The Evolution of States in a Spatial Population Model

The evolution of states in a spatial population model is studied. The model describes an infinite system of point entities in $$\mathbb {R}^d$$Rd which reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The system’s states are probability measures on the space of configurations, and their evolution is obtained from a hierarchical chain of differential equations for the corresponding correlation functions derived from the Fokker–Planck equation for the states. Under natural conditions imposed on the model parameters it is proved that the correlation functions evolve in a scale of Banach spaces in such a way that at each moment of time the correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are described.


Posing
The development of a mathematical theory of complex living systems is a challenging task of modern mathematics [4]. In many cases of such systems, one deals with birth-and-death B Yuri Kozitsky jkozi@hektor.umcs.lublin.pl Yuri Kondratiev kondrat@math.uni-bielefeld.de 1 Fakutät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany 2 Interdisciplinary Center for Complex Systems, Dragomanov University, Kiev, Ukraine 3 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland processes the theory of which traces back to works by Kolmogorov and Feller, see [9,ter XVII] and e.g. [16,30] for a more recent account on the related concepts and results. In the simplest models of this kind, the system is finite and the state space is N 0 := N ∪ {0}. Then the time evolution of the probability of having n particles in the system is obtained from the Kolmogorov equation in which the generator is a tridiagonal infinite matrix containing the birth and death rates λ n and μ n , respectively. If their increase is controlled by affine functions of n, the evolution is obtained with the help of a stochastic semigroup, see e.g. [3,22] and the papers quoted in these works. However, if λ n and μ n increase faster than n, one would not expect that the evolution takes place, for all t > 0, in one and the same Banach space and thus is described by a C 0 -semigroup of operators acting in this space. For infinite systems, the situation is much more complex as the very definition of the Kolmogorov equation cannot be performed directly (for λ ∞ and μ ∞ are infinite in such cases). The main result of the present paper consists in constructing the evolution of states of an infinite birth-and-death system of particles placed in R d with 'rates' that roughly speaking increase as n 2 , cf. (3.9) below. This evolution takes place in an ascending sequence of Banach spaces and is obtained by a method developed in the paper. To the best of our knowledge, this is the first construction of this type.
We continue, cf. [10,12,13,23], studying the model introduced in [6,7,24]. It describes an infinite evolving population of identical point entities (particles) distributed over R d , d ≥ 1, which reproduce themselves and die, also due to competition. This is one of the most important individual-based models in studying large ecological communities (e.g. of perennial plants), see [27] and [25, page 1311]. As is now commonly adopted [6,7,27], the appropriate mathematical context for studying models of this kind is provided by the theory of random point fields in R d in which populations are modeled as point configurations constituting the set = γ ⊂ R d : |γ ∩ | < ∞ for any compact ⊂ R d , (1.1) where | · | denotes cardinality. It is equipped with a σ -field of measurable subsets that allows one to consider probability measures on as states of the system. To characterize such states one employs observables -appropriate functions F : → R. Their evolution is obtained from the Kolmogorov equation (1.2) where the generator L specifies the model. The states' evolution is then obtained from the Fokker-Planck equation related to that in (1.2) by the duality The generator for the model studied in this paper is where E ± (x, γ ) := y∈γ a ± (x − y). (1.5) The first summand in (1.4) corresponds to the death of the particle located at x occurring independently at rate m ≥ 0 (intrinsic mortality) and under the influence of the other particles in γ -at rate E − (x, γ \ x) ≥ 0 (competition). The second term in (1.4) describes the birth of a particle at y ∈ R d occurring at rate E + (y, γ ) ≥ 0. In the sequel, we call a − and a + competition and dispersal kernels, respectively. The particular case of (1.4) with a − ≡ 0 is the continuum contact model studied in [18,21]. Having in mind the results of these works, along with purely mathematical tasks we aim at understanding the ecological consequences of the competition taken into account in (1.4). The problem of constructing spatial birth and death processes in infinite volume was first studied by Holley and Stroock in their pioneering work [15], where a special case of nearest neighbor interactions on the real line was considered. For more general versions of continuum birth-and-death systems, the few results known by this time were obtained under severe restrictions imposed on the birth and death rates. This relates to the construction of a Markov process in [14], as well as to the result obtained in [12] in the statistical approach (see below). In the present work, we make an essential step forward in studying the model specified in (1.4) assuming only that the kernels a ± satisfy some rather natural condition.
The set of finite configurations 0 is a measurable subset of . If μ is such that μ( 0 ) = 1, then the considered system is finite in this state. If μ 0 in (1.3) has such a property, the evolution μ 0 → μ t can be obtained directly from (1.3), see [23]. In this case μ t ( 0 ) = 1 for all t > 0. States of infinite systems are mostly such that μ( 0 ) = 0, which makes direct solving (1.3) with an arbitrary initial state μ 0 rather unaccessible for the method existing at this time, cf. [20]. In this work we continue following the statistical approach [5,10,12,13,20] in which the evolution of states is described by means of the corresponding correlation functions. To briefly explain its essence let us consider the set of all compactly supported continuous functions θ : R d → (−1, 0]. For a probability measure μ on its Bogoliubov functional [11,19] is defined as B μ (θ ) = x∈γ (1 + θ(x))μ(dγ ), (1.6) with θ running through the mentioned set of functions. For π -the homogeneous Poisson measure with intensity > 0, (1.6) takes the form In state π , the particles are independently distributed over R d with density . The set of sub-Poissonian states P sP is then defined as that containing all the states μ for which B μ can be continued, as a function of θ , to an exponential type entire function on L 1 (R d ). This exactly means that B μ can be written down in the form where k (n) μ is the n-th order correlation function corresponding to μ. It is a symmetric element of L ∞ ((R d ) n ) for which with some C > 0 and α ∈ R. This guarantees that B μ is of exponential type. One can also consider a wider class of states, P anal , by imposing the condition that B μ can be continued to a function on L 1 (R d ) analytic in some neighborhood of the origin, see [19]. In that case, the estimate corresponding to (1.8) will contain n!Ce αn in its right-hand side. States μ ∈ P anal are characterized by strong correlations corresponding to 'clustering'. In the contact model the clustering does take place, see [18,21] and especially [10,Eq. (3.5), page 303]. Namely, in this model for each t > 0 and n ∈ N the correlation functions satisfy the following estimates where the left-hand inequality holds if all x i belong to a ball of sufficiently small radius. If the mortality rate m is big enough, then C t → 0 as t → +∞. That is, in the continuum contact model the clustering persists even if the population asymptotically dies out. With this regard, a paramount question about the model (1.4) is whether the competition contained in L can suppress clustering. In short, the answer given in this work is in affirmative provided the competition and dispersal kernels satisfy a certain natural condition. They do satisfy if a − is strictly positive in some vicinity of the origin, and a + has finite range.

Presenting the Result
(i) Under the condition on the kernels a ± formulated in Assumption 1 we prove in Theorem 3.3 that the correlation functions evolve k t is the correlation function of a unique sub-Poissonian measure μ t . (ii) We give examples of the kernels a ± which satisfy Assumption 1. These examples include kernels of finite range -both short and long dispersals (Proposition 3.7), and also Gaussian kernels (Propositions 3.8). (iii) For the whole range of values of the intrinsic mortality rate m, in Theorem 3.4 we obtain the following bounds for the correlation functions holding for all t ≥ 0: where a + is the L 1 -norm of a + , C δ and C ε are appropriate positive constants, whereas δ < m and ε ∈ ( a + , m) take any value in the mentioned sets. By (1.7) these estimates give upper bounds for the type of B μ t . We describe also the pure death case where a + = 0.
More detailed comments and comparison with the previous results on this model are given in Sect. 3.3 below.

The Basic Notions
A detailed description of various aspects of the mathematical framework of this paper can be found in [1,5,10,12,13,17,18,21,26]. Here we present only some of its aspects and indicate in which of the mentioned papers further details can be found. By B(R d ) and B b (R d ) we denote the set of all Borel and all bounded Borel subsets of R d , respectively.

The Configuration Spaces
The space defined in (1.1) is endowed with the weakest topology that makes continuous all the maps Here C 0 (R d ) stands for the set of all continuous compactly supported functions f : R d → R. The mentioned topology on admits a metrization which turns it into a complete and separable metric (Polish) space. By B( ) we denote the corresponding Borel σ -field. For n ∈ N 0 := N ∪ {0}, the set of n-particle configurations in R d is For n ≥ 1, (n) can be identified with the symmetrization of the set which allows one to introduce the topology on (n) related to the Euclidean topology of R d and hence the corresponding Borel σ -field B( (n) ). The set of finite configurations is endowed with the topology of the disjoint union and with the corresponding Borel σ -field B( 0 ). It is a measurable subset of . However, the topology just mentioned and that induced on 0 from do not coincide. For ∈ B b (R d ), the set := {γ ∈ : γ ⊂ } is a Borel subset of 0 . We equip with the topology induced by that of 0 . Let B( ) be the corresponding Borel σ -field. It can be proved, see [ It is known [1, page 451] that B( ) is the smallest σ -field of subsets of such that all the projections Remark 2.1 From the latter discussion it follows that 0 ∈ B( ) and Hence, a probability measure μ on B( ) with the property μ( 0 ) = 1 can be considered also as a measure on B( 0 ).

Functions and Measures on Configuration Spaces
A Borel set ϒ ⊂ is said to be bounded if the following holds for some ∈ B b (R d ) and N ∈ N. In view of (2.2), each bounded set is in B( 0 ). A function G : 0 → R is measurable if and only if, for each n ∈ N, there exists a symmetric Borel function G (n) : The set of all such functions is denoted by B bs ( 0 ). For a given G ∈ B bs ( 0 ), by N (G) we denote the smallest N with the property as in (b).
Clearly, such a map F is measurable. By F cyl ( ) we denote the set of all cylinder functions. For γ ∈ , by writing η γ we mean that η ⊂ γ and η is finite, i.e., η ∈ 0 . For G ∈ B bs ( 0 ), we set As proved in [17], which has to hold for all G ∈ B bs ( 0 ), cf. (2.3). Note that B bs ( 0 ) is a measure defining class. Clearly, λ(ϒ) < ∞ for each bounded ϒ ∈ B( 0 ). With the help of (2.5), we rewrite (1.7) in the following form In the sequel, by saying that something holds for all η we mean that it holds for λ-almost all η ∈ 0 . This relates also to (2.3). Let P( ), resp. P( 0 ), stand for the set of all probability measures on B( ), resp. B( 0 ). Note that P( 0 ) can be considered as a subset of P( ), see Remark 2.1. For a given μ ∈ P( ), the projection μ is defined as where p −1 (A) := {γ ∈ : p (γ ) ∈ A}, see (2.1). The projections of the Lebesgue-Poisson measure λ are defined in the same way. Recall that P anal (resp. P sP ) denotes the set of all those μ ∈ P( ) for each of which B μ defined in (1.6), or (2.6), admits continuation to a function on L 1 (R d ) analytic in some neighborhood of zero (resp. exponential type entire function). The elements of P sP are called sub-Poissonian states. One can show [17,Proposition 4.14] that for each ∈ B b (R d ) and μ ∈ P sP , μ is absolutely continuous with respect to λ . The Radon-Nikodym derivative 8) and the correlation function k μ satisfy which holds for all ∈ B b (R d ). Note that (2.9) ties R μ with the restriction of k μ to . The fact that these are the restrictions of one and the same function k μ : 0 → R corresponds to the Kolmogorov consistency of the family {μ } ∈B b (R d ) .
By (2.4), (2.7), and (2.9) we get which holds for each G ∈ B bs ( 0 ) and μ ∈ P sP . Here and in the sequel we use the notation By [17, Theorems 6.1 and 6.2 and Remark6.3] we know that the following holds.

Proposition 2.3
Let a measurable function k : 0 → R have the following properties: property (iii) holding for some C > 0. Then there exists a unique μ ∈ P sP for which k is the correlation function.
Finally, we mention the convention a∈∅ φ a := 0, a∈∅ ψ a := 1 which we use in the sequel and the integration rule, see, e.g. [10], valid for appropriate functions H .

Spaces of Functions
For each μ ∈ P sP , the correlation function satisfies (1.8) in view of which we introduce the following Banach spaces. For α ∈ R, we set It is a norm that can also be written as follows. As in (2.3), each k : 0 → R is defined by its restrictions to (n) . Let k (n) : (R d ) n → R be a symmetric Borel function such that k (n) (x 1 , . . . , x n ) = k(η) for η = {x 1 , . . . , x n }. We then assume that k (n) ∈ L ∞ ((R d ) n ), n ∈ N, cf. (1.8), and define that yields the same norm as in (2.14). Obviously, is a Banach space. For α < α , we have k α ≤ k α . Hence, Here and in the sequel X → Y denotes continuous embedding. For α ∈ R, we define, cf.
(2.11) and (2.10), It is a subset of the cone By Proposition 2.3 we have that each k ∈ K α with the property k(∅) = 1 is the correlation function of a unique μ ∈ P sP . We also put 20) and equip this set with the inductive topology. Finally, we define

The Model
As was already mentioned, the model is specified by the expression given in (1.4). Regarding the kernels in (1.5) we suppose that and thus define and We also denote where m is the same as in (1.4).
In addition to the standard assumptions (3.1) we shall use the following b|η| Note that the case of point-wise domination (3.6) cf. [13,Eq. (3.11)], corresponds to (3.5) with b = 0. In Sect. 3.4 below we give examples of the kernels a ± which satisfy (3.5). To exclude the trivial case of a + = a − = 0 we also assume that a − > 0.

The Operators
In view of the relationship between states and correlation functions discussed in Sect. 2.3, we describe the system's dynamics in the following way. First we obtain the evolution k μ 0 → k t by proving the existence of a unique solution of the Cauchy problem of the following type where the action of L is calculated from (1.4). Thereafter, we show that each k t has property k t (∅) = 1 and lies in K α for some α ∈ R. Hence, it is the correlation function of a unique μ t ∈ P sP . This yields in turn the evolution μ 0 → μ t .
To describe the action of L in a systematic way we write it in the following form, see [10,13], where see also (3.3), (3.4), and The key idea of the method that we use to study (3.7) is to employ the scale of spaces (2.16) in which A and B act as bounded operators from K α , to any K α with α > α , cf. (2.17). For such α and α , by (2.14) and (2.15) we have, see (3.9), which by (2.15) and (3.2) yields where we have used the estimate In a similar way, we obtain from (3.10) the following estimate, see (3.2), Thus, by means of (3.8) -(3.10), and then by (3.11) and (3.13), for each α, α ∈ R, α < α, one can define a continuous operator (3.14) Let L(K α , K α ) stand for the set of all bounded linear operators K α → K α . The operator norm of L αα can be estimated by means of the above formulas. Thus, the family {L αα } α,α determines a continuous linear operator L : with the inclusion holding for each α < α, see (3.11), (3.13), and (3.8). The operators such introduced are related to each other in the following way:

The Statements
Now we can make precise which equations we are going to solve. One possibility is to consider (3.7) in a given Banach space, K α .

Definition 3.1
Given α ∈ R and T ∈ (0, +∞], by a solution of the Cauchy problem Another possibility is to define (3.7) in the locally convex space (2.20).

Definition 3.2
By a global solution of the Cauchy problem (3.7) in K ∞ with a given k 0 ∈ K ∞ we mean a map [0, +∞) t → k t ∈ K ∞ , continuously differentiable on [0, +∞) and such that (3.7) is satisfied for all t ≥ 0.
According to Definition 3.2, for each T < +∞, there exist α 0 , α ∈ R, α 0 < α, for which the mentioned k t is a solution as in Definition 3.1 with k 0 ∈ K α 0 . Our main results are contained in the following two statements. (3.5) hold true, and μ 0 be an arbitrarily sub-Poissonian state. Then the problem (3.7) with k 0 = k μ 0 has a unique global solution k t ∈ K ∞ with property k t (∅) = 1. Therefore, for each t ≥ 0 there exists a unique sub-Poissonian measure μ t such that k t = k μ t .

Theorem 3.3 Let (b, ϑ)-assumption
The next statement describes the solutions in more detail.
Then the solution k t as in Theorem 3.3, corresponding to this k μ 0 , for all t ≥ 0, satisfies the following estimates.
is a stationary solution.

Comments on the Basic Assumption
By means of the function This resembles the stability condition (with stability constant b ≥ 0) for the interaction potential φ ϑ used in the statistical mechanics of continuum systems of interacting particles, see [31,Chapter 3]. Below we employ some techniques developed therein to prove that important classes of the kernels a ± have this property, see Propositions 3.7 and 3.8. The (b, ϑ) assumption holds with b = 0 if and only if (3.6) does. In this case, the dispersal kernel a + decays faster than the competition kernel a − (short dispersal). It can be characterized as the possibility for each daughter-entity to kill her mother-entity, or to be killed by her. In the previous works on this model [10,12,13] the results were based on this short dispersal condition. The novelty of the result of Proposition 3.7 is that it covers also the case of long dispersal where the range of a + is finite but can be bigger than that of a − . Noteworthy, by our Proposition 3.7 it follows that the interaction potential used in [29] is stable, which was unknown to the authors of that paper, cf. [29, page 146]. Proposition 3.8 describes Gaussian kernels, for which the basic assumption is valid also for both long and short dispersals. In this paper, we restricted our attention to the classes of kernels described in Propositions 3.7 and 3.8. Extensions beyond this classes, which we plan to realize in a separate work, can be made by means of the corresponding methods of the statistical mechanics of interacting particle systems.

Comments on the Results
An important feature of Theorems 3.3 and 3.4 is that the intrinsic mortality rate m ≥ 0 can be arbitrary. Theorem 3.3 gives a general existence of the evolution μ 0 → μ t , t > 0, in the class of sub-Poissonian states through the evolution of the corresponding correlation functions. Its ecological outcome is that the competition in the form as in (1.4), (1.5) excludes clustering provided the kernels satisfy (3.5). A complete characterization of the evolution k 0 → k t is then given in Theorem 3.4. By means of it this evolution is 'localized' in the spaces K α in Corollary 3.5. According to Theorem 3.4, for m < a + , or m ≤ a + and b > 0 in (3.24), the evolution described in Theorem 3.3 takes place in an ascending sequence {K α T } T ≥0 of Banach spaces, see (2.14) -(2.17), and also (3.22). If m > a + , the evolution holds in one and the same space, see Corollary 3.5. The only difference between the cases of b > 0 and b = 0 is that one can take δ = m in the latter case. This yields different results for m = a + , where the evolution takes place in the same space K α with α = log C m . Note also that for m = 0, one should take δ < 0. For m > a + , it follows from (3.19) that the population dies out: for a + > 0, the following holds for some ε ∈ (0, m − a + ), almost all (x 1 , . . . , x n ), and each n ∈ N. For m > 0 and a + = 0, by (3.20) we get This means that k . , x n ) → 0 as n → +∞ for sufficiently big t > 0. This phenomenon does not follow from (3.19). Finally, we mention that (3.21) corresponds to a special case of (3.6) and m = b = 0.

Comparison
Here we compare Theorems 3.3 and 3.4 with the corresponding results obtained for this model in [10,12] (where it was called BDLP model), and in [13]. Note that these are the only works where the infinite particle version of the model considered here was studied. In [10,12], the model was supposed to satisfy the conditions, see [12,Eqs. (3.38) and (3.39)], which in the present notations can be formulated as follows: (a) (3.6) holds with a given ϑ > 0; (b) m > 16 a − /ϑ holding with the same ϑ. Under these conditions the global evolution k 0 → k t was obtained in K α with some α ∈ R by means of a C 0 -semigroup. No information was available on whether k t is a correlation function and hence on the sign of k t .
In [13], the restrictions were reduced just to (3.6). Then the evolution k 0 → k t was obtained in a scale of Banach spaces K α as in Theorem 3.3, but on a bounded time interval. Like in [10,12], also here no information was obtained on whether k t is a correlation function. Until the present work no results on the extinction as in (3.19) and on the case of a + ≡ 0 were known.

Kernels Satisfying the Basic Assumption
Our aim now is to show that the assumption (3.5) can be satisfied in the most of 'realistic' models. We begin, however, by establishing an important property of the kernels satisfying (3.5). To this end we rewrite (3.5) in the form which yields the proof.
In the following two propositions we give examples of the kernels with the property (3.5).
In the first one, we assume that the dispersal kernel has finite range, which is quite natural in many applications. The competition kernel in turn is assumed to be just nontrivial.

Proposition 3.7
In addition to (3.1) and (3.2) assume that the kernels a ± have the following properties: (a) there exist positive c − and r such that a − (x) ≥ c − for |x| < r; (b) there exist positive c + and R such that a + (x) ≤ c + for |x| < R and a + (x) = 0 for |x| ≥ R.
Then for each b > 0, there exists ϑ > 0 such that (3.24) holds for these b and ϑ.
Proof For r ≥ R, (3.24) holds with b = 0 and ϑ = c − /c + . Thus, it remains to consider the case r < R. For |η| = 0 and |η| = 1, (3.24) trivially holds with each b > 0 and ϑ > 0. For |η| = 2, (3.24) holds whenever ϑ ≤ b/c + . For |η| > 2, we apply an induction in |η|, similarly as it was done in [2]. For x ∈ η, we define Then the next estimate holds true for each x ∈ η: Given n > 2 and positive ϑ and b, assume that U ϑ (η) ≥ 0 for each |ϑ| = n − 1. Then to make the inductive step by means of (3.25) we have to show that, for each η such that Ifn = 0, then η is such that |y − z| ≥ r for each distinct y, z ∈ η. In this case, the balls is the maximum number of rigid spheres of radius r/2 packed in a ball of radius R +r/2, and (d) is the density of the densest packing of equal rigid spheres in R d , see e.g. [8,Chapter 1]. We apply this in (3.25) and x such that the balls B x and B y i , i = 1, . . . , s, realize the densest possible packing of the ball of radius R + r/2 centered at x. Then s ≤ (d, r, R) − 1 and, for each y ∈ ξ + x , one finds i such that |y − y i | < r . Otherwise B y would not overlap each B y i , and thus the mentioned packing is not the densest one. Therefore, the balls . Now we apply this in (3.25) .
Thus, the inductive step can be done, which yields the proof.
As an example of kernels with infinite range we consider the Gaussian kernels where c ± > 0 and σ ± > 0 are parameters. Proposition 3.8 Let a ± be as in (3.27). Then for each b > 0, there exists ϑ such that (3.5) holds for these ϑ and b. Proof Then (3.24), and thus (3.5), hold for such ϑ and all b ≥ 0. For σ − < σ + , we can write, see (3.23), For Thus, ϑ 0 satisfies (3.24) with stability constant b 0 = φ ϑ 0 (0). Then we apply Proposition 3.6 and obtain that (3.24) holds for which completes the proof.

Evolution of Correlation Functions and States
Our proof of Theorems 3.3 and 3.4 may roughly be divided into the following three steps. First, we show that for each k 0 ∈ K α 1 and any α 2 > α 1 , cf. (2.17), the problem in (3.17) has a unique solution in K α 2 on the time interval [0, T (α 2 , α 1 )) with an explicitly computed T (α 2 , α 1 ) < ∞, see Lemma 4.8. To this end, in Lemma 4.5 we construct a family of bounded operators, indexed by t ∈ [0, T (α 2 , α 1 )) and acting from K α 1 to K α 2 , which gives the solution in question in the way resembling the action of a C 0 -semigroup. Here we employ a combination of the usual Ovcynnikov method, as in e.g. [5], based on the estimates in (4.18) and (4.19), and a substochastic semigroup constructed in the pre-dual space in Lemma 4.2.
The construction employs Assumption 1 and a perturbation result of [32] applied to the operator pre-dual to A b given in (4.15). In this way, we avoid the consequences of the righthand sides of (3.11) related to (α − α ) 2 , bad for using Ovcynnikov's method. However, due to the term e α 2 in (4.20) the length of the time interval T (α 2 , α 1 )) is bounded by some τ (α 1 ) < ∞. This and the fact that τ (α 1 ) → 0 as α 1 → +∞ do not allow one to increase T (α 2 , α 1 )) ad infinitum just by increasing the space K α 2 containing the solution. To overcome this difficulty, and thus to construct the global solution, we make another two steps. In Lemma 4.9, we show that the constructed solution k t lies in the cone defined in (2.18), and hence is the correlation function of a unique state μ t , see Proposition 2.3. The relevance of this fact is twofold. First of all, it implies that the evolution k μ 0 → k t corresponds to the uniquely determined evolution of states -the main aim of this work. At the same time, by Lemma 4.9 we obtain that k t (η) ≥ 0. By the comparison made in Lemma 4.10 based on this positivity we get rid of the mentioned term e α 2 , cf. the second line in (4.20). This finally allows us to continue the solution k t to all t > 0 -the third step -and thereby to construct the solution as claimed in Theorem 3.3. The estimates as in Theorem 3.4 are obtained by the mentioned comparison. We begin by constructing auxiliary semigroups used to make (in Sect. 4.2) the first step of the construction outlined above.

Auxiliary Semigroups
For a given α ∈ R, the space predual to K α , defined in (2.16), is in which the norm is, cf. (2.5), Clearly, |G| α ≤ |G| α for α < α, which yields cf. (2.17). One can show that this embedding is also dense. Recall that by m ≥ 0 we denote the mortality rate, see (1.4). For b ≥ 0 as in (3.5) we set Here E − (η) and E(η) are as in (3.3) and (3.4), respectively. For the same b, let the action of A b on functions G : 0 → R be as follows (4.5) Our aim now is to define A b as a closed unbounded operator in G α the domain of which contains G α for any α > α.
For each α > α, D α contains G α and hence is dense in G α , see (4.3). Then the first summand in A b turns into a closed and densely defined operator By (2.13) and (3.5) one gets Then for α > − log ϑ, we have that e −α /ϑ < 1, and hence A 2 is A 1,b -bounded. This means that (A b , D α ) is closed and densely defined in G α , see (4.5).
In the proof of Lemma 4.2 below we employ the perturbation theory for positive semigroups of operators in ordered Banach spaces developed in [32]. Prior to stating the lemma we present the relevant fragments of this theory in spaces of integrable functions. Let E be a measurable space with a σ -finite measure ν, and X := L 1 (E → R, dν) be the Banach space of ν-integrable real-valued functions on X with norm · . Let X + be the cone in X consisting of all ν-a.e. nonnegative functions on E. Clearly, f + g = f + g for any f, g ∈ X + , and X = X  Then for all r ∈ [0, 1), the operator A 0 + r P, D(A 0 ) is the generator of a substochastic C 0 -semigroup in X .

Lemma 4.2 For each
Proof We apply Proposition 4.1 with E = 0 , X = G α as in (4.1), and A 0 = A 1,b . For r > 0 and A 2 as in (4.5), we set P = r −1 A 2 . For such A 0 and P, and for G ∈ D + α , the left-hand side of (4.8) takes the form, cf. (4.7), For a fixed α > − log ϑ, pick r ∈ (0, 1) such that r −1 (e −α /ϑ) < 1. Then, for such α and r , we have which holds in view of (3.5). Since r −1 A 2 is a positive operator, by Proposition 4.1 we have Now we turn to constructing the semigroup 'sun-dual' to that mentioned in Lemma 4.2. Let A * b be the adjoint of (A b , D α ) in K α with domain, cf. (3.13), For each k ∈ Dom(A * b ), the action of A * b on k is described in (3.9) with E replaced by E b , see (4.4). By (3.11) we then get K α ⊂ Dom(A * b ) for each α < α. Let Q α stand for the closure of Dom(A * b ) in · α . Then Note that Q α is a proper subset of K α . For each t ≥ 0, the adjoint S * (t) of S(t) is a bounded operator in K α . However, the semigroup {S * (t)} t≥0 is not strongly continuous. For t > 0, let S α (t) denote the restriction of S * (t) to Q α . Since {S(t)} t≥0 is the semigroup of contractions, for k ∈ Q α and all t ≥ 0, we have that cf. [28,Definition 10.3,page39]. The continuity in question follows by the C 0 -property of the semigroup {S α (t)} t≥0 and (4.10).

The Main Operators
For E b as in (4.4), we set and A 2 being as in (3.9). We also set Here B 1 and B 2 are as in (3.10). Note that The expressions in (4.15) and (4.16) can be used to define the corresponding continuous operators acting from K α to K α , α < α, cf. (3.14), and hence the elements of L(K α , K α ) the norms of which are estimated by means of the analogies of (3.11) and (3.13). For these operators, we use notations (B b ) αα and (B 2,b ) αα . Then (B b ) αα will stand for the operator norm, and thus (3.13) can be rewritten in the form For fixed α > α > − log ϑ, we construct continuous operators Q αα (t; B) : K α → K α , t > 0, which will be used to obtain the solution k t as in Theorem 3.3 and to study its properties. Here B will be taken in the following two versions: (a) B = B b ; (b) B = B 2,b , see (4.16). In both cases, for each α 1 , α 2 ∈ [α , α] such that α 1 < α 2 , cf. (4.18), the following holds For t > 0 and α 1 , α 2 as above, let α 2 α 1 (t) : K α 1 → K α 2 be the restriction of S α 2 (t) to K α 1 , cf. (4.12) and (4.13). Note that the embedding K α 1 → K α 2 . can be written as α 2 α 1 (0), and hence Also, for each α 3 > α 2 , we have Here and in the sequel, we omit writing embedding operators if no confusing arises. In view of (4.11), it follows that α 2 α 1 (t) ≤ 1. (4.23)
Proof of Lemma 4.5 Take any T < T (α 2 , α 1 ; B) and then pick α ∈ (α 1 , α 2 ] and a positive δ < α − α 1 such that For this δ, take (l) αα 1 as in (4.30), and then for set t, t 1 , . . . , t l ; B)dt l · · · dt 1 , n ∈ N. holding for all l = 1, . . . , n. This yields This proves claim (i) of the lemma. The proof of claim (ii) follows by the fact that the mentioned above convergence is uniform on [0, T ]. The estimate (4.26) readily follows from that in (4.34). Now by (4.30) and (4.32) we obtain Then the continuous differentiability of the limit and (4.27) follow by standard arguments.

The Proof of Theorem 3.3
First we prove that the problem (3.17) has a unique solution on a bounded time interval.
Proof α 1 , B b )). Then by claim (i) of Lemma 4.5 and (3.15) Hence, k t is a solution of (3.17), see (3.16). Moreover, k t (∅) = 1 since k 0 (∅) = 1, see (2.12), and d dt k t (∅) = L α k t (∅) = 0, see (3.8) -(3.10). To prove the stated uniqueness assume thatk t ∈ D α 2 is another solution of (3.17) with the same initial condition. Then for each α 3 > α 2 , v t := k t −k t is a solution of (3.17) in K α 3 with the zero initial condition. Here we assume that t and α 3 are such that t < T (α 3 , α 1 ; B b ). Clearly, v t also solves (3.17) in K α 2 . Thus, it can be written down in the following form where v t on the left-hand side (resp. v s on the right-hand side) is considered as an element of K α 3 (resp. K α 2 ) and α ∈ (α 2 , α 3 ). Indeed, one obtains (4.38) by integrating the equation, see (4.17), in which the second summand is considered as a nonhomogeneous term, see (4.32). Let us show that for all t < T (α 2 , α 1 ; B b )), v t = 0 as an element of K α 2 . In view of the embedding K α 2 → K α 3 , cf. (2.17), this will follow from the fact that v t = 0 as an element of K α 3 . For a given n ∈ N, we set = (α 3 − α 2 )/2n and α l = α 2 + l , l = 0, . . . , 2n. Then we repeatedly apply (4.38) and obtain Similarly as in (4.34) we then get from the latter, see (4.19), (4.20), and (4.23), This implies that v t = 0 for t < (α 3 − α 2 )/2β(α 3 ; B b ). To prove that v t = 0 for all t of interest one has to repeat the above procedure appropriate number of times.
To make the next step we need the following result, the proof of which will be done in Sect. 5 below. (2.18) and Lemma 4.5.

Lemma 4.9 (Identification Lemma) For each
In the light of Proposition 2.3, Lemma 4.9 claims that for t ∈ [0, τ (α 2 , α 1 )], the solution k t as in Lemma 4.8 is the correlation function of a unique sub-Poissonian state μ t whenever k 0 = k μ 0 for some μ 0 ∈ P sP . To complete the proof of Theorem 4.2 we need the following result. Recall that K α ⊂ K + α , α ∈ R, see (2.19).

Case a + > 0 and m ∈ [0, a + ]
The proof will be done by picking the corresponding bounds for u t defined in (4.41) with k 0 = k μ 0 ∈ K α 0 . Recall that, for α 1 > α 0 , u t ∈ K α 1 for t < T (α 1 , α 0 ; B 2,b ). For a given δ ≤ m, let us choose the value of C δ . The first condition is that Next, if (3.5) holds with a given ϑ > 0 and b = 0, we take any δ ≤ m and C δ ≥ 1/ϑ such that also (4.45) holds. If (3.5) holds with b > 0, we take any δ < m and then C δ ≥ b/(m − δ)ϑ such that also (4.45) holds. In all this cases, by Proposition 3.6 we have that Let r t (η) denote the right-hand side of (3.18). For α 1 > α 0 , we take α, α ∈ (α 0 , α 1 ), α < α and then consider The operator D in (4.47) is The latter inequality holds for all s ∈ [0, τ 2 ], see (4.46), and all m ∈ [0, a + ] and δ < m. Then by (4.36) we obtain from (4.41), the first line of (4.47), and (4.45) that Then by the second line of (4.47) and (4.49) we get that for t ≤ τ 2 , see (4.48), the following holds The continuation of the latter inequality to bigger values of t is straightforward. This completes the proof for this case.

The Proof of the Identification Lemma
To prove Lemma 4.9 we use Proposition 2.3. Note that the solution mentioned in Lemma 4.8 already has properties (ii) and (iii) of (2.12), cf. (2.14). Thus, it remains to prove that also (i) holds. We do this as follows. First, we approximate the evolution k 0 → k t established in Lemma 4.8 by evolutions k 0,app → k t,app such that k t,app has property (i). Then we prove that for each G ∈ B bs ( 0 ), G, k t,app → G, k t as the approximations are eliminated. The limiting transition is based on the representation G, k t,app = G t , k 0,app in which we use the so called predual evolution G → G t . Then we just show that G t , k 0,app → G t , k 0 .

The Predual Evolution
The aim of this subsection is to construct the evolution B loc ( 0 ) G 0 → G t ∈ G α 1 , see (4.1) and (4.2), such that, for each α > α 1 and k 0 ∈ K α 1 , the following holds, cf. (4.37), where b ≥ 0 and B b are as in (3.5) and (4.16), respectively. Let us define the action of B b on appropriate G : 0 → R via the duality Similarly as in (4.16) we then get For α 2 > α 1 , let (B b ) α 1 α 2 be the bounded linear operator from G α 2 to G α 1 the action of which is defined in (5.2). As in estimating the norm of B b in (4.18) one then gets For the same α 2 and α 1 , let S α 1 α 2 (t) be the restriction to G α 2 of the corresponding element of the semigroup mentioned in Lemma 4.2. Then S α 1 α 2 (t) acts as a bounded contraction from G α 2 to G α 1 . Now for a given l ∈ N and α, α 1 as in (5.1), let δ and α s , s = 0, . . . , 2l + 1, be as in (4.29). Then for t > 0 and (t, t 1 , . . . , t l ) ∈ T l , see (4.28), we define, cf. (4.30), As in Proposition 4.6, one shows that the map is continuous. Define  1 ; B b )), see (4.24) and (4.20), the sequence of operators defined in (5.4) converges in L(G α , G α 1 ) to a certain H α 1 α (t) uniformly on [0, T ], and for each G 0 ∈ G α and k 0 ∈ K α 1 the following holds Proof For the operators defined in (5.4), similarly as in (4.35) we get the following estimate which yields the convergence stated in the lemma. By direct inspection one gets that see (4.33). Then (5.5) is obtained from the latter in the limit n → +∞. Similarly as in (4.26), for the limiting operator the following estimate holds

An Auxiliary Model
The approximations mentioned at the beginning of this section employ also an auxiliary model, which we introduce and study now. For this model, we construct three kinds of evolutions. The first one is k 0 → k t ∈ K α obtained as in Lemma 4.8. Another evolution q 0 → q t ∈ G ω is constructed in such a way that q t is positive definite in the sense that G, q t ≥ 0 for all G ∈ B bs ( 0 ). These evolutions, however, take place in different spaces. To relate them to each other we construct one more evolution, u 0 → u t , which takes place in the intersection of the mentioned Banach spaces. The aim is to show that k t = u t = q t and thereby to get the desired property of k t . Thereafter, we prove the convergence mentioned above.

The Model
The function has the following evident properties The model we need is characterized by L as in (1.4) with E + (x, η), cf. (1.5), replaced by

The Evolution in K α
For the new model (with E + σ as in (5.9)), the operator L ,σ corresponding to L takes the form, cf. (5.10) Here where A 1 , B 1 , and A 1,b are the same as in (3.9), (3.10), and (4.15), respectively, and define the corresponding bounded operators acting from K α to K α for each real α > α . As in (3.15) we then set (5.13) and thus define the corresponding operator (L ,σ α , D ,σ α ). Along with (3.17) we also consider d dt By the literal repetition of the construction used in the proof of Lemma 4.5 one obtains the operators Q σ αα (t; (4.25), the norm of which satisfies, cf. (4.26), which is uniform in σ .

Lemma 5.2
Let α 1 and α 2 be as in Lemma 4.8. Then for a given k 0 ∈ K α 1 , the unique solution of (5.14) in K α 2 is given by Proof Repeat the proof of Lemma 4.8.
see (5.18). Now we can state the following analog of Lemma 4.8.
Proof Fix T < T (α 2 , α 1 ; B b ) and find α ∈ (α 1 , α 2 ) such that also T < T (α , α 1 ; B b ). Then, cf. (4.37), is the solution in question, which can be checked by means of (5.23). Its uniqueness can be proved by the literal repetition of the corresponding arguments used in the proof of Lemma 4.8.

Corollary 5.4
Let k t be the solution of the problem (5.14) with k 0 ∈ U σ,α 1 mentioned in Lemma 5.2. Then k t coincides with the solution mentioned in Lemma 5.3.

The Evolution in G ω
We recall that the space G α was introduced in (4.1), (4.2), where we used it as a predual space to K α . Now we employ G α to get the positive definiteness mentioned at the beginning of this subsection. Here, however, we write G ω to show that we use it not as a predual space.
which is an isometry on G ω . Then q t solves (5.25) if and only if w t solves the following equation By Proposition 4.1 we prove that the operator defined by the first two summands in (5.28) with domain D ω generates a substochastic semigroup, {V ω (t)} t≥0 , acting in G ω . Indeed, in this case the condition analogous to that in (4.8) takes the form, cf. (4.9), which certainly holds for each ω > 0 and an appropriate r < 1. For each ω ∈ (0, ω), we have that G ω → G ω , and the second two summands in (5.28) define a bounded operator, W ω ω : G ω → G ω , the norm of which can be estimated as follows, cf. (5.3), . (5.29) Assume now that (5.28) has two solutions corresponding to the same initial condition w 0 (η) = (−1) |η| q 0 (η). Let v t be their difference. Then it solves the following equation, cf. (4.38), where v t on the left-hand side is considered as an element of G ω and t > 0 will be chosen later. Now for a given n ∈ N, we set = (ω − ω )/n and then ω l := ω − l , l = 0, . . . , n. Thereafter, we iterate (5.30) and get Similarly as in (4.39), by (5.29) this yields the following estimate The latter implies that v t = 0 for t < (ω − ω )/ a + (e ω + 1). To prove that v t = 0 for all t of interest one has to repeat the above procedure appropriate number of times.
Recall that each U σ,α is continuously embedded into each G ω , see (5.26).

Corollary 5.6
For each ω > 0, the problem (5.25) with q 0 ∈ U σ,α 0 has a unique solution q t which coincides with the solution u t ∈ U σ,α mentioned in Lemma 5.3.
Proof By (5.27) u t is a solution of (5.25). Its uniqueness follows by Lemma 5.5.

Local Evolution
In this subsection we pass to the so called local evolution of states of the auxiliary model (5.10), (5.11). For this evolution, the corresponding 'correlation function' q t ∈ G ω has the positive definiteness in question. Then we apply Corollaries 5.4 and 5.6 to get the same for the evolution in K α . Thereafter, we pass to the limit and get the proof of Lemma 4.9.

The Evolution of Densities
In view of (2.2), each state with the property μ( 0 ) = 1 can be redefined as a probability measure on B( 0 ), cf. and We solve (5.32) in the Banach spaces G 0 = L 1 ( 0 , dλ), cf. (4.1). For n ∈ N we denote by G 0,n the subset of G 0 consisting of all those R : 0 → R for which R d |η| n |R(η)| λ(dη) < ∞.
Let also G + ω stand for the cone of positive elements of G ω . Set D 0 = {R ∈ G 0 : σ R ∈ G 0 }.

The Evolution of Local Correlation Functions
For a given μ ∈ P sP , the correlation function k μ and the local densities R μ , ∈ B b (R d ), see (2.8), are related to each other by (2.9). In the first formula of (5.36) we extend R 0 to the whole 0 . Then the corresponding integral as in (2.9) coincides with k μ 0 only on . The truncation made in the second formula in (5.36) diminishes R 0 . Its aim is to satisfy (5.37). Thus, with a certain abuse of the terminology we call that can be obtained by (2.13). Since R ,N 0 ∈ G β for any β > 0, see (5.37), we can take β = β + 1 which maximizes T (β , β) given in (5.35). Then for each β > 0, we have that for t < τ(β) := e −β e a + . (5.44) Hence, q ,N t ∈ G ω whenever R ,N t ∈ G β with β such that e β = 1 + e ω , cf. (5.43). Moreover, for such ω and β the right-hand side of (5.41) defines a continuous map from G β to G ω .

Taking the Limits
Note that (5.49) holds for with α ∈ (α 1 , α 2 ) dependent on t, see (5.16). In this subsection, we first pass in (5.49) to the limit σ ↓ 0, then we get rid of the locality imposed in (5.36).