Competing Particle Systems Evolving by I.I.D. Increments

We consider competing particle systems in R d , i.e. random locally ﬁnite upper bounded conﬁgurations of points in R d evolving in discrete time steps. In each step i.i.d. increments are added to the particles independently of the initial conﬁguration and the previous steps. Ruzmaikina and Aizenman characterized quasi-stationary measures of such an evolution, i.e. point processes for which the joint distribution of the gaps between the particles is invariant under the evolution, in case d = 1 and restricting to increments having a density and an everywhere ﬁnite moment generating function. We prove corresponding versions of their theorem in dimension d = 1 for heavy-tailed increments in the domain of attraction of a stable law and in dimension d ≥ 1 for lattice type increments with an everywhere ﬁnite moment generating function. In all cases we only assume that under the initial conﬁguration no two particles are located at the same point. In addition, we analyze the attractivity of quasi-stationary Poisson point processes in the space of all Poisson point processes with almost surely inﬁnite, locally ﬁnite and upper bounded conﬁgurations.


Introduction
Recently, evolutions of point processes on the real line by discrete time steps were successfully analyzed for quasi-stationary states, i.e. demanding the stationarity of the distances between the points rather than the positions of the points, see e.g. [2], [14]. In particular, the processes for which the joint distribution of the gaps stays invariant under the evolution were determined in the cases that Gaussian or i.i.d. increments having a density and an everywhere finite moment generating function are added to the particles. In the i.i.d. case Ruzmaikina and Aizenman proved that these quasi-stationary point processes are of a particularly simple form, given by superpositions of Poisson point processes with exponential densities. In the context of spin glass models their result says that quasi-stationary states in the free energy model starting with infinitely many pure states and adding a spin variable in each time step are given by superpositions of random energy model states introduced in [13]. The connection between the cavity method in the theory of spin glasses and quasi-stationary measures of evolutions of points on the real line is explained in full extent in [1] and [2]. For an introduction to spin glass models see for instance [11], [15].
The crucial assumption in [14] is that the distribution of the increments possesses a density and has an everywhere finite moment-generating function. In particular, the increments are in the domain of attraction of a normal law. Although this is the case in the context of the Sherrington-Kirkpatrick model of spin glasses, it is of interest to determine the quasi-stationary states for more general increments. Here we treat increments in the domain of attraction of a stable law and multidimensional evolutions with increments having exponential moments, thus being in the domain of attraction of a multidimensional normal law. The results for lattice type and heavytailed increments in dimension d = 1 may be as well applicable in the context of non-Gaussian spin glass models. The resulting quasi-stationary measures are superpositions of Poisson point processes, whose intensities vary with the type of the increments considered. In addition to the Ruzmaikina-Aizenman type quasi-stationary states we find completely new quasi-stationary states in the case of lattice type increments with either exponential moments or heavy tails. We also prove attractivity of the quasi-stationary Poisson point processes in the space of all Poisson point processes in d with almost surely infinite, locally finite and upper bounded configurations.
To determine the quasi-stationary measures in the case of increments with heavy tails we observe that the Poissonization Theorem of [14] can be generalized to apply in our context. Hence, we are able to write each quasi-stationary measure as a weak limit of superpositions of Poisson point processes. Subsequently, we present a direct argument in which we evaluate the limit in the Generalized Poissonization Theorem in order to conclude that it is itself a superposition of Poisson point processes. In the case of increments in the domain of attraction of a normal law we follow the approach of [14]. In our case we use a version of the Bahadur-Rao Theorem which gives the sharp asymptotics of large deviations for infinite rectangles in d and is an analog of the results in [10] for smooth domains in d . This allows us to perform a compactness argument similiar to the one in [14] allowing us to pass to the limit in the Poissonization Theorem through a subsequence. One of the main obstacles hereby is the lack of a natural total order on d . After proving that in both cases the quasi-stationary measures are given by superpositions of Poisson point processes we show that the intensities of the latter are solutions of Choquet-Deny type equations. This is done by extending the steepness relation on tail distribution functions to the multidimensional setting and generalizing the monotonicity argument in [14]. In our more general setting we find new intensities in addition to the ones in [14]. To prove the attractivity of certain quasi-stationary Poisson point processes in the space of all Poisson point processes with almost surely infinite, locally finite and upper bounded configurations we analyze the corresponding evolution of intensity measures and exploit the fact that the weak convergence of intensity measures implies the weak convergence of the Poisson point processes.
To define the evolution in d in full generality we consider the partial order ≥ on d where a ≥ b when a j ≥ b j , 1 ≤ j ≤ d. Let l ⊂ d be any line in d for which ≥ is a total order and which contains infinitely many lattice points in the case that the increments are of lattice type. Moreover, let p : d → l be the affine map which assigns to every point x the closest point y on l with x ≥ y. Finally, we set a b if p(a) > p(b) or if a = b and define a b in an arbitrary, but deterministic and measurable way for which a ≥ b implies a b if p(a) = p(b), a = b (one can use induction on d to prove that this is possible). Note that is a total order on d in agreement with the partial order ≥ and its level sets are infinite rectangles up to a modification of the boundary. We consider competing particle systems with a random µ-distributed starting configuration (x n ) n≥1 ordered by and evolving by i.i.d. increments (π n ) n≥1 of distribution π, i.e. each step of the evolution is described by the mapping where ↓ denotes the sequence rearranged in non-ascending order .
From now on all considered evolutions will satisfy one of the following two assumptions: assumption 1.1 in the case of a one-dimensional evolution with heavy-tailed increments belonging to the domain of attraction of a stable law and assumption 1.2 in the case of increments being in the domain of attraction of a (possibly multidimensional) normal law. Assumption 1.1. d = 1 and there exist sequences (a n ) n≥1 , (b n ) n≥1 of real numbers such that S n − a n b n ≡ n i=1 π i − a n b n converges in distribution to an α-stable law with α ∈ (0, 2). Further, the initial distribution µ is simple, i.e.: and both the evolution with increments distributed according to π and the one with increments distributed according to the corresponding α-stable law make sense (which means that the particle configuration can be reordered with probability 1 after each step of the evolution). Finally, without loss of generality [π n ] = 0 and the π n are not almost surely equal to 0.
An example of a robust condition on µ and π which assures that the evolution makes sense is the following. Denote by λ the intensity measure of µ, i.e. define for Borel sets A ⊂ . If λ * π is finite on all intervals of the type [x, ∞), then the particle configuration can be reordered with probability 1, since it can be checked by a direct computation that λ * π is the intensity measure of the point process resulting from µ after one step of the evolution.
Assumption 1.2. The sequence of i.i.d. d -valued random variables (π n ) n≥1 which describes the increments of the evolution satisfies and each component of the π n is of positive variance. For d > 1 assume further that the π n have a density or take values in a lattice A d + b for a fixed real d × d matrix A and a vector b ∈ d . Moreover, the initial measure µ on particle configurations is simple and such that µ-a.s. where x i n is the i-th coordinate of x n . Finally, without loss of generality [π n ] = 0.
In the case d = 1 assumption 1.2 allows us to deal with the evolutions considered in [14], as well as various lattice-type evolutions of interest, e.g. π n being Bernoulli {−1, 1}-valued or following a signed Poisson distribution. Moreover, assumption 1.2 ensures that starting with a locally finite, upper bounded configuration we get a configuration of the same type after each step of the evolution (apply the remark in section 1.2 of [2] to each of the d coordinate processes). We will denote the space of such configurations by Ω and equip it with the σ-algebra which is generated by the shift invariant functions measurable with respect to the σ-algebra generated by occupancy numbers of finite boxes. For the sake of full generality we include the case of configurations with finitely many particles by allowing the x n to take the value (−∞, . . . , −∞). Our main result is the following: Then the joint distribution of the gaps of N after n evolutions converges for n tending to infinity to the corresponding quantity for a Poisson point process with intensity measure λ ∞ .
Remark. The quasi-stationary Poisson point processes with exponential intensities dλ = r e −rτ d r, τ > 0 found in [14] are quasi-stationary also in the case that π is a lattice type distribution. They can be recovered from Theorem 1.3 (c) as the special case n = 1, dα(s, y) = dδ τ (s)τe −τ y d y.
This is due to the computation where one has to observe that the sum is a geometric series and the integrand takes only two different values.
A crucial step in the proof of Theorem 1.3 consists of writing quasi-stationary measures as weak limits of superpositions of Poisson point processes which is called Poissonization in [14]. More precisely, we use the following generalization of the Poissonization Theorem of [14]: where (x n ) n≥1 is a fixed starting configuration of the particles. Then for any non-negative continuous function with compact support f ∈ C + c ( ) it holds Here, G µ denotes the modified probability generating functional of µ given by and G F N denotes the modified probability generating functional of the Poisson point process on with intensity measure λ N uniquely determined by To prove Theorem 1.5 it suffices to observe that the proof of the Poissonization Theorem in [14] can be adapted to our context by applying the spreading property in the form of Lemma 11.4.I of [5] which we state now for the sake of completeness: The paper is organized as follows. We prove part (

Quasi-stationary measures of the evolution with heavy-tailed increments
In this section as well as sections 3 and 4 we restrict to the case d = 1 for the sake of a simpler notation and prove Theorem 1.3 in the one-dimensional setting. Subsequently, we show in section 5 how our arguments extend to the case d > 1. In this section we present the proof of Theorem 1.3 (a) for an evolution satisfying assumption 1.1. We explain the main part of the proof first and defer the technical issue of approximating the distribution of the increments by an α-stable law to the end of the proof. The proof uses the Generalized Poissonization Theorem (Theorem 1.5) in deducing that every quasi-stationary µ satisfying assumption 1.1 is a superposition of Poisson point processes. converges to an α-stable law. Since we are only interested in the joint distribution of the gaps between the particles, we may assume that the particle configuration starts at x 1 = 0. We will shift it subsequently to the left by numbers c N depending on the initial configuration (x n ) n≥1 and tending monotonously to infinity for N → ∞. The resulting configuration of particles will be denoted by We note that Since a shift of the particle configuration by c N does not affect the value of G µ and (c N ) N ≥1 can be chosen to converge to infinity fast enough, the functions F N can be replaced by in the statement of the Generalized Poissonization Theorem where C = C(α) is a constant and (e N ) N ≥1 is an increasing sequence in + depending on the initial configuration (x n ) n≥1 , converging to infinity and satisfying e N ≤ c N 2 . The approximation by an α-stable law used here is justified in steps 2 to 4. We remark at this point that the right-hand side is finite due to the assumption 1.1, since it corresponds to the evolution with α-stable increments. Next, note that which follows by conditioning the Poisson point process on its leader and was shown in [14]. Here, the integrals are taken with respect to the infinite positive measures induced by the corresponding non-increasing functions. Hence, again referring to steps 2 to 4 for the justification of the approximation by an α-stable law we may conclude Setting K(N ) ≡ C L(N ) α N , the Generalized Poissonization Theorem yields: Recalling that x n (N ) was defined as x n − c N we may rewrite the inner integral as Next, we enlarge the shift parameters c N , if necessary, to have Note that this is possible, because the sum on the right-hand side is finite for the original choice of (c N ) N ≥1 due to assumption 1.1 and, moreover, the sequence (c N ) N ≥1 can be adjusted separately for each starting configuration (x n ) n≥1 . For the sake of shorter notation we introduce positive measures α N , α on defined by Now, we would like to interchange the limit N → ∞ with the µ-integral on the right-hand side of the equation for G µ ( f ). To this end, we remark that the Dominated Convergence Theorem may be applied, since the integrands are dominated by and the right-hand side can be made uniformly bounded in N by enlarging the c N , if necessary. By interchanging the limit with the µ-integral we deduce But α N and α were defined in such a way that α is the weak limit of the α N . Thus, the Poisson point processes with the intensity measures α N converge weakly to the Poisson point process with the intensity measure α (see Theorem 11.1.VII in [5] for more details). In particular, their modified probability generating functionals converge. Thus, we may pass to the limit and deduce In other words, µ is a superposition of Poisson point processes with intensities α(d x) mixed according to µ itself. This proves that each quasi-stationary measure of the evolution is a superposition of Poisson point processes given that the distribution of the increments can be approximated by an α-stable law in a suitable sense.
2) With the notation θ n,N (x) ≡ x−x n (N ) we need to justify that we are allowed to replace the appearing on the right-hand side of the statement of the Generalized Poissonization Theorem by which plays the same role on the right-hand side of the corresponding Generalized Poissonization Theorem for increments following an α-stable law. To this end, by the second remark on page 260 of [9] which characterizes domains of attraction of α-stable laws we can find constants d N ∈ [0, 1], functions ǫ N : → + and slowly varying functions s N : 3) Suppose first that inf N d N > 0. Choosing the shift parameters c N introduced in step 1 to be large enough, we can achieve by taking the absolute value inside the sum and using the monotonicity of x → x d N +x . Under the assumption inf N d N > 0 we have for any test function f ∈ C + c ( ) and functions F N defined by using the approximation of G F by functionals continuous in F presented in the proof of Theorem 6.1 in [14]. This and the fact that the F N 's differ from the corresponding expressions in step 1 only by the constants d N and the slowly varying functions s N , which both can be dominated by an appropriate choice of the sequence (c N ) N ≥1 , justify the approximation by an α-stable law in the case inf N d N > 0. We observe that this reasoning goes through also under the weaker assumption of lim inf N →∞ d N > 0. 4) Now, let lim inf N →∞ d N = 0. We may even assume lim N →∞ d N = 0, since we may pass to the limit in the Generalized Poissonization Theorem through any subsequence. Since 1 d N F N is a multiple of the expected number of particles on [x, ∞) after N steps in the evolution with α-stable increments, the measure induced by 1 d N F N is not only locally finite, but also finite on intervals of the type [x, ∞). Hence, by enlarging the shift parameters c N to make 1 for each x, we can achieve that the measures induced by F N converge weakly to the zero measure on for N tending to infinity. In addition, we have the estimate The rightmost expression converges to 0 for N → ∞ which shows that the approximation by an α-stable law may be applied with C = 0. This follows again by the same approximation of G F as in the proof of Theorem 6.1 in [14]. We observe that this case corresponds to the quasi-stationary measure in which the configuration with no particles occurs with probability 1.

Quasi-stationary measures of the evolution with increments in the domain of attraction of a normal law
In this section we show that a quasi-stationary measure µ of an evolution satisfying assumption 1.2 is a superposition of Poisson point processes. The main difference to the proof of Theorem 6.1 in [14] is that we apply a multidimensional version of the Bahadur-Rao Theorem which applies to any distribution π as in assumption 1.2. This leads to the replacement of Laplace transforms by modified Laplace transforms and of normalizing shifts of the whole configuration by particle dependent shifts due to the fact that the Bahadur-Rao Theorem gives only information on probabilities of large deviations for lattice points in case that π is a lattice type distribution. The version of the Bahadur-Rao Theorem we use is an analog of the results in [10] where we replace smooth domains by infinite rectangles.
Lemma 3.1 (Multidimensional Bahadur-Rao Theorem). Let (π n ) n≥1 be as in assumption 1.2 and set S N ≡ N n=1 π n . Then in case that d = 1 and the π n are non-lattice or d > 1 and the π n have a density we have for all x ∈ d and d ∋ q ≥ 0: uniformly in q where ∼ means that the quotient of the two expressions tends to 1, η = η(q) is the unique solution of Proof. 1) In the case d = 1 both asymptotics and their uniformity follow directly from Lemma 2.2.5 and the proof of the one-dimensional Bahadur-Rao Theorem in [6].
2) From now on let d > 1 and set where ≥ and min are meant componentwise. With an abuse of notation let be the distribution of the π n on d and following [10] define the q-centered conjugate by Next, choose ν to be the minimizer of γ over Γ and let ν N be the corresponding minimizer over Γ ∧ Γ N . The representation formula for large deviations of [10] implies .
Note further that since η(ν N ) solves ∇γ(ν N ) = η(ν N ) and ν N is the boundary point of Γ ∧ Γ N where the level set of γ touches Γ ∧ Γ N , it follows that η(ν N ) is the inward normal to Γ ∧ Γ N in ν N in case that ν N = min q + x N , q and a vector pointing inward Γ ∧ Γ N otherwise. Hence, in both cases the integrands in the numerator and denominator are bounded by 1, because Γ, Γ N ⊂ Γ ∧ Γ N by definition. Next, let V be the covariance matrix of (. ; η(ν)) and ϕ 0,V be the Gaussian density with mean 0 and covariance V . Applying the expansion in Lemma 1.1 of [10] and its analog for the lattice case in section 2.6 of the same paper and using the boundedness of the integrands we deduce which proves the theorem.
Next, we define modified Laplace transforms.

Definition 3.2.
Let be the space of finite measures on (0, ∞) and M be the Borel σ-algebra on for the weak topology. Moreover, denote by the Laplace transform of a measure ̺ ∈ and by R ̺ its modified Laplace transform given by Proof. 1) We introduce again the functions F N defined by with S N = N n=1 π n and a starting configuration (x n ) n≥1 . In order to deduce Proposition 3.3 from Theorem 1.5 we want to find measures ̺ N ∈ such that their modified Laplace transforms R ̺ N are close to the functions F N in a suitable sense. Since R ̺ N (0) = 1, we will normalize the functions F N such that F N (0) will be close to 1. For this purpose define numbers z N by Moreover, let z n,N = z N for all n if the distribution of the π n is non-lattice and let z n,N ≥ z N be closest number to z N satisfying z n,N − x n N ∈ p + r if the distribution of the π n is supported in p + r. Lastly, define functions H N which may be viewed as the normalized versions of the functions F N by Note that in the lattice case each function H N is piecewise constant with jumps on a subset of p . Applying Lemma 3.1 we deduce that for an appropriate K > 0 and all n for which x n ≥ −K N it holds for all x ∈ in the non-lattice case and for all x ∈ p in the lattice case. Moreover, sup n |ǫ n,N | → N →∞ 0.
Hence, with high probability H N (x) can be written as 2) In this step we will prove that tends to 0 for N → ∞ and an appropriately chosen K.
In case that lim η→∞ Λ ′ (η) < ∞, we conclude that the support of the distribution of the π n is bounded from above. Hence, the expression above vanishes for a fixed large enough K and N tending to infinity (provided that z n,N is bounded from below by an affine function of N uniformly in n which will be proven in the next step).
It remains to consider the case lim η→∞ Λ ′ (η) = ∞. As in the proof of the Bahadur-Rao Theorem in [6] we define ψ N (η) ≡ η N Λ ′′ (η) and let F q N be the distribution function of .
In the same way as in [6] we deduce for all n with x n < −K N that with A, B independent of n which will be proven in the next step. Next, choose K such that which is possible because γ is convex with γ ′ (q) = η(q) → q→∞ ∞. Thus, for N large enough for µ-a.e. (x n ) n≥1 and where the convergence follows from assumption 1.2.
3) We will bound z n,N from below by an affine function in N uniformly in n, i.e. find uniform constants A, B such that for all n, N . To this end note that by the Central Limit Theorem we have for all x ∈ in the non-lattice case and for all x ∈ p in the lattice case. It follows directly that for all such x we can find positive numbers δ N tending to 0 for N tending to infinity such that Recalling that in the lattice case the functions H N are piecewise constant having jumps only on a subset of p we may write the same inequality in terms of the functions R ̺ N and get with δ N → N →∞ 0 and ǫ N (x) → N →∞ 0 for all x ∈ in both cases. We will use this estimate in order to rewrite the equation of Theorem 1.5 in terms of the functions R ̺ N . To this end for each N define a measurable transformation T N of the space of configurations by and let µ N be the measure on Ω induced by µ via T N . Then by the definition of µ N and by the invariance of G under the shift of particles by K N + z N we may conclude The explicit representation of the modified probability generating functional of a Poisson point process implies Taking into account the bounds for non-negative x, y and it follows that To be precise, one can estimate and then estimate the difference between the two integrands in a similar way. 5) Define by ν N the probability measure on induced by µ N through the measurable mapping T N given by Step 4 and the definition of ν N imply Our next claim is that the sequence (ν N ) N ≥1 is tight. To this end we will show that for each δ > 0 there exists a function M such that for all N it holds The compactness of {̺ ∈ | R ̺ (x) ≤ M (x)} will then imply the claim. Note that because of the monotonicity of R ̺ we may replace R ̺ by R ̺ in the last inequality without loss of generality. Recalling the definition of ν N we observe that the inequality corresponds to The upper bound on |H N (x) − R ̺ N (x)| and the definition of µ N allows us to replace R ̺ N by H N and subsequently to rewrite the inequality as We can get an upper bound on n π (x n +S N ≥ x + K N +z N ) which is uniform in N by adapting the estimates of step 2 to the present situation and enlarging the constant K if necessary. More precisely, in from below by ζ 1 (x + K N + z N − x n ) and subsequently estimate z N by AN + B leading to Finally, the right-hand side is bounded by e −ζ 1 (x+B) n e ζ 1 x n which is finite µ-a.s. The claim follows now by choosing M (x) satisfying Strictly speaking we may have to choose a larger M (x), because the estimate above is only valid for N ≥ N 0 (x) with probability tending to 1 for N 0 (x) tending to infinity. Hence, we can choose N 0 (x) such that the probability is larger than 1 − δ 2 and fix an M (x) large enough for our purposes afterwards. The claim readily follows. 6) Define ν to be the limit point of a converging subsequence of (ν N ) N ≥1 . Then the same approximation of G F by functionals continuous in F as in [14] implies which proves the proposition.

Poisson intensities
Up to this point we have shown that a quasi-stationary measure of a one-dimensional evolution satisfying the assumption 1.1 or the assumption 1.2 is given by a superposition of Poisson point processes. In this section we provide the exact shape of the Poisson intensity measures both in case that assumption 1.1 and in case that assumption 1.2 is satisfied. This is done by exploiting the properties of the steepness relation defined next.
Remark. Note that since the measures ̺ ∈ are supported on (0, ∞), their Laplace transforms R ̺ and also their modified Laplace transforms R ̺ are elements of .
The main tool in the characterization of the Poisson intensities is the following result.

Lemma 4.2.
Let ̺ be a measure in , F = R ̺ ∈ and λ be the unique positive measure on with If the π n satisfy assumption 1.2, π is the probability distribution of each of the π n and G ∈ is the unique function with then G is steeper than F .
Proof. Let G(a) = F (b) for some a, b ∈ . Without loss of generality we may assume b = 0, because otherwise we can replace ̺ by ̺(du) ≡ e su ̺(du) yielding shifted versions F , G of F , G with F (0) = F (b) for a suitable s. By performing the same argument as below for F , G instead of F , G we can conclude that G is steeper than F . Thus, G is steeper than F by the invariance of the steepness relation under shifts. Furthermore we may assume that a ≥ 0, because otherwise we can replace π by a shifted version of itself and note that the following argument does not depend on the fact that the expectation of π is zero. We need to show G(a + v) ≤ F (v) for any v > 0. By Fubini's Theorem and integration by parts we can estimate G(x) for x ≥ 0 in the following way: A calculation for G(a + v) similar to that for G(x) together with the calculation for F (x) imply where we have used the inequality of Lemma 7.2 in [14] which relies only on the fact that π is a probability measure and not on its shape.
The last tool we need is the following classical version of the Choquet-Deny Theorem (Theorem 3 in [7]).

Lemma 4.3 (Choquet-Deny Theorem).
Let π be a probability measure on a locally compact abelian group X . Then the positive measures λ on X satisfying λ * π = λ are given by where Γ is a set containing exactly one representative of each coset of the subgroup Y generated by the support of π in X , the set is given by = { f |∀g 1 , g 2 ∈ Y : f (g 1 + g 2 ) = f (g 1 ) f (g 2 )}, the measure ω is a Haar measure on Y and ν is a positive Radon measure on × Γ.
The Choquet-Deny Theorem in this general form allows us to finish the proof of Theorem 1.3 in the one-dimensional setting which is one of the main results of the paper.
Proof of Theorem 1.3 in the one-dimensional setting. 1) We have shown above that under assumption 1.1 or 1.2 each quasi-stationary measure is a superposition of Poisson point processes.
To prove that the intensity measures of the latter have the desired form, we will restrict to the case that assumption 1.2 is satisfied. The other case is completely analogous and requires only a replacement of the space over which the superposition is taken. From now on we consider an evolution satisfying assumption 1.2 and let F , G be functions constructed as in Lemma 4.2 where ̺ will vary over , so that F varies over the space of the modified Laplace transforms of measures in . Making N + 1 steps of the evolution instead of N in the proof of Proposition 3.3 shows that From here it follows that λ * π is a translate of λ with the notation used in Lemma 4.2. This can be shown as in the proof of Theorem 8.1 in [14], applying our Lemma 4.2 instead of their Theorem 7.3. 4) In this last step we will prove that a Poisson point process with the intensity measure λ is a simple quasi-stationary measure for our evolution (hence, superpositions of such processes are also simple and quasi-stationary). The simplicity follows from the corresponding remarks in step 3. For quasi-stationarity note that λ was constructed as the solution of a Choquet-Deny equation which by Lemma 7.4 in [14] implies Here F , G denote probabilities associated with Poisson point processes corresponding to F , G, respectively. Thus, the distribution of the first gap is invariant under the evolution. A similar calculation proves that this holds for any finite number of gaps which proves quasi-stationarity.

Quasi-stationary measures of the multidimensional evolution
This section contains a sketch of the proof of Theorem 1.3 under the assumption 1.2 for any dimension d in which we explain how the arguments of sections 3 and 4 generalize to the multidimensional case.
Proof of Theorem 1.3 under assumption 1.2. 1) For the extension of the Generalized Poissonization Theorem define functions F N in the same way as for d = 1 with instead of ≥ and let measures λ N be defined analogously to the case d = 1 by setting λ N ([a, b)) for any finite box [a, b) to be the alternating sum of values of F N at the vertices of the box. Note that λ N is a positive measure, because every summand π (x n + S N x) in the definition of F N defines a probability measure on d . Moreover, for test functions f ∈ C + c ( d ) let the modified probability generating functional G µ ( f ) be defined by where the configurations are arranged in the non-ascending order . Define G F N as the corresponding quantity for the d-dimensional Poisson point process with intensity measure λ N . Then performing the proof of the Poissonization Theorem of [14] with ≥ replaced by and < replaced by and applying the spreading property (Lemma 1.6) to each of the components to bound π (S N x, S N x + D) uniformly in x one deduces 2) To prove that µ is a superposition of Poisson point processes we generalize the proof of where now denotes the space of finite measures on ( + ) d and R ̺ denotes the Laplace transform of a ̺ ∈ with the argument modified to the closest lattice point from above if necessary. Observe that R ̺ is a tail distribution function of a locally finite positive measure for ̺ ∈ , because for any x ≥ 0 the function e −u·x is the tail distribution function of a product of exponential distributions on . By exactly the same argument as in section 3 it follows that the sequence ν N is tight and we let ν be a subsequential limit of it. By the approximation of the functional G F by functionals continuous in F as done in [14] for d = 1 one may conclude so µ is a superposition of Poisson point processes.
3) To extend section 4 to the case d > 1 we extend first the steepness relation to the space of tail distribution functions F of positive measures on d which satisfy F (λx) → λ→∞ 0 for each x ∈ d − ( − ) d . We call G steeper than F if F (x) = G( y) for some x, y ∈ d implies F (x + a) ≥ G( y + a) for all a ∈ ( + ) d . The same calculation as before for each of the d coordinates yields that functions become steeper if convolved with probability measures in the sense of Lemma 4.2. Finally, the same monotonicity argument as in section 4 shows that for intensity measures λ of quasi-stationary Poisson point processes λ * π has to be a translate of λ. Finally, the Poisson point process with intensity measure λ has to be supported on upper bounded configurations and be simple, hence λ has no point masses.

Attractivity
In this concluding section we prove attractivity of certain quasi-stationary Poisson point processes in the space of all Poisson point processes with almost surely infinite, locally finite and upper bounded configurations by analyzing the corresponding evolution of intensity measures. The latter will be assumed locally finite satisfying for a c ∈ d depending on λ which means precisely that the corresponding Poisson point processes are supported on infinite, locally finite and upper bounded configurations. As before we will denote the points of such configurations by x 1 , x 2 , . . . in descending order and call the Poisson point processes and intensity measures of this type regular. In particular, it turns out that the space of regular Poisson point processes is invariant under evolutions with i.i.d. increments. For the increments (π n ) n≥1 we assume for this section that [π n ] = 0 and the π n are not almost surely equal to 0. Obviously, the two assumptions can be made without loss of generality since a recentering of the increments does not affect the joint distribution of the gaps of the evolved process which will be the only quantity of interest. Lemma 6.1 and Lemma 6.2 are the key to the attractivity result. The first is taken from section 11.4 of [5], so we omit the proof and the proof of the second is given below. Lemma 6.1. Let N be a regular Poisson point process on d with corresponding intensity measure λ and configurations (x n ) n≥1 . Define N to be the point processes with configurations ( y n ) n≥1 ≡ (x n + π n ) n≥1 where π n are i.i.d. random variables with distribution π which are independent of N . Then N is a Poisson point process with intensity measure λ * π. Lemma 6.2. Let N be a regular Poisson point process with intensity measure λ and suppose that there exists a measure λ ∞ corresponding to a regular Poisson point process and satifying λ * π * n w → n→∞ λ ∞ (VI.16) where π is the distribution of the increments of the evolution as in Lemma 6.1. Then the joint distribution of the gaps of N after n evolutions converges to the corresponding quantity for the Poisson point process with intensity measure λ ∞ for n → ∞.
Proof. From the multidimensional version of the Levy Continuity Theorem it can be deduced that the convergence of the joint distribution of the gaps follows from the convergence of the corresponding modified probability generating functionals for test functions f ∈ C + c ( d ), so it suffices to show the convergence of the latter. To this end let N k be the point process resulting from N after k steps of the evolution. By Lemma 6.1 it is a Poisson point process with intensity measure λ * π * k . By the general formula for modified probability generating functionals of Poisson point processes the modified probability generating functional of N k is given by