Evolution of states in a continuum migration model

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbbm {R}}^d$$\end{document} in which the constituents appear (immigrate) with rate b(x) and disappear, also due to competition. For this model, we prove the existence of the evolution of states μ0↦μt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _0 \mapsto \mu _t$$\end{document} such that the moments μt(NΛn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _t(N_\Lambda ^n)$$\end{document}, n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbbm {N}}$$\end{document}, of the number of entities in compact Λ⊂Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset {\mathbbm {R}}^d$$\end{document} remain bounded for all t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}. Under an additional condition, we prove that the density of entities and the second correlation function remain point-wise bounded globally in time.

intrinsic disappearance rate m(x) ≥ 0 and the part related to the interaction with the existing community, which is interpreted as competition between the entities. The phase space is the set of all subsets γ ⊂ R d such that the set γ := γ ∩ is finite whenever ⊂ R d is compact. For each such , one defines the counting map γ → |γ | := #{γ ∩ }, where the latter denotes cardinality. Thereby, one introduces the subsets ,n := {γ ∈ : |γ | = n}, n ∈ N 0 , and equips with the σ -field generated by all such ,n . This allows for considering probability measures on as states of the system. Among them there are Poissonian states in which the entities are independently distributed over R d , see [6,Chapter 2]. They may serve as reference states for studying correlations between the positions of the entities. For the nonhomogeneous Poisson measure π with density : R d → R + := [0, +∞), n ∈ N 0 , and every compact , one has π ( ,n ) = n exp (− ) /n!, which implies π (N ) = |γ |π (dγ ) = , (1.2) where N is the number of entities contained in . In the case of ≡ > 0, one deals with the homogeneous Poisson measure π . The counting map γ → |γ | can also be defined for = R d . Then the set of finite configurations 0 := n∈N 0 {γ ∈ : |γ | = n} (1.3) is clearly measurable. In a state with the property μ( 0 ) = 1, the system is (μ-almost surely) finite. By (1.1) one gets that either π ( 0 ) = 1 or π ( 0 ) = 0, depending on whether or not is globally integrable. If π ( 0 ) = 0, the system in state π is infinite. The use of infinite configurations for modeling large finite populations is as a rule justified, see, e.g., [2], by the argument that in such a way one gets rid of the boundary and size effects. Note that a finite system with dispersal-like the one studied in [5,7]-being placed in a noncompact habitat always disperse to fill its empty parts, and thus is developing. Infinite configurations are supposed to model developed populations. In this work, we shall consider infinite systems and hence deal with states μ such that μ( 0 ) = 0.
To characterize states on one employs observables-appropriate functions F : → R. Their evolution is obtained from the Kolmogorov equation (1.4) where the generator L specifies the model. The states' evolution is then obtained from the Fokker-Planck equation (1.5) related to that in (1.4) by the duality μ t (F 0 ) = μ 0 (F t ), where The model that we study in this work is specified by the following Here, for appropriate γ ∈ and x ∈ R d , by writing γ \ x we mean γ \ {x}. Likewise, γ ∪ x stands for γ ∪ {x}. In (1.6), b(x) is the immigration rate, m(x) ≥ 0 is the intrinsic emigration (mortality) rate, and a ≥ 0 is the competition kernel. The model parameters are supposed to satisfy the following. If one sets in (1.6) a ≡ 0, the model becomes exactly soluble, see Sect. 2.3 below. This means that the evolution can be constructed explicitly for each initial state μ 0 . In this case, assuming that μ 0 (N n ) < ∞ for all n ∈ N, one can get the information about the time dependence of such moments. Namely, if m(x) ≥ m * > 0 for all x ∈ R d , then for each compact , the following holds Otherwise, all the moments μ t (N n ) are increasing ad infinitum as t → +∞. If the initial state is π 0 , then μ t = π t with t (x) = 0 (x) + b(x)t for all x such that m(x) = 0, cf (2.22) below. In [3], for the model (1.6) with m ≡ 0 and a nonzero a satisfying a certain (quite burdensome) condition, it was shown that μ t (N )/V( ) ≤ C for large enough values of the Euclidean volume V( ), provided the evolution of states μ 0 → μ t exists.
In this article, assuming that the initial state μ 0 is sub-Poissonian, see Definition 2.1 below, we prove that the evolution of states μ 0 → μ t , t > 0, exists (Theorem 2.4) and is such that μ t (N n ) ≤ C (n) for each t > 0 (Theorem 2.5) and all m, including the case m ≡ 0. Moreover, if the correlation functions k (n) μ 0 , n ∈ N, of the initial state are continuous, see Sect. 2.1 below, then k (n) μ t , n ∈ N, are also continuous and the following holds with some positive C 1 and C 2 .
The structure of the article is as follows. In Sect. 2, we introduce the necessary technicalities and then formulate the results: Theorems 2.4 and 2.5. Thereafter, we make a number of comments to them. In Sects. 3 and 4, we present the proofs of Theorems 2.4 and 2.5, respectively.

Preliminaries and the results
We begin by presenting some facts on the subject-a more detailed description of them can be found in [5,7,8] and in the literature quoted therein.
By B(R d ) and B b (R d ) we denote the sets of all Borel and bounded Borel subsets of R d , respectively. The configuration space is equipped with the vague topology, see [8], and thus with the corresponding Borel σ -field B( ), which makes it a standard Borel space. Note that B( ) is exactly the σ -field generated by the sets ,n , mentioned in Introduction. By P( ) we denote the set of all probability measures on ( , B( )).

Correlation functions
Like in [5,7], the evolution of states will be described by means of correlation functions without the direct use of (1.5). To explain the essence of this approach let us consider the set Θ of all compactly supported continuous functions θ : R d → (−1, 0]. For a state, μ, its Bogoliubov functional, cf. [9,12], is For the homogeneous Poisson measure π , it takes the form

Definition 2.1
The set of sub-Poissonian states P exp ( ) consists of all those states μ ∈ P( ) for which B μ can be continued, as a function of θ , to an exponential type entire function on L 1 (R d ).
It can be shown that a given μ belongs to P exp ( ) if and only if its functional B μ can be written down in the form μ is the n-th order correlation function of μ, which is a symmetric element with some C > 0 and ϑ ∈ R. Note that k (n) π (x 1 , . . . , x n ) = n . Note also that (2.2) resembles the Taylor expansion of the characteristic function of a probability measure. In view of this, k (n) μ are also called (factorial) moment functions, cf. e.g., [11]. [5], if and only if, for each n ∈ N, there exists a symmetric Borel function G (n) : there exists N ∈ N 0 such that G(η) = 0 whenever |η| > N . By (G) and N (G) we denote the smallest and N with the properties just mentioned. By B bs ( 0 ) we denote the set of all bounded functions with bounded support.
The Lebesgue-Poisson measure λ on ( 0 , B( 0 )) is defined by the following formula We also set k μ (∅) = 1. With the help of the measure introduced in (2.5), the expressions for B μ in (2.1) and (2.2) can be combined into the following formulas Thereby, one can transform the action of L on F, see (1.6), to the action of L on k μ according to the rule This will allow us to pass from (1.5) to the corresponding Cauchy problem for the correlation functions By (2.7) the action of L is as follows and In the next subsection, we introduce the spaces where we are going to define (2.8).

The Banach spaces
By (2.2) and (2.6), it follows that μ ∈ P exp ( ) implies holding for λ-almost all η ∈ 0 , some C > 0, and ϑ ∈ R. In view of this, we set where Clearly, (2.12) and (2.13) define a Banach space with the usual point-wise linear operations. In the following, we use the ascending scale of such spaces K ϑ , ϑ ∈ R, with the property K ϑ → K ϑ , ϑ < ϑ , (2.14) where → denotes continuous embedding.
For G ∈ B bs ( ), we set (2.15) where indicates that the summation is taken over all finite subsets. It satisfies, see Definition 2.2, The latter means that μ(K G) < ∞ for each μ ∈ P exp ( ). By (2.6) this yields By [8, Theorems 6.1 and 6.2 and Remark 6.3] one can prove the next statement.

Proposition 2.3
Let a measurable function k : 0 → R have the following properties:

18)
with (c) holding for some C > 0 and λ-almost all η ∈ 0 . Then there exists a unique state μ ∈ P exp ( ) for which k is the correlation function.
Define, cf. (2.17), This is clearly a subset of the cone By Proposition 2.3 it follows that each k ∈ K ϑ with the property k(∅) = 1 is the correlation function of a unique state μ ∈ P exp ( ). Then we define As a sum of Banach spaces, the linear space K is equipped with the corresponding inductive topology that turns it into a locally convex space.

Without competition
The version of (1.6) with a ≡ 0 is known as the Surgailis model, see [14] and the discussion in [3]. This model is exactly soluble. This means that the solution of (2.8) can be written down explicitly in the following form The corresponding state μ t has the Bogoliubov functional which one obtains from (2.6) and (2.21). This formula can be used to extend the evolution μ 0 → μ t to all μ 0 ∈ P( ). Indeed, for each t > 0 and θ ∈ Θ, cf (2.1), we have that θψ t ∈ Θ, and hence B μ 0 (θ ψ t ) is the Bogoliubov functional of a certain state. 1 The same is true for the left-hand side of (2.23), and the state μ t contained therein can be condidered as a weak solution of the corresponding Fokker-Planck equation (1.5).
If the initial state is Poissonian with density 0 (x), by (2.23) the state μ t is also Poissonian with the density Otherwise, the solution in (2.21) is unbounded in t. If, for some compact , m(x) = 0 for x ∈ , then by (2.21) and (2.22) we get

The statements
For each ϑ ∈ R and ϑ > ϑ, the expressions in (2.9) and (2.10) can be used to define the corresponding bounded linear operators L ϑ ϑ acting from K ϑ to K ϑ . Their operator norms can be estimated similarly as in [7, eqs. (3.11), (3.13)], which yields, cf. (1.7), By means of the collection {L ϑ ϑ } with all ϑ ∈ R and ϑ > ϑ we introduce a continuous linear operator acting on K, denoted also as L , and thus define the corresponding Cauchy problem (2.8) in this space. By its (global in time) solution we will mean a continuously differentiable function [0, +∞) t → k t ∈ K such that both equalities in (2.8) hold. Our results are given in the following statements, both based on Assumption 1.1.

Theorem 2.4 (Existence of evolution)
For each μ 0 ∈ P exp ( ), the problem in (2.8) with L : K → K as in (2.9), (2.10) and (2.26) has a unique solution which lies in K and is such that k t (∅) = 1 for all t > 0. Therefore, for each t > 0, there exists a unique state μ t ∈ P exp ( ) such that k t = k μ t . Moreover, for all t > 0, the following holds where φ t and ψ t are as in (2.22). If the intrinsic mortality rate satisfies m(x) ≥ m * > 0 for all x ∈ R d , then for all t > 0 the solution k t lies in K ϑ * with ϑ * given in (2.24).

Theorem 2.5 (Global boundedness)
The states μ t , t ≥ 0, mentioned in Theorem 2.4 have the property: for every n ∈ N and compact ⊂ R d , the following holds μ 0 is a continuous function, then so is k (n) μ t for all n ∈ N and t > 0. Moreover, k (1) μ t and k (2) μ t have the properties as in (1.8).

Comments and comparison
By (2.25) it follows that the global in time boundedness in the Surgailis model is possible only if m(x) ≥ m * > 0 for all x ∈ R d . As follows from our Theorem 2.5, adding competition to the Surgailis model with the zero intrinsic mortality rate yields the global in time boundedness. In this case, the competition rate a(0) appears to be an effective mortality, see (4.19) below and the comments following the proof of Theorem 2.5. Note also that the global boundedness as in Theorem 2.5 does not mean that the evolution k μ 0 → k t holds in one and the same K ϑ with sufficiently large ϑ. It does if m(x) ≥ m * > 0. Since Theorem 2.4 covers also the case a ≡ 0, the solution in (2.21) is unique in the same sense. A partial result on the global boundedness in the model discussed here was obtained in [3, Theorem 1]. Therein, under quite a strong condition imposed on the competition kernel a (which, in particular, implies that it has infinite range), and under the assumption that the evolution of states μ 0 → μ t exists, there was proved the fact which in the present notations can be formulated as μ t (N ) ≤ C .

The existence of the evolution of states
We follow the line of arguments used in proving Theorem 3.3 in [7] and perform the following three steps: (i) Defining the Cauchy problem (2.8) with k μ 0 ∈ K ϑ 0 in a given Banach space K ϑ with ϑ > ϑ 0 , see (2.12) and (2.14), and then showing that this problem has a unique solution k t ∈ K ϑ on a bounded time interval [0, T (ϑ, ϑ 0 )) (Sect. 3.1). (ii) Proving that the mentioned solution k t has the properties (a) and (b) in (2.18) ((c) follows by the fact that k t ∈ K ϑ ). Then k t ∈ K ϑ and hence also in K + ϑ , see (2.20) and (2.19). By Proposition 2.3 it follows that k t is the correlation function of a unique state μ t (Sect. 3.2). (iii) Constructing a continuation of k t from [0, T (ϑ, ϑ 0 )) to all t > 0 by means of the fact that k t ∈ K + ϑ (Sect. 3.3).
From (3.11) one can get that the family mentioned in Proposition 3.2 enjoys the following 'semigroup' property holding whenever (ϑ, ϑ , t + s), (ϑ , ϑ , t), and (ϑ, ϑ , s) are in Θ. Now we make precise which Cauchy problem we are going to solve. Set where L is as in (2.9). This defines an unbounded linear operator L ϑ : D ϑ → K ϑ , being the extension of the operators L ϑ ϑ 0 : and all ϑ 0 < ϑ, cf. (2.14). Then we consider the Cauchy problem (2.8) in K ϑ 1 with this operator L ϑ 1 and k μ 0 ∈ K ϑ 0 . By its classical solution we understand the corresponding map t → k t ∈ D ϑ 1 , continuously differentiable in K ϑ 1 .

Remark 3.4
As in the proof of Lemma 3.3 one can show that, for each t ∈ [0, T (ϑ 1 , ϑ 0 )), the following holds: see (3.14), and Now we construct the evolution of functions G 0 → G t such that, for k ∈ K ϑ , the following holds, cf. (2.16), To this end, we introduce, cf. (2.12) and (2.13), Clearly, G ϑ → G ϑ for ϑ < ϑ ; hence, we have introduced another scale of Banach spaces, cf. (2.14). As in (3.1) and (3.3), we define the corresponding multiplication operators A ϑϑ and S ϑϑ (t), and also C ϑϑ ∈ L(G ϑ , G ϑ ) which acts as For this l ϑϑ , one can get the properties analogous to those stated in Proposition 3.1. Next, for n ∈ N, we define, cf. (3.12), As in the proof of Proposition 3.2, by means of (3.19) we then show that the sequence {H holding for each G ∈ G ϑ and k ∈ K ϑ .

The identification
Our next step is based on the following statement.

Lemma 3.5 Let {Q ϑ ϑ (t) : (ϑ, ϑ , t) ∈ Θ} be the family as in Proposition 3.2. Then,
for each ϑ and ϑ and t ∈ [0, T (ϑ , ϑ)/2), we have that Q ϑ ϑ (t) : We prove this lemma in a number of steps. First we introduce auxiliary models, indexed by σ > 0, for which we construct the families of operators Q σ ϑ ϑ (t) similar as in Proposition 3.2. Then we prove that these families have the property stated in Lemma 3.5. Thereafter, we show that holding for each G ∈ B bs ( 0 ) and k t as in Lemma 3.3 with t ∈ [0, T (ϑ , ϑ)/2), see (2.17). By Proposition 2.3 this yields the fact we wish to prove.

Auxiliary models
For σ > 0, we set Let also L ,σ stand for L as in (2.9) with b replaced by b σ . Note that b σ ≤ b . Clearly, for this L ,σ , we can perform the same construction as in the previous subsection and obtain the family {Q σ ϑ ϑ (t) : (ϑ, ϑ , t) ∈ Θ} as in Proposition 3.2 with Θ and T (ϑ , ϑ) given in (3.10) and (3.9), respectively. Note also that Q σ ϑ ϑ (t) satisfy, cf. (3.11) and Remark 3.4, Like in (3.15) we then set Also as above, we construct the operators H σ ϑϑ (t) such that, cf. (3.21), holding for appropriate G and k.
Proof Take ϑ = (ϑ 1 + ϑ 0 )/2 and then pick ϑ ∈ (ϑ, ϑ 1 ) such that (3.27) which is possible in view of the continuous dependence of T (ϑ , ϑ) on both its arguments, see (3.9). For t < T (ϑ 1 , ϑ 0 )/2, by (3.15) and (3.25) we get that see (2.9) and k σ s lies in K ϑ , which is possible since see (3.27). Take G ∈ B bs . Since it lies in each G ϑ , and hence in G ϑ 1 , we can get (3.27). For this G, by (3.21) and (3.28) we have To get the latter line we also used (3.29). Recall that here G t−s ∈ G ϑ and k σ s ∈ K ϑ with ϑ < ϑ . Let us prove that The latter convergence follows by (3.31) and the Lebesgue dominated convergence theorem. This completes the proof.

Auxiliary evolutions
Now we turn to proving that the assumption of Proposition 3.6 holds true. For a compact , by we denote the set of configurations η contained in . It is a measurable subset of 0 , i.e., ∈ B( ). Recall that B( ) can be generated by the cylinder sets ,n with all possible compact and n ∈ N 0 . Let B( ) denote the sub-σ -field of B( ) consisting of A ⊂ . For A ∈ B( ), we set C (A) = {γ ∈ : γ ∈ A}. Then, for a state μ, we define μ by setting μ (A) = μ(C (A)); thereby, μ is a probability measure on B( ). It is possible to show, see [8], that for each compact and μ ∈ P exp ( ), the measure μ has density with respect to the Lebesgue-Poisson measure defined in (2.5), which we denote by R μ . Moreover, the correlation function k μ and the density R μ satisfy (3.32) Let μ 0 ∈ P exp ( ) be the initial state as in Lemma 3.3. Fix some compact and N ∈ N, and then, for η ∈ 0 , set Proposition 3.7 The operator (L † , D) defined in (3.34) and (3.36) is the generator of a substochastic semigroup Proof In this statement we mean that We use the Thieme-Voigt theorem in the form of [10, Propositions 3.1 and 3.2]. By this theorem the proof amounts to checking the validity of the following inequalities: (3.38) holding for some positive C and ε. Recall that σ is defined in (3.35). By direct inspection we get from (3.34) that the left-hand side of the first line in (3.38) equals zero for each R ∈ D. Proving the second inequality in (3.38) reduces to showing that, for each ϑ > 0, the function is bounded from above, which is obviously the case.
The second auxiliary evolution is supposed to be constructed in G ϑ . It is generated by the operator L ϑ the action of which coincides with that of L ,σ , see (2.9) and (2.10) with b replaced by b σ . The domain of this operator is Proof As in the proof of Lemma 5.5 in [7], we pass from q to w by setting w(η) = (−1) |η| q(η), and hence to L ϑ defined on the same domain (3.39) by the relation Then we just prove that ( L ϑ , D ϑ ) generates a C 0semigroup on G ϑ . In view of (2.10) -(2.9), we have The latter estimate allows us to apply here the Thieme-Voigt theorem, see [10, Proposition 3.1] by which A + B generates a substochastic semigroup in G ϑ . Thus, L ϑ generates a C 0 -semigroup since C is bounded. This completes the proof. Now for R ,N 0 defined in (3.33), we set where β > 0 is to satisfy e β = 1 + e ϑ . Hence, q ,N 0 ∈ G ϑ for each ϑ > 0. In view of this, q ,N 0 ∈ D ϑ for each ϑ > 0, see (3.39). Consider the problem in G ϑ (3.42) Proposition 3.9 For each ϑ > 0, the problem in (3.42) has a unique global solution q t ∈ D ϑ such that, for each G ∈ B bs ( 0 ), the following holds G, q t ≥ 0.
On the other hand, this solution can be sought in the form where S † is the semigroup constructed in Proposition 3.7. Indeed, by direct inspection one verifies that q t in this form satisfies (3.42), cf. the proof of Lemma 5.8 in [7]. Then, cf. (2.15), which yields (3.43). The inequality in (3.46) follows by the fact that the semigroup S † is substochastic, see (3.37). This completes the proof.
By (3.41) it follows that q ,N 0 ∈ K ϑ 0 , hence we may use it in (3.25) and obtain Proof A priori k ,N t and q t lie in different spaces: K ϑ 1 and G ϑ , respectively. Note that the latter ϑ can be arbitrary positive. The idea is to construct one more evolution q ,N 0 → u t in some intersection of these two spaces, related to the evolutions in (3.47) and (3.44). Then the proof will follow by the uniqueness as in Proposition 3.9.

Proof of Lemma 3.5
We have to show that the assumption of Proposition 3.6 holds true for each σ > 0, which is equivalent to proving that k σ t given in (3.25) has the property holding for all t < T (ϑ 1 , ϑ 0 ) and G 0 ∈ B bs ( 0 ). By definition, a cofinal sequence of { n } n∈N is a sequence of compact subsets n ⊂ R d such that n ⊂ n+1 , n ∈ N, and each x ∈ R d is contained in a certain n . Let { n } n∈N be such a sequence. Fix σ > 0 and then, for given n and N ∈ N, obtain q n ,N 0 from k μ 0 ∈ K ϑ 0 by (3.33), (3.40). As in [1,Appendix] one can show that, for each G ∈ G ϑ 0 , the following holds lim

Proof of Theorem 2.4
To complete proving the theorem we have to construct the continuation of the solution (3.15) to all t ≥ 0 and prove the upper bound in (2.27). The lower bound follows by the fact that k t ∈ K . This will be done by comparing k t with the solution of the equation (2.8) for the Surgailis model given in (2.21). If we denote the latter by v t , then For k μ 0 ∈ K ϑ 0 , by (2.13) and (1.7) we get from the latter which holds also in the case m ≡ 0. Thus, for a given T > 0, W (t) with t ∈ [0, T ] acts as a bounded operator W ϑ T ϑ 0 (t) from K ϑ 0 to K ϑ T with For ϑ ∈ R, we set, cf. (3.9), For ϑ 1 = ϑ 0 + 1, let k t be given in (3.15) with t ∈ [0, τ (ϑ 0 )). Fix some κ ∈ (0, 1/2) and set T 1 = κτ (ϑ 0 ). By Lemmas 3.3 and 3.5 we know that k t = Q ϑ 1 ϑ 0 (t)k μ 0 exists and lies in K ϑ 1 for all t ∈ [0, T 1 ]. Take ϑ ∈ (ϑ 0 , ϑ 0 + 1) such that T 1 < T (ϑ, ϑ 0 ). Then take ϑ > ϑ and set, cf. (3.58), For t ∈ [0, where k s belongs to K ϑ , whereas v t and k t belong to Kθ 1 . By (3.56) and (3.1) the action of D in (3.60) is Set T 2 = κτ (ϑ T 1 ), ϑ 2 = ϑ T 1 + 1 and consider k (2) (3.13), and hence is a continuation of k t to [T 1 , T 2 ]. Now we repeat this procedure due times and obtain where ϑ n = ϑ T n−1 + 1 and The continuation to all t > 0 will be obtained if we show that n≥1 T n = +∞. Assume that this is not the case. From the second line in (3.62) we get T n−1 = (e ϑ Tn − e ϑ T n−1 )/ b . Hence
Proof of Theorem 2.5 By means of the evident monotonicity we conclude that it is enough to prove the statement for: finds τ (ε, ), dependent also on μ 0 , such that, for all x ∈ and t ≥ τ (ε, ), the following holds k (1) That is, after some time the density at each point of approaches a certain level, independent of the initial distribution of the entities in .