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 $\mathds{R}^d$ 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 $\mu_0 \mapsto \mu_t$ such that the moments $\mu_t(N_\Lambda^n)$, $n\in \mathds{N}$, of the number of entities in compact $\Lambda \subset \mathds{R}^d$ remain bounded for all $t>0$. Under an additional condition, we prove that the density of entities and the second correlation function remain bounded globally in time.


Introduction
We study the Markov dynamics of an infinite system of point entities placed in R d , d ≥ 1, which appear (immigrate) with space-dependent rate b(x) ≥ 0, and disappear. The rate of disappearance of the entity located at a given x ∈ R d is the sum of the 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 Γ of all subsets γ ⊂ R d such that the set γ Λ := γ ∩ Λ is finite whenever Λ ⊂ R d is compact. For each such Λ, one defines the counting map Γ ∋ γ → |γ Λ | := #{γ ∩ Λ}, where the latter denotes cardinality. Thereby, one introduces the subsets Γ Λ,n := {γ ∈ Γ : |γ Λ | = n}, n ∈ N 0 , and equips Γ with the σ-field generated by all such Γ Λ,n . This allows for considering probability measures on Γ as states of the system. Among them there are Poissonian states in which the entities are independently distributed over R d , see [7,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 Λ, one has π ̺ (Γ Λ,n ) = ̺ n Λ exp (− ̺ Λ ) /n!, ̺ Λ := Λ ̺(x)dx. (1.1) By (1.1) one readily gets the π ̺ -expected value of the number N Λ of entities contained in Λ in the form The case of ̺ ≡ κ > 0 corresponds to the homogeneous Poisson measure π κ . The counting map Γ ∋ γ → |γ| can also be defined for Λ = R d . Then the set of finite is measurable. In a state with the property µ(Γ 0 ) = 1, the system is (µ-almost surely) finite. By (1.1) one gets that either π ̺ (Γ 0 ) = 1 or π ̺ (Γ 0 ) = 0, depending on whether or not ̺ is globally integrable. As π κ (Γ 0 ) = 0, the system in state π κ 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 [5,6,8] 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.
To characterize states on Γ one employs observables -appropriate functions F : Γ → 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 The model that we study in this work is specified by the following Here b(x) is the immigration rate, m(x) ≥ 0 is the intrinsic emigration (mortality) rate, and a ≥ 0 is the competition kernel. If one takes in (1.6) a ≡ 0, the model becomes exactly soluble, see subsection 2.3 below. This means that the evolution can be constructed explicitly for each initial state µ 0 . Assuming that µ 0 (N n Λ ) < ∞ for all n ∈ N, one can get the information about the time dependence of such moments. For m(x) ≥ m * > 0 for all x ∈ R d , one obtains that Λ , holding for each compact Λ. Otherwise, all the moments µ t (N n Λ ) 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 Λ ) ≤ CV(Λ) for an appropriate constant and large enough values of the Euclidean volume V(Λ), 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 Λ ) ≤ C (n) Λ 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 subsection 2.1 below, then all k (n) µt , n ∈ N are also continuous and such that holding (with some positive C 1 and C 2 ) for all t > 0 and all values of the spatial variables.
The structure of the article is as follows. In Section 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 Sections 3 and 4, we present the proofs of Theorems 2.4 and 2.5, respectively.

Preliminaries and the Results
We begin by outlining some technical aspects of this work -a more detailed description of them can be found in [5,6,8,9] and in the literature quoted therein.
By B(R) we denote the sets of all Borel subsets of R. The configuration space Γ is equipped with the vague topology, see [9], and thus with the corresponding Borel σ-field B(Γ), which makes it a standard Borel space. Note that B(Γ) is exactly the σ-field generated by the sets Γ Λ,n , mentioned in Introduction. By P(Γ) we denote the set of all probability measures on (Γ, B(Γ)).

Correlation functions.
Like in [6,8], 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. [10], is For the homogeneous Poisson measure π κ , it takes the form Definition 2.1. The set of states P exp (Γ) is defined as that containing all those states µ ∈ P(Γ) for which B µ can be continued, as a function of θ, to an exponential type entire function on L 1 (R d ). The elements of P exp (Γ) are called sub-Poissonian states.
It can be shown that a given µ belongs to P exp (Γ) if and only if its functional B µ can be written down in the form where k (n) µ is the n-th order correlation function of µ. It is a symmetric element of with some C > 0 and ϑ ∈ R. Note that k (n) πκ (x 1 , . . . , x n ) = κ 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., [12]. Recall that Γ 0 -the set of all finite γ ∈ Γ defined in (1.3) -is an element of B(Γ). A function G : Γ 0 → R is B(Γ)/B(R)-measurable, see [6], if and only if, for each n ∈ N, there exists a symmetric Borel function G (n) : (R d ) n → R such that G(η) = G(η) = G (n) (x 1 , . . . , x n ), for η = {x 1 , . . . , x n }. (2.4) exists N ∈ N 0 such that G(η) = 0 whenever |η| > N . By Λ(G) and N (G) we denote the smallest Λ and N with the properties just mentioned. By B bs (Γ 0 ) we denote the set of all such functions.
holding for all G ∈ B bs (Γ 0 ). Like in (2.4), we introduce k µ : 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 d dt and In the next subsection, we introduce the spaces where we are going to define (2.8).
2.2. The Banach spaces. By (2.2) and (2.6), it follows that µ ∈ P exp (Γ) implies holding for λ-almost all η ∈ Γ 0 , some C > 0, and ϑ ∈ R. In view of this, we set where k ϑ = ess sup Clearly, (2.12) and (2.13) define a Banach space. In the following, we use the ascending scale of such spaces K ϑ , ϑ ∈ R, with the property where ֒→ denotes continuous embedding.
where ⋐ indicates that the summation is taken over all finite subsets. It satisfies, see . The latter means that µ(KG) < ∞ for each µ ∈ P exp (Γ). By (2.6) this yields (2.17) By [9, Theorems 6.1 and 6.2 and Remark 6.3] one can prove the next statement. Proposition 2.3. Let a measurable function k : Γ 0 → R have the following properties: with (c) holding for some C > 0 and λ-almost all η ∈ Γ 0 . Then there exists a unique state µ ∈ P exp (Γ) for which k is the correlation function.
Set, cf (2.17), which is a subset of the cone By Proposition 2.3 it follows that each k ∈ K ⋆ ϑ such that k(∅) = 1 is the correlation function of a unique state µ ∈ P exp (Γ). 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.
2.3. 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, which 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(Γ). Indeed, for each t > 0 and θ ∈ Θ, cf (2.1), we have that θψ t ∈ Θ, and hence B µ 0 (θψ 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 can be considered 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 If m(x) ≥ m * > 0 for some m * and all x ∈ R d , then the solution in (2.21) lies in K ϑ * for all t > 0. Here (2.24) Otherwise, the solution in (2.21) is unboundedly increasing in t. If, for some compact Λ, m(x) = 0 for x ∈ Λ, then by (2.21) and (2.22) we get that by (1.2), (2.15) and (2.16) yields 2.4. 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 [8, 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 (Γ), 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 (Γ) such that k t = k µt . 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 * is given in (2.24).
Theorem 2.5 (Global boundedness). The states µ t mentioned in Theorem 2.4 have the property: for every n ∈ N and compact Λ ⊂ R d , the following holds µt for all n ∈ N and t > 0. Moreover, k (1) µt and k (2) µt have the properties as in (1.8). 2.5. 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 the proof of Theorem 2.5 and (4.19) in particular. 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 Λ ) ≤ C Λ .

The Existence of the Evolution of States
We follow the line of arguments used in proving Theorem 3.3 in [8] 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 )) (subsection 3.1). (ii) Proving that the mentioned solution k t has 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 (subsection 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 (subsection 3.3). 3.1. Solving the Cauchy problem. We begin by rewriting L ∆ (given in (2.9), (2.10)) in the following form For ϑ ∈ R and ϑ ′ > ϑ, let L(K ϑ , K ϑ ′ ) be the Banach space of all bounded linear operators acting from Now we set, see (2.11), and then introduce the corresponding S ϑ ′ ϑ (t) ∈ L(K ϑ , K ϑ ′ ), t ≥ 0. By the first estimate in (3.2) one shows that the map is continuous and such that holding for each ϑ ′′ ∈ (ϑ, ϑ ′ ). Note that (3.3) may be used to define a bounded multiplication operator, S ϑ (t) ∈ L(K ϑ ) := L(K ϑ , K ϑ ). However, in this case the map [0, +∞) ∋ t → S ϑ (t) ∈ L(K ϑ ) would not be continuous. For ϑ and ϑ ′ > ϑ as above, we fix some δ < ϑ ′ − ϑ. Then, for a given l ∈ N, we divide the interval [ϑ, ϑ ′ ] into subintervals with endpoints ϑ s , s = 0, 1, . . . , 2l + 1, as follows.
3.2. The identification. Our next step is based on the following statement.
Proof. Take ϑ = (ϑ 1 + ϑ 0 )/2 and then pick ϑ ′ ∈ (ϑ, ϑ 1 ) such that 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 where (2.9), 29) 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 lies in L 1 (R d ) for each s ≤ T . Indeed, by (2.5) and (3.18) we have We use this in (3.30) to get The latter convergence follows by (3.31) and the Lebesgue dominated convergence theorem. This completes the proof.

3.2.2.
Auxiliary evolutions. Now we turn to proving that the assumption of Proposition 3.6 holds true. For a compact Λ, by Γ Λ we denote the set of configurations η contained in Λ. It is a measurable subset of Γ 0 , i.e., Γ Λ ∈ B(Γ). Recall that B(Γ) can be generated by the cylinder sets Γ Λ,n with all possible compact Λ and n ∈ N 0 . Let B(Γ Λ ) denote the sub- Then, for a state µ, we define µ Λ by setting µ Λ (A) = µ(C Λ (A)); thereby, µ Λ is a probability measure on B(Γ Λ ). It is possible to show, see [9], that for each compact Λ and µ ∈ P exp (Γ), the measure µ Λ has density with respect to the Lebesgue-Poisson measure defined in (2.5), which we denote by R Λ µ . Moreover, the correlation function k µ and the density R Λ µ satisfy Let µ 0 ∈ P exp (Γ) be the initial state as in Lemma 3.3. Fix some compact Λ and N ∈ N, and then, for η ∈ Γ 0 , set Let us now consider the auxiliary model specified by L ∆,σ , and also by L σ which one obtains by replacing in (1.6) b by b σ , see (3.23). Then the equation for the densities obtained from the Fokker-Planck equation (1.5) takes the form Proof. In this statement we mean that (c) S † (t) : G + ϑ → G + ϑ , for all ϑ > 0. We use the Thieme-Voigt theorem in the form of [11, Propositions 3.1 and 3.2]. By this theorem the proof amounts to checking the validity of the following inequalities: holding for some positive C and ε. Recall that Ψ 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 (3.39) Proposition 3.8. For each ϑ > 0, the operator ( L ϑ , D ϑ ) is the generator of a C 0semigroup S ϑ := { S ϑ (t)} t≥0 of bounded operators on G ϑ .

By (3.18) we get
hence C is a bounded operator. For w ∈ G + ϑ , we have The latter estimate allows us to apply here the Thieme-Voigt theorem, see [11, 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 Λ,N 0 defined in (3.33), we set where β > 0 is to satisfy e β = 1 + e ϑ . Hence, q Λ,N 0 ∈ G ϑ for each ϑ > 0. In view of this, q Λ,N 0 ∈ D ϑ for each ϑ > 0, see (3.39). Consider the problem in G ϑ d dt q t = L ϑ q t , q t | t=0 = q Λ,N 0 .
(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 (Γ 0 ), the following holds G, q t ≥ 0. (3.43) Proof. By Proposition 3.8 the problem in (3.42) has a unique global solution given by 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 [8]. 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 Λ,N 0 ∈ K ϑ 0 , hence we may use it in (3.25) and obtain Proof. A priori k Λ,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 Λ,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.
By (4.7) it follows that We apply this in (4.17) and obtain, cf (4.13) and (4.12), For l = 1, by (4.9) and (4.14) we get from the latter that (4.16) holds. Now we assume that (4.16) holds for a given l − 1. It yields in (4.18) d dt q (l) from which by (4.14) we obtain that (4.16) holds also for l.
Proof of Theorem 2.5. By means of the evident monotonicity we conclude that it is enough to prove the statement for: