefore natural to conjecture that these polynomials also occur as duality functions in the Immediate Exchange Model of [5], relating this model to a simpler discrete dual model. In this paper we show that this is indeed the case, and we generalize the Immediate Exchange Model to the case in which the random fraction of wealth the agents exchange is Beta(s, t) distributed. In this more general setting, the invariant measure shows to be a product of Gamma(s + t) distributions. As in [4], using duality we are able to directly infer basic properties of the time-dependent expected wealth, together with an ergodic result.
The rest of our paper is organized as follows: in Section 2 we describe the Immediate Exchange Model when the economy is just made up of two agents and prove duality with a discrete two-agent model. In Section 3 we extend the model to the case of many agents and we give some relevant consequences of duality. In Section 4 a further generalization is proposed, by assuming Beta(s, t)-distributed exchanged fractions of wealth; also for this generalized model we obtain duality with a discrete model and stationary product measures which are Gamma with shape parameter s + t. In Section 5 we study various properties of the discrete dual process, which is an interesting model in itself. We characterize its reversible product measures and prove that in an appropriate scaling limit it scales to a simple variation of the original continuum model. Finally, in section 6 we show self-duality of the discrete model for the general case via a Lie algebraic approach, where we actually obtain the full SU (1, 1) symmetry of the discrete model, and, as a further consequence, of the continuous model, too. Self-duality then follows by acting with an appropriate symmetry on the so-called cheap duality function obtained from the reversible product measure [2].
2 The Immediate Exchange model with two agents and its dual

Definition of the model
We start by considering a toy economy with just two agents, as given in [5] and [6]. More complex models can be built by addition of two-agent generators along the edges of a graph. Most properties such as duality and self-duality transfer immediately from the two-agent model to the many agent models. More formally, we write (x, y) ∈ Ω, with Ω = [0, ∞) 2 . With s = x + y we indicate the total wealth in the economy. Then the dynamics of two agents is described as follows, starting from an initial state (X 0 , Y 0 ) = (x, y), after an exponential waiting time (with mean one), an exchange of wealth occurs, whereby the wealth configuration (x, y) is updated towards (x , y ), with where U and V are two i.i.d. U nif orm(0, 1) random variables. This gives a continuous-time Markov jump process (X t , Y t ) for which the total wealth The infinitesimal generator of this exchange process is defined on bounded continuous functions f via Notice that L can be rewritten as P −I, where P is the discrete-time Markov transition operator and I is the identity. We denote (X 0 , Y 0 ) = (x, y) to be the initial wealth configuration of the two agents, and (X t , Y t ) indicates the wealth of the two agents at time t ≥ 0.

Duality for the two-agent model
We will now first define a discrete wealth distribution model, i.e., where wealth can only be a nonnegative integer quantity. See the figure for the continuous model and its discrete dual. This model will be related to the original one via a duality relation.
Hence, in the discrete model the couple (x, y) ∈ Ω is replaced by a couple (n, m) ∈ N 2 , where N denotes the set of non-negative integers (including zero). On this couple we define a continuous-time Markov process with generator L f (n, m) = n k=0 m l=0 1 n + 1 In this process, when initiated at (n, m), for a given k, l with 0 ≤ k ≤ n, 0 ≤ l ≤ m, the wealth configuration changes from (n, m) to (n−k+l, m−l+k) at rate 1 (n+1)(m+1) . We denote this discrete state space continuous-time Markov process by (N t , M t ), with (N 0 , M 0 ) = (n, m). We will then show that the processes (X t , Y t ) and (N t , M t ) are related via duality. To introduce this, we need some further notation. Define, for x ∈ [0, ∞), n ∈ N, the polynomial d(n, x) = x n Γ(2) Γ(2 + n) = x n (n + 1)! (4) and D(n, m; x, y) = d(n, x)d(m, y).
The d(n, ·) polynomials are naturally associated to the Gamma distribution ν θ with shape parameter 2 and scale parameter θ, i.e.
With a slight abuse of notation, we will denote by ν θ (dxdy) the product measure with marginals ν θ .
We are now ready to state the first main result.
The processes (X t , Y t ) and (N t , M t ) are each others dual with duality function given by (5).

PROOF.
To prove (6) it is sufficient to show the same relation at the level of the generators. In other words, we have to show that LD(n, m; x, y) = L D(n, m; x, y), for all (x, y) ∈ [0, ∞) 2 and (n, m) ∈ N 2 , and where L works on (x, y), and L on (n, m). We compute P D(n, m; x, y) = Therefore, we indeed find that LD(n, m; x, y) = n k=0 m l=0 1 n + 1 As a consequence of duality, and thanks to the relation between the duality functions and the measure ν θ , we obtain relevant information about the invariant measures. Let us denote by P f the set of probability measures on [0, ∞) 2 with finite moments of all order and which are such that their finite moments determine the probability measure uniquely. I.e., two measures in P f with identical moments are equal. We say that such a measure satisfies the "finite moments condition". Similarly for a probability measure on [0, ∞) we say that it satisfies the "finite moments condition" if it has finite moments of all order and which are such that these finite moments determine the probability measure uniquely. This is e.g. assured by the Carleman's moment growth condition. We will focus from now on only probability measures in this set P f . for all n, m ∈ N. In particular the product measures ν θ (dxdy) are invariant for the process (X t , Y t ).
Combining this with duality and Fubini's theorem we obtain As a result, we find that ν is invariant if and only if To show the invariance of the ν θ measures, just notice that and recall that in the process (N t , M t ) the sum N t + M t is conserved.
Another consequence of duality is the ergodicity of the process (X t , Y t ). i.e., starting from any initial condition (x, y) the process converges to a unique stationary distribution determined by the conserved sum x + y. Indeed, the dual process starting from (n, m) is an irreducible continuous-time Markov chain on the finite set {(k, l) : k+l = n+m} and therefore converges to a unique stationary distribution denoted by ν n+m , given by where For all (n, m) ∈ N 2 we can therefore observe that It then follows from an easy computation using (11) that k,l:k+l=n+m where Z n+m is given by (12) i.e., the limit in the r.h.s. of (13) only depends on x+y. On the other hand, in the process (X t , Y t ) we know that S = x+y is conserved. Therefore, the conditional measure obtained by conditioning the stationary product measure ν θ on the sum being equal to s is an invariant measure concentrating on the set This measure is exactly the distribution of (s , s(1 − )), with being Beta(2, 2). If we combine this fact with (13), we obtain the following ergodic theorem and complete characterization of the set of invariant measures satisfying the finite moment condition.
b) The set of invariant measures contained in P f is given by the distributions of the form (S , S(1 − )) where S is an arbitrary distribution on [0, ∞) satisfying the finite moments condition and ∼ Beta(2, 2).

Generalization to many agents
Consider now an economy populated by many agents. Let us assume that the economy can be represented as a graph S, where each vertex represents an agent. Consider now an irreducible symmetric random walk kernel p(i, j) on S, i.e., such that p(i, j) = p(j, i) ≥ 0, j p(i, j) = 1, and for all i, j ∈ S there exists n with p (n) (i, j) > 0. In this setting, the wealth configuration of the economy is an element of the set Ω = [0, ∞) S . For x ∈ Ω (from now on simply x), we denote with x i the wealth of the agent i, that is of vertex i. We then define the generator of the model via with Accordingly, the dual process has state space N S and the elements of this state space are denoted by ξ (from now on just ξ), where ξ i is the number of "dual units" at vertex i. A configuration ξ is called finite if |ξ| = i ξ i is finite. The generator of the dual process is then with The relation between these duality polynomials and the product measure In the many agents economy model, the duality relation between both processes is then given by the following theorem. Its proof is direct from the two agents case, because the generator is a sum of two agents generators.
As a consequence, the product measures Notice that when S is finite, the product measures ⊗ i∈S ν θ (dx i ) can never be ergodic because the total wealth is conserved. However, for infinite S, we have ergodicity under an additional condition. Let us denote by p t (ξ, ξ ) the probability to go from the finite configuration ξ ∈ N S to the finite configuration ξ in time t > 0, in the dual process with generator (15). Assume that lim for all ξ, ξ ∈ N S . As an example we have S = Z d and p(i, j) symmetric nearest neighbor random walk. PROPOSITION 3.1. Let S be infinite and let p(i, j) be such that (20) holds. Then the product measure ⊗ i∈S ν θ (dx i ) is ergodic PROOF. Abbreviate ν := ⊗ i∈S ν θ (dx i ). Because ergodicity is implied by mixing, it suffices to show that because linear combinations of the polynomials D(ξ, x) are dense in L 2 (ν θ ). To prove (21) denote ξ ⊥ ξ if the support of ξ and ξ are disjoint, i.e., if there are no vertices i ∈ S which contain both particles from ξ and ξ . If ξ ⊥ ξ then under the measure ν θ , the polynomials D(ξ, ·) and D(ξ , ·) are independent. Because of (20) it then follows, using duality and conservation of the total number of particles in the dual process: Notice that, for a single dual particle, that is to say when ξ = δ i , we have In the dual process, the motion of single dual particle is a continuous-time random walk jumping with rate p(i, j) from i to j. If we denote by p t (i, j) the time t > 0 transition probability of this walk, then duality with a single dual particle implies the following "random walk" spread of the expected wealth at time t > 0. PROPOSITION

3.2.
In the model with generator (15), for all x ∈ Ω and i ∈ S we have

Generalized immediate exchange model
Consider the update rule (1) and assume that U and V are now independent and B(s, t) distributed (the original model is then recovered for s = t = 1).
In other words, we consider the generator is the probability density of two independent B(s, t) distributed random variables.
As before, the generator can be rewritten as L = P − I, where I is the identity and P the discrete Markov transition operator In this generalized setting, the polynomials which we need for duality are now given by and D s,t (n, m; x, y) = d s,t (n, x)d s,t (m, y).
These polynomials are associated to the Gamma distribution ν s+t θ (dx) with shape parameter s + t, As before, with a slight abuse of notation we also denote ν s+t θ (dxdy) the product measure with marginals ν s+t θ (dx). The same computation as the one following (7) now yields that for a given k, l with 0 ≤ k ≤ n, 0 ≤ l ≤ m, the dual process will jump from (n, m) towards (n − k + l, m − l + k), at rate r s,t (n, m; k, l) = n!m! B(s, t) 2 where the factorials are to be interpreted as x! = Γ(x + 1), when x is noninteger. Notice that as before in (8) we have that the rates sum up to one n k=0 m l=0 r s,t (n, m; k, l) = 1.
This follows via rewriting r s,t (n, m; k, l) = w s,t (n, k)w s,t (l, m) with w s,t (n, k) = n!(k + s − 1)!(n − k + t − 1)! B(s, t)(s + t + n − 1)!k!(n − k)! and recognizing the probability mass function of the Beta binomial distribution with parameters n, s, t, given by as a consequence one has n k=0 w s,t (n, k) = 1 We can then state the generalized duality result, and its consequences, as in theorem 2.1. The dual process when initiated at (n, m) is once more an irreducible continuous-time Markov chain on the finite set {(k, l) : k +l = n + m} which converges to unique stationary distribution which we denote by denote ν s+t n+m (k, l) and is given by Notice now that we have the analogue of (14), i.e., if we the product measure ν s+t θ (k, l) conditioned on k + l = n + m then is only a function of x + y. As a consequence, we obtain the following result in the generalized model. 1. The processes (N t , M t ) and (X t , Y t ) with generator (22) and rates (26) are dual with duality function (24). This means that, for all (n, m) ∈ N 2 and (x, y) ∈ [0, ∞) 2 , we have E s,t x,y D s,t (n, m; X t , Y t ) = E s,t n,m D s,t (N t , M t , x, y).
2. As a consequence, the product measure ν s,t θ (dxdy) is invariant.
3. Moreover, starting from any initial state (x, y), the process (X t , Y t ) converges in distribution to (S , S(1 − )) where is Beta(s + t, s + t)distributed, and S = x + y = X 0 + Y 0 .
4. The invariant measures with finite moments are of the form (S , S(1− )), with is Beta(s + t, s + t)-distributed.
We can then build the analogue of this model for many agents associated to the vertices of a graph S, as in equations (15) and (16). First notice that for a single dual particle, when ξ = δ i , we get Just as before, the motion of single dual particle in the dual process is a continuous-time random walk, jumping with rate p(i, j) from i to j. If we denote by p t (i, j) the time t > 0 transition probability of this walk, we then have the following result.

Properties of the discrete dual process
The discrete dual process is a redistribution model of independent interest. It is a discrete redistribution model of the same type as the original continuous model in the spirit of the KMP process and its analogues of [1], where also a similar discrete process is the dual of the original continuous redistribution model. It is therefore useful to understand better the discrete dual process and its connection to the original process.

Reversible measures
Define the discrete Gamma distribution with shape parameter s+t and scale parameter 0 < θ < 1 as the probability measure on N with probability mass function where Z θ = (1 − θ) −s−t is the normalizing factor. We first recall that the dual process has generator (32) where the rates are given by (26). It is important to notice here that this generator can be rewritten as follows is Beta binomial distributed with parameters n, s, t and X 2 = X (m) 1 independent Beta binomial with parameters m, s, t, and E is expectation w.r.t. these variables.  For all θ ∈ (0, 1), the product probability measures with marginals ν s+t θ (n) are reversible for the process with generator (32).
PROOF. The reversibility of ν s+t θ for the generator L follows from a standard detailed balance computation. Indeed, fix two configurations (n, m) and (n , m ) ∈ N 2 with n + m = n + m ; now, for any 0 ≤ k ≤ n and 0 ≤ l ≤ m such that n = n − k + l and m = m − l + k, it trivially follows that l ≤ n = n − k + l and k ≤ m = m − l + k and n = n − l + k, m = m − k + l. In other words, for each redistribution of (n, m) according to r(n, m; k, l), we can find a "reverse" redistribution of (n , m ) according to r(n , m ; l, k). Furthermore, these two redistributions are indeed reversible, as one may see by explicit computation, combining (26) and (31) that r(n, m; k, l)ν s+t θ (n)ν s+t θ (m) = r(n+l−k, m+k−l; l, k)ν s+t θ (n+l−k)ν s+t θ (m+k−l) which implies detailed balance and thus reversibility.

Scaling limit
The fact that the rescaled Beta Binomial converges to the Beta distribution (by the law of large numbers) provides a connection between the discrete dual process and the continuous process. The continuous process arises as a limit of the discrete dual process where the number of initial "coins" is suitably rescaled to infinity. This is expressed in the following result. THEOREM 5.1. Let n K , m K be a sequence of integers indexed by K ∈ N, and such that as K → ∞. Then we have that the corresponding processes n K (t)/K, m K (t)/K, with generator (32) converge to the continuous process with generator (22), starting from (x, y).
PROOF. Define a number A > x + y. Because convergence of generators on a core implies convergence of the processes, it suffices to show that for smooth f : where f K (n, m) = f (n/K, m/K), L is given by (32), and L s,t by (22). Consider X (n K ) Beta binomial with parameters n K , s, t, and X (m K ) independent Beta binomial with parameters m K , s, t. By the law of large numbers it follows that with Y s,t , Y s,t being independent B(s, t) distributed. Therefore, by smoothness of f and dominated convergence, as K → ∞ we have which shows (34).
6 Self duality and SU (1, 1) symmetry of the dual process In this section we show self-duality with the self-duality polynomials which are naturally associated to the reversible discrete Gamma distributions. More precisely, we define the following discrete polynomials: where negative factorials are defined to be infinite. These polynomials are naturally connected to the discrete reversible Gamma distribution via n d s,t (k, n)ν s+t θ (n) = ρ(θ) k with ρ(θ) = θ/(1 − θ). Next we have the associated polynomial in two variables: D s,t (k, l; n, m) = d s,t (k, n)d s,t (l, m) Notice that in the case n = N x , m = N y , divided by N k+l , and in the limit N → ∞, these discrete polymials converge to the duality polynomials (24). We recall that the dual process has a generator of the form To prove self-duality of the process with generator (32), we show that it commutes with a SU (1, 1) raising operator K + 1 + K + 2 , from which we can generate the self-duality function via the strategy described in [2], namely by acting with e K + 1 +K + 2 on a cheap self-duality function coming from the reversible product measure.
In order to proceed with this, we introduce the SU (1, 1) raising operators [9], For a function f (n, m) of two discrete variables, we denote K + 1 , resp. K + 2 the operator K + defined in (38) working on the first (resp. second) variable.
Similarly we have the lowering and diagonal operators Together, the K − , K + , K 0 generate a discrete (left) representation of SU (1, 1); i.e. they satisfy the SU (1, 1) commutation relations where [A, B] = AB − BA denotes the commutator. We will show in this subsection that the generator L defined in (32) has SU (1, 1) symmetry and that the self-duality follows as a consequence, in the spirit of [9], [1]. We start by noticing that by reversibility of the measure ν s+t θ , the function D(n , m ; n, m) = δ n ,n δ m ,m n!Γ(s + t) Γ(s + t + n) m!Γ(s + t) Γ(s + t + m) is a "cheap" self-duality function [9], [2]. Furthermore, we remark that the claimed self-duality polynomials can be obtained via D(n , m ; n, m) = e K + 1 +K + 2 D(n , m ; n, m) where the operator e K + 1 +K + 2 is working on the n , m variables. Therefore, in order to prove that self-duality holds with the claimed polynomials (35), (37), it suffices to prove that K + 1 + K + 2 commutes with the generator. Indeed, then from the general theory developed in [9], see also [2], it follows that e K + 1 +K + 2 D(k, l; n, m), which arises from the action of a symmetry (an operator commuting with the generator) on a self-duality function, is again a self-duality function. THEOREM 6.1. The generator L in (32) and the operator K + 1 + K + 2 commute, i.e., for all f : N 2 → R we have c ; z is defined via the following series expansion Note that for all n, k ∈ N and t ∈ R + , and Moreover, as a particular case of Gauss's summation theorem ([8], Theorem 2), we can state that Some useful formulas are listed below: n k=0 Γ(s + k) Γ(1 + k) PROOF. Let us prove that for all functions f : N 2 → R and (n, m) ∈ N 2 By straightforward computations and substitutions, if we adopt the notation a b s,t := Γ(a + s + t) , a ≥ b ≥ 0, s, t > 0, the l.h.s. rewrites (K + := K + 1 + K + 2 ) while the r.h.s K + P f (n, m) = (s + t + n)P f (n + 1, m) + (s + t + m)P f (n, m + 1) Let us introduce another shortcut: As it is enough to show the identity only for the functions f : N 2 → R in the form f (n, m) := θ n 1 θ m 2 , θ 1 , θ 2 ∈ (0, 1), (n, m) ∈ N 2 , we can recast (42) as follows: and similarly forM andM , we can continue with Note that, as in Remark 6.1, we can rewrite these quantities N ,N etc., in terms of hypergeometric functions as follows In conclusion, if we go back and plug the latter expression into (44), we can rewrite the l.h.s. in (43) as By simply replacing n by m, θ 1 by θ 2 etc. and exchanging the sign in the latter expression, one simply obtains the explicit form of the r.h.s. in (43), which indeed proves identity (42).
We extend now the commutation of the generator with K + 1 + K + 2 to full SU (1, 1) symmetry of both the discrete and the continuous model. For this we need some additional notation. Denoting the operators K α working on we have that the algebra generated by K α forms a (right) representation of SU (1, 1), i.e., satisfy the commutation relations (40) with opposite sign. Moreover, this continuous right representation is linked with the discrete left representation used before via the duality polynomials (23), i.e., where K works on x, and K on n (see e.g. [2] for the proof). We now first formulate a simple lemma, showing that θ −1 K − is the adjoint of K + in L 2 (ν θ ). LEMMA 6.1. Let ν s+t θ be the reversible measure for the discrete dual process, defined in (31). We have in L 2 (ν s+t θ ) where K α are the operators introduced in (38),(39).
PROOF. We start with the discrete process. Because the sum of the wealths is conserved, L trivially commutes with K 0 1 + K 0 2 . We showed in (41) that it commutes with K + 1 + K + 2 . To show that it commutes with K − 1 + K − 2 we use lemma 6.1 and the fact that L is self-adjoint in L 2 (ν s+t θ ) by the reversibility of ν s+t θ .
[L , K − 1 + K − 2 ] = θ[L * , (K + 1 + K + 2 ) * ] = −θ [L , (K + 1 + K + 2 )] * = 0 We then turn to the continuous model, using (48). We show the commutation with K + 1 +K + 2 , the other cases are similar. We consider D s,t (n, m; x, y), the duality polynomial defined in (24), (25), and abbreviate it simply by D, where in what follows we tacitly understand that operators of the form K are working on x, y and of the form K on n, m. In this notation, remark that operators working on different variables always commute (e.g. K commutes with L , etc.). We can then proceed as follows, using duality which reads L D = LD.
On the other hand, via (48) where in the third equality we used the commutation of L with K + 1 + K + 2 . Combination of these computations then gives indeed on the functions D, and then by standard arguments on all f in L 2 (ν s+t θ ).