Statistical dynamics of continuous systems: perturbative and approximative approaches

We discuss general concept of Markov statistical dynamics in the continuum. For a class of spatial birth-and-death models, we develop a perturbative technique for the construction of statistical dynamics. Particular examples of such systems are considered. For the case of Glauber type dynamics in the continuum we describe a Markov chain approximation approach that gives more detailed information about statistical evolution in this model.

(R d ). With this operator we can relate two evolution equations, namely Kolmogorov backward equation for observables and Kolmogorov forward equation on probability measures on the phase space (R d ) (macroscopic states of the system). The latter equation is also known as Fokker-Planck equation in the mathematical physics terminology. Compared with the usual situation in stochastic analysis, there is an essential technical difficulty: the corresponding Markov process in the configuration space may be constructed only in very special cases. As a result, a description of Markov dynamics in terms of random trajectories is absent for most models under considerations.
As an alternative approach we use a concept of statistical dynamics that substitutes the notion of a Markov stochastic process. A central object now is an evolution of states of the system that will be defined by mean of the Fokker-Planck equation. This evolution equation w.r.t. probability measures on (R d ) may be reformulated as a hierarchical chain of equations for correlation functions of considered measures. Such kind of evolution equations are well known in the study of Hamiltonian dynamics for classical gases as BBGKY chains but now they appear as a tool for construction and analysis of Markov dynamics. As an essential technical step, we consider related pre-dual evolution chains of equations on the so-called quasi-observables. As it will be shown in the paper, such hierarchical equations may be analyzed in the framework of semigroup theory with the use of powerful techniques of perturbation theory for the semigroup generators, etc. Considering the dual evolution for the constructed semigroup on quasi-observables we then introduce the dynamics on correlation functions. The described scheme of the dynamics construction looks quite surprising because any perturbation techniques for initial Kolmogorov evolution equations one cannot expect. The point is that states of infinite interacting particle systems are given by measures which are, in general, orthogonal to each other. As a result, we cannot compare their evolutions or apply a perturbative approach. But under quite general assumptions they have correlation functions and corresponding dynamics may be considered in a common Banach space of correlation functions. A proper choice of this Banach space means, in fact, that we find an admissible class of initial states for which the statistical dynamics may be constructed. There we see again a crucial difference with the framework of Markov stochastic processes where the initial distribution evolution is defined for any initial data.
The structure of the paper is as follows. In Sect. 2 we discuss general concept of statistical dynamics for Markov evolutions in the continuum and introduce necessary mathematical structures. Then, in Sect. 3, this concept is applied to an important class of Markov dynamics of continuous systems, namely to birth-and-death models. Here general conditions for the existence of a semigroup evolution in a space of quasi-observables are obtained. Then we construct evolutions of correlation functions as dual objects. It is shown how to apply general results to the study of particular models of statistical dynamics coming from mathematical physics and ecology.
Finally, in Sect. 4 we describe an alternative technique for the construction of solutions to hierarchical chains evolution equations by means of an approximative approach. For concreteness, this approach is discussed in the case of the so-called Glauber-type dynamics in the continuum. We construct a family of Markov chains on configuration space in finite volumes with concrete transition kernels adopted to the Glauber dynamics. Then the solution to the hierarchical equation for correlation functions may be obtained as the limit of the corresponding object for the Markov chain dynamics. This limiting evolution generates the state dynamics. Moreover, in the uniqueness regime for the corresponding equilibrium measure of Glauber dynamics which is, in fact, Gibbs, dynamics of correlation functions is exponentially ergodic. This paper is based on a series of our previous works [26,[28][29][30]34,53], but certain results and constructions are detailed and generalized, in particular, in more complete analysis of the dual dynamics on correlation functions.

Complex systems in the continuum
In recent decades, different branches of natural and life sciences have been addressing to a unifying point of view on a number of phenomena occurring in systems composed of interacting subunits. This leads to formation of an interdisciplinary science which is referred to as the theory of complex systems. It provides reciprocation of concepts and tools involving wide spectrum of applications as well as various mathematical theories such that statistical mechanics, probability, nonlinear dynamics, chaos theory, numerical simulation, and many others.
Nowadays complex systems theory is a quickly growing interdisciplinary area with a very broad spectrum of motivations and applications. For instance, having in mind biological applications, Levin [61] characterized complex adaptive systems by such properties as diversity and individuality of components, localized interactions among components, and the outcomes of interactions used for replication or enhancement of components. We will use a more general informal description of a complex system as a specific collection of interacting elements which have so-called collective behavior. This means appearance of properties of the system which are not peculiar to inner nature of each element itself. The significant physical example of such properties is thermodynamical effects which were a basis for creation by Boltzmann of statistical physics as a mathematical language for studying complex systems of molecules.
We assume that all elements of a complex system are identical by properties and possibilities. Thus, one can model these elements as points in a proper space whereas the complex system will be modeled as a discrete set in this space. Mathematically this means that for study of complex systems a proper language and techniques are delivered by the interacting particle models which form a rich and powerful direction in modern stochastic and infinite dimensional analysis. Interacting particle systems have a wide use as models in condensed matter physics, chemical kinetics, population biology, ecology (individual based models), sociology, and economics (agent based models). For instance, a population in biology or ecology may be represented by a configuration of organisms located in a proper habitat.
In spite of completely different orders of numbers of elements in real physical, biological, social, and other systems (typical numbers start from 10 23 for molecules and, say, 10 5 for plants) their complexities have analogous phenomena and need similar mathematical methods. One of them consists in mathematical approximation of a huge but finite real-world system by an infinite system realized in an infinite space. This approach was successfully applied to the thermodynamic limit for models of statistical physics and appeared quite useful for the ecological modeling in the infinite habitat to avoid boundary effects in a population evolution.
Therefore, our phase space for the mathematical description should consist of countable sets from an underlying space. This space itself may have discrete or continuous nature that leads to segregation of the world of complex systems on two big classes. Discrete models correspond to systems whose elements can occupy some prescribing countable set of positions, for example, vertices of the lattice Z d or, more generally, of some graph embedded to R d . These models are widely studied and the corresponding theories were realized in numerous publications, see, e.g. [62,63] and the references therein. Continuous models, or models in the continuum, were studied not so intensively and broadly. We concentrate our attention exactly on continuous models of systems whose elements may occupy any points in Euclidean space R d . (Note that most part of our results may be easily transferred to much more general underlying spaces). Having in mind that real elements have physical sizes we will consider only the so-called locally finite subsets of the underlying space R d that means that in any bounded region we assume to have finite number of elements. Another restriction will be prohibition of multiple elements at the same position of the space.
We will consider systems of elements of the same type only. The mathematical realization of considered approaches may be successfully extended to multi-type systems; meanwhile such systems will have richer qualitative properties and will be an object of interest for applications. Some particular results can be found, e.g. in [21,22,39].

Mathematical description for a complex systems
We proceed to the mathematical realization of complex systems.
The configuration space over space R d consists of all locally finite subsets (configurations) of R d , namely Here | · | means the cardinality of a set, and γ := γ ∩ . We may identify each γ ∈ with the non-negative where δ x is the Dirac measure with unit mass at x, x∈∅ δ x is, by definition, the zero measure, and M(R d ) denotes the space of all non-negative Radon measures on B(R d ).
This identification allows to endow with the topology induced by the vague topology on M(R d ), i.e. the weakest topology on with respect to which all mappings are continuous for any f ∈ C 0 (R d ) that is the set of all continuous functions on R d with compact supports. It is worth noting the vague topology may be metrizable in such a way that becomes a Polish space (see, e.g. [50] and references therein).
Corresponding to the vague topology the Borel σ -algebra B( ) appears the smallest σ -algebra for which all mappings are measurable for any ∈ B b (R d ), see, e.g. [1]. This σ -algebra may be generated by the sets Among all measurable functions F : →R := R ∪ {∞} we mark out the set F 0 ( ) consisting of such of them for which |F(γ )| < ∞ at least for all |γ | < ∞. The important subset of F 0 ( ) formed by cylindric functions on . Any such a function is characterized by a set Functions on are usually called observables. This notion is borrowed from statistical physics and means that typically in course of empirical investigation we may estimate, check, and see only some quantities of a whole system rather then look on the system itself. otherwise.
Then one can consider the so-called energy function Clearly, E φ ∈ F 0 ( ). However, even for φ with a compact support, E φ will not be a cylindric function.
As we discussed before, any configuration γ represents some system of elements in a real-world application. Typically, investigators are not able to take into account exact positions of all elements due to huge number of them. For quantitative and qualitative analysis of a system researchers mostly need some its statistical characteristics such as density, correlations, spatial structures, and so on. This leads to the so-called statistical description of complex systems when people study distributions of countable sets in an underlying space instead of sets themselves. Moreover, the main idea in Boltzmann's approach to thermodynamics based on giving up the description in terms of evolution for groups of molecules and using statistical interpretation of molecules motion laws. Therefore, the crucial role for studying of complex systems plays distributions (probability measures) on the space of configurations. In statistical physics these measures usually called states that accentuates their role for description of considered systems.
We denote the class of all probability measures on , B( ) by M 1 ( ). Given a distribution μ ∈ M 1 ( ) one can consider a collection of random variables N (·), ∈ B b (R d ) defined in (2.2). They describe random numbers of elements inside bounded regions. The natural assumption is that these random variables should have finite moments. Thus, we consider the class M 1 fm ( ) of all measures from M 1 ( ) such that Then the Poisson measure π σ with intensity measure σ is defined on B( ) by This formula is nothing but the statement that the random variables N have Poissonian distribution with mean value σ ( ), Note that by the Rényi theorem [47,74] a measure π σ will be Poissonian if (2.6) holds for n = 0 only. In the case then dσ (x) = ρ(x) dx one can say about nonhomogeneous Poisson measure π ρ with density (or intensity) ρ. This notion goes back to the famous Campbell formula [15,16] which states that if only the right-hand side of (2.7) is well defined. The generalization of (2.7) is the Mecke identity which holds for all measurable nonnegative functions h : R d × → R. Here and in the sequel we will omit brackets for the one-point set {x}. In [65], it was shown that the Mecke identity is a characterization identity for the Poisson measure. In the case ρ(x) = z > 0, x ∈ R d one can say about the homogeneous Poisson distribution (measure) π z with constant intensity z. We will omit sub-index for the case z = 1, namely π := π 1 = π dx . Note that the property (2.5) is followed from (2.8) easily.

Example 2.4
Let φ be as in Example 2.2 and suppose that the energy given by (2.4) is stable: there exists B ≥ 0 such that, for any |γ | < ∞, E φ (γ ) ≥ −B|γ |. An example of such φ my be given by the expansion where φ + ≥ 0, whereas φ p is a positive defined function on R d (the Fourier transform of a measure on R d ), see, e.g. [40,75]. Fix any z > 0 and define the Gibbs measure μ ∈ M 1 ( ) with potential φ and activity parameter z as a measure which satisfies the following generalization of the Mecke identity: The identity (2.10) is called the Georgii-Nguyen-Zessin identity, see [45,67]. If potential φ is additionally satisfied the so-called integrability condition then it can checked that the condition (2.5) for the Gibbs measure holds. Note that under conditions zβ ≤ (2e) −1 there exists a unique measure on , B( ) which satisfies (2.10). Heuristically, the measure μ may be given by the formula where Z is a normalizing factor. To give rigorous meaning for (2.13) it is possible to use the so-called DLRapproach (named after R. L. Dobrushin, O. Lanford, D. Ruelle), see, e.g. [2] and references therein. As was shown in [67], this approach gives the equivalent definition of the Gibbs measures which satisfies (2.10).
Note that (2.13) could have a rigorous sense if we restrict our attention on the space of configuration which belong to a bounded domain ∈ B b (R d ). The space of such (finite) configurations will be denoted by ( ). The σ -algebra B( ( )) may be generated by family of mappings ( ) fm ( ) is called locally absolutely continuous with respect to the Poisson measure π if for any ∈ B b (R d ) the projection of μ onto ( ) is absolutely continuous with respect to (w.r.t.) the projection of π onto ( ). More precisely, if we consider the projection mapping p : → ( ), p (γ ) := γ then μ := μ • p −1 is absolutely continuous w.r.t. π := π • p −1 .
Remark 2.5 Having in mind (2.13), it is possible to derive from (2.10) that the Gibbs measure from Example 2.4 is locally absolutely continuous w.r.t. the Poisson measure, see, e.g. [24] for the more general case.
By, e.g. [48], for any μ ∈ M 1 fm ( ) which is locally absolutely continuous w.r.t the Poisson measure there exists the family of (symmetric) correlation functions k (n) μ : (R d ) n → R + := [0, ∞) which is defined as follows. For any symmetric function f (n) : (R d ) n → R with a finite support the following equality holds for n ∈ N, and k ρ(x i ), (2.15) in particular, Remark 2.7 Note that if potential φ from Example 2.4 satisfies to (2.9), (2.12), then, by [76], there exists C = C(z, φ) > 0 such that for μ defined by (2.10) The inequality (2.17) is referred to as the Ruelle bound.
We dealt with symmetric function of n variables from R d ; hence, they can be considered as functions on n-point subsets from R d . We proceed now to the exact constructions. The We put (0) (Y ) := {∅}. As a set, (n) (Y ) may be identified with the symmetrization of Hence, one can introduce the corresponding Borel σ -algebra, which we denote by This space is equipped with the topology of the disjoint union. Let B 0 (Y ) denote the corresponding Borel σ -algebra. In the case of Y = R d we will omit the index Y in the previously defined notations, namely The restriction of the Lebesgue product measure (dx) n to (n) , B( (n) ) we denote by m (n) . We set m (0) := δ {∅} . The Lebesgue-Poisson measure λ on 0 is defined by (2.20) For any ∈ B b (R d ) the restriction of λ to 0 ( ) = ( ) will be also denoted by λ. Functions on 0 will be called quasi-observables. Any B( 0 )-measurable function G on 0 , in fact, is defined by a sequence of functions G (n) n∈N 0 where G (n) is a B( (n) )-measurable function on (n) . We preserve the same notation for the function G (n) considered as a symmetric function on (R d ) n . Note that The set of bounded measurable functions on 0 with bounded support we denote by B bs ( 0 ), i.e. G ∈ B bs ( 0 ) iff G 0 \M = 0 for some bounded M ∈ B( 0 ). For any G ∈ B bs ( 0 ) the functions G (n) have finite supports in (R d ) n and may be substituted into (2.14). But, additionally, the sequence of G (n) vanishes for big n. Therefore, one can summarize equalities (2.14) by n ∈ N 0 . This leads to the following definition. Let G ∈ B bs ( 0 ); then we define the function K G : → R such that see, e.g. [48,59,60]. The summation in (2.21) is taken over all finite subconfigurations η ∈ 0 of the (infinite) configuration γ ∈ ; we denote this by the symbol, η γ . The mapping K is linear, positivity preserving, and invertible, with By [48], for any G ∈ B bs ( 0 ), The expression (2.21) can be extended to the class of all nonnegative measurable G : 0 → R + , in this case, evidently, K G ∈ F 0 ( ). Stress that the left-hand side (l.h.s.) of (2.22) has a meaning for any F ∈ F 0 ( ), moreover, in this case (K K −1 F)(γ ) = F(γ ) for any γ ∈ 0 .
For G as above we may summarize (2.14) by n and rewrite the result in a compact form: (2. 24) As was shown in [48], the equality (2.21) may be extended on all functions G such that the l.h.s. of (2.24) is finite. In this case (2.21) holds for μ-a.a. γ ∈ and (2.24) holds too.

Remark 2.9
The equality (2.24) may be considered as definition of the correlation function k μ . In fact, the definition of correlation functions in statistical physics, given by Bogolyubov in [7], based on a similar relation. More precisely, consider for a B(R d )-measurable function f the so-called coherent state, given as a function on 0 by Then for any f ∈ C 0 (R d ) we have the point-wise equality As a result, the correlation functions of different orders may be considered as kernels of a Taylor-type expansion for some A, C > 0, δ ∈ (0, 1] independent on n, then the l.h.s. of (2.27) may be estimated by the expression

Suppose that k is a positive definite function that means that for any G
for all γ ∈ the following inequality holds: Suppose also that k(∅) = 1. Then there exists at least one measure μ ∈ M 1 fm ( ) such that k = k μ . 2. For any n ∈ N, ∈ B b (R d ), we set Suppose that for all m ∈ N, Then there exists at most one measure μ ∈ M 1 fm ( ) such that k = k μ .
Remark 2.13 1. In [58,60], the wider space of multiple configurations was considered. The adaptation for the space was realized in [57]. 2. It is worth noting also that the growth of correlation functions k (n) up to (n!) 2 is admissible to have (2.31). 3. Other conditions for existence and uniqueness for the moment problem on were studied in [4,48].

Statistical descriptions of Markov evolutions
Spatial Markov processes in R d may be described as stochastic evolutions of configurations γ ⊂ R d . In course of such evolutions points of configurations may disappear (die), move (continuously or with jumps from one position to another), or new particles may appear in a configuration (that is birth). The rates of these random events may depend on whole configuration that reflect an interaction between elements of the our system.
The construction of a spatial Markov process in the continuum is highly difficult question which is not solved in a full generality at present, see, e.g. a review [71] and more detail references about birth-and-death processes in Sect. 3. Meanwhile, for the discrete systems the corresponding processes are constructed under quite general assumptions, see, e.g. [62]. One of the main difficulties for continuous systems includes the necessity to control number of elements in a bounded region. Note that the construction of spatial processes on bounded sets from R d are typically well solved, see, e.g. [41].
The existing Markov process γ → X γ t ∈ , t > 0 provides solution to the backward Kolmogorov equation for bounded continuous functions: where L is the Markov generator of the process X t . The question about existence and properties of solutions to (2.32) in proper spaces itself is also highly nontrivial problem of infinite-dimensional analysis. The Markov generator L should satisfy the following two (informal) properties: (1) to be conservative, that is These properties might yield that the semigroup, related to (2.32) (provided it exists), will preserve constants and positive functions, correspondingly.
To consider an example of such L let us consider a general Markov evolution with appearing and disappearing of groups of points (giving up the case of continuous moving of particles). Namely, let F ∈ F cyl ( ) and set (2.33) Heuristically, it means that any finite group η of points from the existing configuration γ may disappear and simultaneously a new group ξ of points may appear somewhere in the space R d . The rate of this random event is equal to c(η, ξ, γ \η) ≥ 0. We need some minimal conditions on the rate c to guarantee that at least 3 for a particular case). The term in the sum in (2.33) with η = ∅ corresponds to a pure birth of a finite group ξ of points whereas the part of integral corresponding to ξ = ∅ (recall that λ({∅}) = 1) is related to pure death of a finite sub-configuration η ⊂ γ . The parts with |η| = |ξ | = 0 corresponds to jumps of one group of points into another positions in R d . The rest parts present splitting and merging effects. In the present paper the technical realization of the ideas below is given for one-point birth-and-death parts only, i.e. for the cases |η| = 0, |ξ | = 1 and |η| = 1, |ξ | = 0, correspondingly. As we noted before, for most cases appearing in applications, the existence problem for a corresponding Markov process with a generator L is still open. On the other hand, the evolution of a state in the course of a stochastic dynamics is an important question in its own right. A mathematical formulation of this question may be realized through the forward Kolmogorov equation for probability measures (states) on the configuration space , namely we consider the pairing between functions and measures on given by Then we consider the initial value problem where F is an arbitrary function from a proper set, e.g.
In fact, the solution to (2.36) describes the time evolution of distributions instead of the evolution of initial points in the Markov process. We rewrite (2.36) in the following heuristic form: where L * is the (informally) adjoint operator of L with respect to the pairing (2.35).
In the physical literature, (2.37) is referred to the Fokker-Planck equation. The Markovian property of L yields that (2.37) might have a solution in the class of probability measures. However, the mere existence of the corresponding Markov process will not give us much information about properties of the solution to (2.37), in particular, about its moments or correlation functions. To do this, we suppose now that a solution μ t ∈ M 1 fm ( ) to (2.36) exists and remains locally absolutely continuous with respect to the Poisson measure π for all t > 0 provided μ 0 has such a property. Then one can consider the correlation function k t := k μ t , t ≥ 0.
Recall that we suppose (2.34). Then, one can calculate K −1 L F using (2.22), and, by (2.24), we may rewrite (2.36) in the following way: Here the pairing between functions on 0 is given by Let us recall that then, by (2.20), for all G ∈ B bs ( 0 ). Here the operator (2.34). As a result, we are interested in a weak solution to the equation where L * is dual operator to L with respect to the duality (2.39), namely The procedure of deriving the operator L for a given L is fully combinatorial; meanwhile, to obtain the expression for the operator L * we need an analog of integration by parts formula. For a difference operator L considered in (2.33) this discrete integration by parts rule is presented in Lemma 3.4 below.
We recall that any function on 0 may be identified with an infinite vector of symmetric functions of the growing number of variables. In this approach, the operator L * in (2.41) will be realized as an infinite matrix L * n,m n,m∈N 0 , where L * n,m is a mapping from the space of symmetric functions of n variables into the space of symmetric functions of m variables. As a result, instead of Eq. (2.36) for infinite-dimensional objects we obtain an infinite system of equations for functions k (n) t ; each of them is a function of a finite number of variables, namely Of course, in general, for a fixed n, any equation from (2.43) itself is not closed and includes functions k (m) t of other orders m = n; nevertheless, the system (2.43) is a closed linear system. The chain evolution equations for k (n) t consists the so-called hierarchy which is an analog of the BBGKY hierarchy for Hamiltonian systems, see, e.g. [18].
One of the main aims of the present paper was to study the classical solution to (2.41) in a proper functional space. The choice of such a space might be based on estimates (2.17), or more generally, (2.29). However, even the correlation functions (2.16) of the Poisson measures show that it is rather natural to study the solutions to the Eq. (2.41) in weighted L ∞ -type space of functions with the Ruelle-type bounds. Integrable correlation functions are not natural for the dynamics on the spaces of locally finite configurations. For example, it is well known that the Poisson measure π ρ with integrable density ρ(x) is concentrated on the space 0 of finite configurations [since in this case on can consider R d instead of in (2.6)]. Therefore, typically, the case of integrable correlation functions yields that effectively our stochastic dynamics evolves through finite configurations only. Note that the case of an integrable first-order correlation function is referred to zero density case in statistical physics.
In the present paper we restrict our attention to the so-called sub-Poissonian correlation functions. Namely, for a given C > 0 we consider the following Banach space: with the norm In the following, we distinguish two possibilities for a study of the initial value problem (2.41). We may try to solve this equation in one space K C . The well-posedness of the initial value problem in this case is equivalent with an existence of the strongly continuous semigroup (C 0 -semigroup in the sequel) in the space K C with a generator L * . However, the space K C is isometrically isomorphic to the space L ∞ ( 0 , C |·| dλ), whereas by the Lotz theorem [3,64], in a L ∞ space any C 0 -semigroup is uniformly continuous, that is it has a bounded generator. Typically, for the difference operator L given in (2.33), any operator L * n,m , cf. (2.43), might be bounded as an operator between two spaces of bounded symmetric functions of n and m variables, whereas the whole operator L * is unbounded in K C .
To avoid these difficulties we use a trick which goes back to Phillips [72]. The main idea is to consider the semigroup in L ∞ space not itself but as a dual semigroup T * (t) to a C 0 -semigroup T (t) with a generator A in the pre-dual L 1 space. In this case T * (t) appears strongly continuous semigroup not on the whole L ∞ but on the closure of the domain of A * only.
In our case this leads to the following scheme. We consider the pre-dual Banach space to K C , namely for C > 0, The norm in L C is given by Consider the initial value problem, cf. (2.40), (2.41), Then using Philips' result we obtain that the restriction of the dual semigroup T * (t) onto Dom( L * ) will be C 0 -semigroup with generator which is a part of L * (the details see in Sect. 3 below). This provides a solution to (2.41) which continuously depends on an initial data from Dom( L * ). And after we would like to find a more useful universal subspace of K C which is not dependent on the operator L * . The realization of this scheme for a birth-and-death operator L is presented in Sect. 3 below. As a result, we obtain the classical solution to (2.41) for t > 0 in a class of sub-Poissonian functions which satisfy the Ruelle-type bound (2.45). Of course, after this we need to verify existence and uniqueness of measures whose correlation functions are solutions to (2.41), cf. Proposition 2.12 above. This usually can be done using proper approximation schemes, see, e.g. Sect. 4. There is another possibility for a study of the initial value problem (2.41) which we will not touch below, namely one can consider this evolutional equation in a proper scale of spaces {K C } C * ≤C≤C * . In this case we will have typically that the solution is local in time only. More precisely, there exists T > 0 such that for any t ∈ [0, T ) there exists a unique solution to (2.41) and k t ∈ K C t for some C t ∈ [C * , C * ]. We realized this approach in series of papers [5,25,37,38] using the so-called Ovsyannikov method [69,77,78]. This method provides less restrictions on systems parameters; however, the price for this is a finite time interval. And, of course, the question about possibility to recover measures via solutions to (2.41) should be also solved separately in this case.

Microscopic description
One of the most important classes of Markov evolution in the continuum is given by the birth-and-death Markov processes in the space of all configurations from R d . These are processes in which an infinite number of individuals exist at each instant, and the rates at which new individuals appear and some old ones disappear depend on the instantaneous configuration of existing individuals [46]. The corresponding Markov generators have a natural heuristic representation in terms of birth and death intensities. The birth intensity b(x, γ ) ≥ 0 characterizes the appearance of a new point at x ∈ R d in the presence of a given configuration γ ∈ . The death intensity d(x, γ ) ≥ 0 characterizes the probability of the event that the point x of the configuration γ disappears, depending on the location of the remaining points of the configuration, γ \x. Heuristically, the corresponding Markov generator is described by the following expression, cf. (2.33): The study of spatial birth-and-death processes was initiated by Preston [73]. This paper dealt with a solution of the backward Kolmogorov equation (2.32) under the restriction that only a finite number of individuals are alive at each moment of time. Under certain conditions, corresponding processes exist and are temporally ergodic, that is, there exists a unique stationary distribution. Note that a more general setting for birth-anddeath processes only requires that the number of points in any compact set remains finite at all times. A further progress in the study of these processes was achieved by Holley and Stroock in [46]. They described in detail an analytic framework for birth-and-death dynamics. In particular, they analyzed the case of a birth-and-death process in a bounded region.
Stochastic equations for spatial birth-and-death processes were formulated in [42], through a spatial version of the time-change approach. Further, in [43], these processes were represented as solutions to a system of stochastic equations, and conditions for the existence and uniqueness of solutions to these equations, as well as for the corresponding martingale problems, were given. Unfortunately, quite restrictive assumptions on the birth and death rates in [43] do not allow an application of these results to several particular models that are interesting for applications (see, e.g. some of examples below).
A growing interest to the study of spatial birth-and-death processes, which we have recently observed, is stimulated by (among others) an important role which these processes play in several applications. For example, in spatial plant ecology, a general approach to the so-called individual based models was developed in a series of works, see, e.g. [8,9,17,66] and the references therein. These models are described as birth-anddeath Markov processes in the configuration space with specific rates b and d which reflect biological notions such as competition, establishment, fecundity, etc. Other examples of birth-and-death processes may be found in mathematical physics. In particular, the Glauber-type stochastic dynamics in is properly associated with the grand canonical Gibbs measures for classical gases. This gives a possibility to study these Gibbs measures as equilibrium states for specific birth-and-death Markov evolutions [6]. Starting with a Dirichlet form for a given Gibbs measure, one can consider an equilibrium stochastic dynamics [54]. However, these dynamics give the time evolution of initial distributions from a quite narrow class, namely the class of admissible initial distributions is essentially reduced to the states which are absolutely continuous with respect to the invariant measure. Below we construct non-equilibrium stochastic dynamics which may have a much wider class of initial states.
This approach was successfully applied to the construction and analysis of state evolutions for different versions of the Glauber dynamics [28,34,53] and for some spatial ecology models [26]. Each of the considered models required its own specific version of the construction of a semigroup, which takes into account particular properties of corresponding birth and death rates.
In this section, we realize a general approach considered in Sect. 2 to the construction of the state evolution corresponding to the birth-and-death Markov generators. We present conditions on the birth-and-death intensities which are sufficient for the existence of corresponding evolutions as strongly continuous semigroups in proper Banach spaces of correlation functions satisfying the Ruelle-type bounds. Also we consider weaker assumptions on these intensities which provide the corresponding evolutions for finite time intervals in scales of Banach spaces as above.

Expressions for L and L * . Examples of rates b and d
We always suppose that rates d, b : R d × → [0; +∞] from (3.1) satisfy the following assumptions: (3.5) Proposition 3.1 Let conditions (3.2)-(3.5) hold. The for any G ∈ B bs ( 0 ) and F = K G one has L F ∈ F 0 ( ).
We start from the deriving of the expression for L = K −1 L K .

Proposition 3.2
For any G ∈ B bs ( 0 ) the following formula holds: Proof First of all, note that, by (3.3) and (2.22), the expressions In the same way, for x / ∈ γ , we derive and making substitution ξ = ξ ∪ x ⊂ ζ , one may continue Next, for any measurable H : 0 × 0 → R, one has Using this changing of variables rule, we continue: The expression (3.6) shows that the matrix above has on the main diagonal the collection of operators L n,n , n ∈ N 0 which forms the following operator on functions on 0 : where the term in the square brackets is equal, by (2.22), to K −1 b(x, · ∪ (η\y)) ({y}). Next, by (3.6), there exists only one non-zero upper diagonal in the matrix. The corresponding operator is η). The rest part of the expression (3.6) corresponds to the low diagonals.
As we mentioned above, to derive the expression for L * we need some discrete analog of the integration by parts formula. As such, we will use the partial case of the well-known lemma (see, e.g. [56]):

Lemma 3.4 For any measurable function H
if at least one side of the equality is finite for |H |.
In particular, if H (ξ, ·, ·) ≡ 0 if only |ξ | = 1 we obtain an analog of (2.8), namely for any measurable function h : R d × 0 × 0 → R such that both sides make sense. Using this, one can derive the explicit form of L * .

Proposition 3.5
For any k ∈ B bs ( 0 ) the following formula holds:
Proof Using Lemma 3.4, (2.42), (3.6), we obtain for any G ∈ B bs ( 0 ) Applying (3.12) for the second term, we easily obtain the statement. The correctness of using (2.8) and (3.12) follows from the assumptions that G, k ∈ B bs ( 0 ); therefore, all integrals over 0 will be taken, in fact, over some bounded M ∈ B( 0 ). Then, using (3.4), (3.5), we obtain that the all integrals are finite.
Remark 3.6 Accordingly to Remark 3.3 [or just directly from (3.13)], we have that the matrix corresponding to (2.43) has the main diagonal given by where we have used (3.12). Next, this matrix has only one non-zero low diagonal, given by the expression The rest part of expression (3.13) corresponds to the upper diagonals.
Let us consider now several examples of rates b and d which will appear in the following considerations (concrete examples of birth-and-death dynamics, with such rates, important for applications will be presented later). As we see from (3.6), (3.13), we always need to calculate expressions like K −1 a(x, ·∪ξ) (η), η∩ξ = ∅, where a equal to b or d. We consider the following kinds of function a : R d × → R: If we substitute f ≡ 0 into (2.26), we obtain that where as usual 0 0 := 1, and, of course, in this case K −1 a(x, · ∪ ξ )(η) also equal to m0 |η| for any ξ ∈ 0 ; -Linear rate: where c is a potential like in Example 2.2. Any such c for a given x ∈ R d defines a function C x : 0 → R such that C x (η) = 0 for all η / ∈ (1) and, for any η ∈ (1) , y ∈ R d with η = {y}, we have C x (η) = c(x − y). Then, in this case, taking into account (3.17) and the obvious equality (3.20) -Exponential rate: where c as above. Taking into account (3.19) and (2.26), we obtain that in this case -Product of linear and exponential rates: where c 1 and c 2 are potentials as before. Then we have  Then, similarly to (3.26), we easily derive Using the similar arguments one can consider polynomial rates and their compositions with exponents as well.

Semigroup evolutions in the space of quasi-observables
We proceed now to the construction of a semigroup in the space L C , C > 0, see (2.46), which has a generator, given by L, with a proper domain. To define such domain, let us set (3.28) Note that B bs ( 0 ) ⊂ D and B bs ( 0 ) is a dense set in L C . Therefore, D is also a dense set in L C . We will show now that ( L, D) given by (3.6), (3.29) generates C 0 -semigroup on L C if only 'the full energy of death', given by (3.28), is big enough.
and, moreover, This is the reason to demand that a 1 should be not less than 1.
Proof of Theorem 3.7 Let us consider the multiplication operator (L 0 , D) on L C given by We recall that a densely defined closed operators A on L C is called sectorial of angle ω ∈ (0, π 2 ) if its resolvent set ρ(A) contains the sector Sect π 2 + ω := z ∈ C | arg z| < π 2 + ω \{0}, and for each ε ∈ (0; ω) there exists M ε ≥ 1 such that Here and below we will use notation The set of all sectorial operators of angle ω ∈ (0, π 2 ) in L C we denote by H C (ω). Any A ∈ H C (ω) is a generator of a bounded semigroup T (t) which is holomorphic in the sector | arg t| < ω (see, e.g. [19,Theorem II.4.6]). One can prove the following lemma: Lemma 3. 9 The operator (L 0 , D) given by (3.33) is a generator of a contraction semigroup on L C . Moreover, L 0 ∈ H C (ω) for all ω ∈ (0, π 2 ) and (3.34) holds with M ε = 1 cos ω for all ε ∈ (0; ω). Proof of Lemma 3.9 It is not difficult to show that the densely defined operator L 0 is closed in L C . Let 0 < ω < π 2 be arbitrary and fixed. Clear, that for all z ∈ Sect π 2 + ω Therefore, for any z ∈ Sect π 2 + ω the inverse operator R(z, L 0 ) = (z1 1 − L 0 ) −1 , the action of which is given by is well defined on the whole space L C . Moreover, and for any z ∈ Sect π 2 + ω |Im z| = |z|| sin arg z| ≥ |z| sin π 2 + ω = |z| cos ω.
As a result, for any z ∈ Sect π 2 + ω ||R(z, L 0 )|| ≤ 1 |z| cos ω , (3.36) that implies the second assertion. Note also that |D(η) + z| ≥ Re z for Re z > 0; hence, that proves the first statement by the classical Hille-Yosida theorem. For any G ∈ B bs ( 0 ) we define Next Lemma shows that, under conditions (3.30), (3.31) above, the operator L 1 is relatively bounded by the operator L 0 .  L 1 , D) is a well-defined operator in L C such that and (3.40)

Evolutions in the space of correlation functions
In this Subsection we will use the semigroup T (t) acting oh the space of quasi-observables for a construction of solution to the evolution Eq. (2.41) on the space of correlation functions.
We denote dλ C := C |·| dλ; and the dual space As was mentioned before the space (L C ) is isometrically isomorphic to the Banach space K C considered in (2.44)-(2.45). The isomorphism is given by the isometry R C (3.43) Recall, one may consider the duality between the Banach spaces L C and K C given by (2.39) with Let L , Dom( L ) be an operator in (L C ) which is dual to the closed operator L, D . We consider also its image on K C under the isometry R C . Namely, let L * = R C L R C −1 with the domain Dom( L * ) = R C Dom( L ). Similarly, one can consider the adjoint semigroup T (t) in (L C ) and its image T * (t) in K C .
The space L C is not reflexive: hence, T * (t) is not C 0 -semigroup in whole K C . By, e.g. [19, Subsection II.2.5], the last semigroup will be weak*-continuous, weak*-differentiable at 0 and L * will be weak*-generator of T * (t). Therefore, one has an evolution in the space of correlation functions. In fact, we have a solution to the evolution Eq. (2.41), in a weak*-sense. This subsection is devoted to the study of a classical solution to this equation. By, e.g. [19, Subsection II.2.6], the restriction T (t) of the semigroup T * (t) onto its invariant Banach subspace Dom( L * ) (here and below all closures are in the norm of the space K C ) is a strongly continuous semigroup. Moreover, its generator L will be a part of L * , namely Dom( L ) = k ∈ Dom( L * ) L * k ∈ Dom( L * ) (3.44) and L k = L * k for any k ∈ Dom( L ).

45)
Then for any α ∈ 0; 1 Proof In order to show (3.46) it is enough to verify that for any k ∈ K αC there exists k * ∈ K C such that for any G ∈ Dom( L) LG, k = G, k * . (3.47) By the same calculations as in the proof of Proposition 3.5, it is easy to see that (3.47) is valid for any k ∈ K αC with k * = L * k, where L * is given by (3.13), provided k * ∈ K C . To prove the last inclusion, one can estimate, by (3.30), (3.31), (3.45) that Using elementary inequality we have for αν < 1 ess sup The statement is proved.

Then for any
Proof The first and last inclusions are obvious. To prove the second one, we use (3.45), (3.48) and obtain for any The statement is proved.
Then, for any α with a 2 Proof First of all note that the condition on α implies (3.50). Next, we prove that T α (t) G = T (t) G for any G ∈ L C ⊂ L αC . Let L α = ( L, D α ) be the operator in L αC . There exists ω > 0 such that (ω; +∞) ⊂ ρ( L) ∩ ρ( L α ), see, e.g. [19,Section III.2]. For some fixed z ∈ (ω; +∞) we denote by R(z, L) = z1 1 − L −1 the resolvent of ( L, D) in L C and by R(z, L α ) = z1 1 − L α −1 the resolvent of L α in L αC . Then for any Note that for any G ∈ L C ⊂ L αC and for any k where, by the same construction as before, Hence, T * (t)k = T * α (t)k ∈ K αC that proves the statement due to continuity of the family T * (t).
By, e.g. [19,Subsection II.2.3], one can consider the restriction T α (t) of the semigroup T (t) onto K αC . It will be strongly continuous semigroup with the generator L α which is a restriction of L onto K αC . Namely cf. 3.44, (3.53) and L α k = L k = L * k for any k ∈ K αC . In the other words, L α is a part of L * . And now we may proceed to the main statement of this Subsection. has a unique classical solution for any k 0 ∈ Dom( L * . In particular, it means that the solution k t = T (t)k 0 is continuously differentiable in t w.r.t. the norm of Dom( L * that is the norm · K C and also k t ∈ Dom( L ). But by Proposition 3.15, the space K αC is T (t)-invariant. Hence, if we consider now the initial value k 0 ∈ K αC ⊂ Dom( L * we obtain with a necessity that k t = T (t)k 0 = T α (t)k 0 ∈ K αC . Therefore, k t ∈ K αC Dom( L ) = Dom( L α ) (see again [19,Subsection II.2.3]) and, recall, k t is continuously differentiable in t w.r.t. the norm · K C that is the norm in K αC . This completes the proof of the first statement. The second one follows directly now from Proposition 3.15.

Examples of dynamics
We proceed now to describing the concrete birth-and-death dynamics which are important for different application. We will consider the explicit conditions on parameters of systems which imply the general conditions on rates b and d above. For simplicity of notations we denote the l.h.s. of (3.30) and (3.31) by I d (ξ ) and I b (ξ ), ξ ∈ 0 , respectively.
Then, by (3.17) we obtain that Therefore, (3.30), (3.31), (3.32) hold if only Clearly, in this case (3.45) holds with N = 0, ν = 1; therefore, the condition (3.52) is just The case of constant (in space) m and σ was considered in [23]. Similarly to that results, one can derive the explicit expression for the solution to the initial value problem (2.41) considered point-wise in 0 , namely , and s ∈ [0; 1]. Note that in the case m(x) ≡ 1, z(x) ≡ z > 0 and for any s ∈ [0; 1] the operator L is well defined and, moreover, symmetric in the space L 2 ( , μ), where μ is a Gibbs measure, given by the pair potential φ and activity parameter z (see, e.g. [55] and references therein). This gives possibility to study the corresponding semigroup in L 2 ( , μ). If, additionally, s = 0, the corresponding dynamics was also studied in another Banach spaces, see, e.g. [28,34,53]. Below we show that one of the main results of the paper stated in Theorem 3.16 can be applied to the case of arbitrary s ∈ [0; 1] and non-constant m and z. Set and, analogously, taking into account that φ ≥ 0, Therefore, to apply Theorem 3.7 we should assume that there exists σ > 0 such that  [26] and references therein). Let L be given by (3.1) with Then, by (3.17), (3.20), and (2.18)- (2.19), Let us suppose, cf. [26], that there exists δ > 0 such that Then Hence, (3.30), (3.31) hold with that fulfills (3.32). Next, under conditions (3.70), (3.72), we have |ξ |, ξ ∈ 0 , and hence (3.45) holds with ν = 1, which makes (3.51) obvious.

Remark 3.20
It was shown in [26] that, for the case of constant m, ± , the condition like (3.70) is essential. Namely, if m > 0 is arbitrary small the operator L will not be even accretive in L C .

Example 3.21 (Contact model with establishment) Let L be given by (3.1) with d(x, γ ) = m(x) for all γ ∈ and
(3.73)

Stationary equation
In this subsection we study the question about stationary solutions to (2.41). For any s ≥ 0, we consider the following subset of K C We define K to be the closure of K αC in the norm of K C . It is clear that K with the norm of K C is a Banach space.
As a result,k Next, for η = ∅ Hence, This finishes the proof.
that is quite similar of the so-called Kirkwood-Salsburg operator known in mathematical physics (see, e.g. [49,75]). For s = 0 condition (3.74) has form z C e Cβ −1 < 1. Under this condition, the stationary solution to (3.76) is unique and coincides with the correlation function of the Gibbs measure, corresponding to potential φ and activity z. Remark 3.25 It is worth pointing out that b(x, ∅) = 0 in the case of Example 3.19. Therefore, if we suppose [cf. (3.70), (3.72)] that 2 − C < m and 2 + a + (x) ≤ C − a − (x), for x ∈ R d , condition (3.74) will be satisfied. However, the unique solution to (3.76) will be given by (3.79). In the next example we improve this statement.

Example 3.26
Let us consider the following natural modification of BDLP-model coming from Example 3.19: let d be given by (3.68) and where + , a + are as before and κ > 0. Then, under assumptions we obtain for some δ > 0 The latter inequalities imply (3.74). In this case, and the Poisson measure π z with the intensity z will be symmetrizing measure for the operator L. In particular, it will be invariant measure. This fact means that its correlation function k z (η) = z |η| is a solution to (3.76). Conditions (3.83) and (3.84) in this case are equivalent to 4z < C and 2 − C < m. As a result, due to uniqueness of such solution,

Approximative approach for the Glauber dynamics
In this section we consider an approximative approach for the construction of the Glauber-type dynamics described in Example 3.18 for Therefore, in such a case, (3.1) has the form with E φ given by (2.11). Let G ∈ B bs ( 0 ) then F = K G ∈ F cyl ( ). By (3.6), (3.17), (3.22), one has the following explicit form for the mapping L : where e λ is given by (2.25). Let us denote, for any η ∈ 0 , To simplify notation we continue to write C φ for β −1 . In contrast to (3.29), we will not work the maximal domain of the operator L 0 , namely the following statement will be used Proof For any G ∈ L 2C and, by Lemma 3.4,

Description of approximation
In this section we will use the symbol K 0 to denote the restriction of K onto functions on 0 . Let δ ∈ (0; 1) be arbitrary and fixed. Consider for any ∈ B b (R d ) the following linear mapping on where Clearly, P δ is a positive preserving mapping and Operator (4.5) is constructed as a transition operator of a Markov chain, which is a time discretization of a continuous time process with the generator (4.1) and discretization parameter δ ∈ (0; 1). Roughly speaking, according to the representation (4.5), the probability of transition γ → (γ \η) ∪ ω (which describes removing of subconfiguration η ⊂ γ and birth of a new subconfiguration ω ∈ ( )) after small time δ is equal to We may rewrite (4.5) in another manner.

Proposition 4.2
For any F ∈ F cyl ( 0 ) the following equality holds: To rewrite (4.5), we have used also that any η ⊂ γ corresponds to a unique γ \η ⊂ γ . Applying the definition where after changing summation over η ⊂ γ and ζ ⊂ η we have used the fact that for any configuration η ⊂ γ which contains fixed ζ ⊂ γ there exists a unique η ⊂ γ \ζ such that η = η ∪ ζ . But by the binomial formula Combining (4.8), (4.9), (4.10), we get Next, Lemma 3.4 yields which proves the statement.
In the next proposition we describe the image of P δ under the K 0 -transform.

Proposition 4.3
Let P δ = K −1 0 P δ K 0 . Then for any G ∈ B bs ( 0 ) the following equality holds: Proof By (4.7) and the definition of K −1 0 , we have By the definition of the relative energy The well-known equality (see, e.g. [36]) completes the proof.

Construction of the semigroup on L C
By analogy with (4.11), we consider the following linear mapping on measurable functions on 0 : Then P δ , given by (4.12), is a well-defined linear operator in L C , such that P δ ≤ 1. (4.14) Proof Since φ ≥ 0 we have It is easy to see by the induction principle that for φ ≥ 0, ω ∈ 0 , y / ∈ ω Then For the last inequality we have used that (4.13) implies 16) and the statement is proved.

Proposition 4.5 Let the inequality (4.13) be fulfilled and define
where 1 1 is the identity operator in L C . Then for any G ∈ L 2C Proof Let us denote and Now we estimate each of the terms in (4.22) separately. By (4.3) and (4.18), we have But, for any |η| ≥ 2 Next, by (4.4) and (4.20), one can write where we have used Lemma 3.4. Note that for any |ξ | ≥ 1 Then, by (4.13) and (2.12), one may estimate Since n (n − 1) ≤ 2 n , n ≥ 1 and by (4.13), the latter expression can be bounded by Finally, Lemma 3.4, (4.15) and bound e −E φ (y,ω) ≤ 1, imply (we set here Combining inequalities (4.23)-(4.25) we obtain the assertion of the proposition.
We will need the following results in the sequel:  2. For each f ∈ D, there exists f n ∈ L n for each n ∈ N such that f n → p n f and A n f n → p n A f in L n .
And now we are able to show the existence of the semigroup on L C . Moreover, since we proved the existence of the semigroup T t on L C one can apply contractions P δ defined above by (4.12) to approximate the semigroup T t .

Finite-volume approximation of T t
Note that P δ defined by (4.12) is a formal point-wise limit of P δ as ↑ R d . We have shown in (4.16) that this definition is correct. Corollary 4.9 claims additionally that the linear contractions P δ approximate the semigroup T t , when δ ↓ 0. One may also show that mappings P δ have a similar property when Let us fix a system { n } n≥2 , where n ∈ B b (R d ), n ⊂ n+1 , n n = R d . We set Note that any T n is a linear mapping on B bs ( 0 ). We consider also the system of Banach spaces of measurable functions on 0 Let p n : L C → L C,n be a cut-off mapping, namely for any Then, obviously, p n G C,n ≤ G C . Hence, p n : L C → L C,n is a linear bounded transformation with p n = 1. contraction on L C,n , n ≥ 2 (note that for any n ≥ 2, (2.12) implies n 1 − e −φ(x) dx ≤ C φ < ∞). Next, we set A n = n(T n − 1 1 n ) where 1 1 n is a unit operator on L C,n and let us expand T n in three parts analogously to the proof of Proposition 4.5: For any G ∈ L 2C we set G n = p n G ∈ L 2C,n ⊂ L C,n . To finish the proof we have to verify that for any G ∈ L 2C A n G n − p n LG C,n → 0, n → ∞. (4.29) For any G ∈ L 2C A n G n − p n LG C,n ≤ n(T (0) n − 1 1 n )G n − p n L 0 G C,n + nT (1) n G n − p n L 1 G C,n + nT (≥2) n G n C,n . (4.30) Note that p n L 0 G = L 0 G n . Using the same arguments as in the proof of Proposition 4.5 we obtain n(T (0) n − 1 1 n )G n − p n L 0 G C,n + nT (≥2) n G n C,n ≤ 2 n G 2C,n ≤ 2 n G 2C . Next, where we have used (2.12) and (4.13). Using the same estimates as for (4.24) we may continue But by the Lebesgue dominated convergence theorem, Indeed, 1 1 c n (x)|G (ξ ∪ x) | → 0 point-wisely and may be estimated on 0 × R d by |G (ξ ∪ x) | which is integrable: Therefore, by (4.30), the convergence (4.29) holds for any G ∈ L 2C , which completes the proof.

Evolution of correlation functions
Under condition (4.28), we proceed now to the same arguments as in Subsect. 3.4. Namely, one can construct the restriction T (t) of the semigroup of T * (t) onto the Banach space D( L * ) (recall that the closure is in the norm of K C ). Note that the domain of the dual operator to ( L, L 2C ) might be bigger than the domain considered in Subsect. 3.4. Nevertheless, T (t) will be a C 0 -semigroup on D( L * ) and its generator L will be a part of L * , namely (3.44) holds and L * k = L k for any k ∈ D( L ).
The next statement is a straightforward consequence of Proposition 3.12.

Proposition 4.11
For any α ∈ (0; 1) the following inclusions hold: Then, by Proposition 3.5, we immediately obtain that, for k ∈ K αC , The next statement is an analog of Proposition 3.15.
By assumption (4.28), This inequality remains also true if CC φ = ln 2 because of (4.32). Under condition (4.33), the equation f (x) = zC φ has exactly two roots, say, 0 < x 1 < 1 < x 2 < +∞. Then, (4.32) implies If CC φ > 1 then we set α 0 := max 1 2 ; 1 CC φ ; 1 C < 1. This yields 2αCC φ > CC φ and αCC φ > 1 > x 1 . If x 1 < CC φ ≤ 1 then we set α 0 := max 1 2 ; x 1 CC φ ; 1 C < 1 that gives 2αCC φ > CC φ and αCC φ > x 1 . As a result, and 1 < αC < C < 2αC < 2C. The last inequality shows that L 2C ⊂ L 2αC ⊂ L C ⊂ L αC . Moreover, by (4.34), we may prove that the operator ( L, L 2αC ) is closable in L αC and its closure is a generator of a contraction semigroup T α (t) on L αC . The proof is identical to the proofs above. It is easy to see that T α (t)G = T (t)G for any G ∈ L C . Indeed, from the construction of the semigroup T (t) and analogous construction for the semigroup T α (t), we have that there exists family of mappings P δ , δ > 0 independent of α and C, given by (4.12), such that P t δ δ for any t ≥ 0 strongly converges to T (t) and T α (t) in L C and L αC , correspondingly, as δ → 0. Here and below [ · ] means the entire part of a number. Then for any G ∈ L C ⊂ L αC we have that T (t)G ∈ L C ⊂ L αC and T α (t)G ∈ L αC and Note that for any G ∈ L C ⊂ L αC and for any k ∈ K αC ⊂ K C we have T α (t)G ∈ L αC and where, by construction, T * Hence, T * (t)k = T * α (t)k ∈ K αC , k ∈ K αC that proves the statement.

Remark 4.13
As a result, (4.28) implies that for any k 0 ∈ D( L * ) the Cauchy problem in has a unique mild solution: k t = T * (t)k 0 = T (t)k 0 ∈ D( L * ). Moreover, k 0 ∈ K αC implies k t ∈ K αC provided (4.32) is satisfied.
Remark 4.14 The Cauchy problem (4.35) is well-posed inK C = D( L * ), i.e. for every k 0 ∈ D( L ) there exists a unique solution k t ∈K C of (4.35).
Let (4.28) and (4.32) be satisfied and let α 0 be chosen as in the proof of Proposition 4.12 and fixed. Suppose that α ∈ (α 0 ; 1). Then, Propositions 4.11 and 4.12 imply K αC ⊂ D( L * ) and the Banach subspace K αC is T * (t)and, therefore, T (t)-invariant due to the continuity of these operators.
As a result, we obtain an approximation for the semigroup.

Theorem 4.16
Let α 0 be chosen as in the proof of the Proposition 4.12 and be fixed. Let α ∈ (α 0 ; 1) and k ∈ K αC be given. Then in the space K αC with norm · K C for all t ≥ 0 uniformly on bounded intervals.
Proof We may apply Proposition 4.15 to use Lemma 4.7 in the case L n = L = L αC , p n = 1 1, f n = f = k, ε n = δ → 0, n ∈ N.

Positive definiteness
We consider a small modification of the notion of positive definite functions considered in Proposition 2.12. Namely, we denote by L 0 ls ( 0 ) the set of all measurable functions on 0 which have a local support, i.e. G ∈ L 0 ls ( 0 ) if there exists ∈ B b (R d ) such that G 0 \ ( ) = 0. We will say that a measurable function k : 0 → R is a positive defined function if, for any G ∈ L 0 ls ( 0 ) such that K G ≥ 0 and G ∈ L C for some C > 1 the inequality (2.30) holds.
For a given C > 1, we set L ls C = L 0 ls ( 0 ) ∩ L C . Since B bs ( 0 ) ⊂ L ls C , for any C > 1, Proposition 2.12 (see also the second part of Remark 2.13) implies that if k is a positive definite function as above then there exists a unique measure μ ∈ M 1 fm ( ) such that k = k μ be its correlation function in the sense of (2.24). Our aim is to show that the evolution k → T (t)k preserves this property of the positive definiteness. Theorem 4.17 Let (4.28) hold and k ∈ D( L * ) ⊂ K C be a positive definite function. Then k t := T (t)k ∈ D( L * ) ⊂ K C will be a positive definite function for any t ≥ 0.
Let us now consider anyG ∈ L ls C [stress thatG is not necessary equal to 0 outside of ( n )] and suppose that KG (γ ) ≥ 0 for any γ ∈ ( n ). Then By (4.48), settingG = G n ∈ L ls C we obtain, because of (4.49), K T n G n ≥ 0. Next, settingG = T n G n ∈ L ls C we obtain, by (4.49), K T 2 n G n ≥ 0. Then, using an induction mechanism, we obtain that K T m n G n γ n ≥ 0, m ∈ N 0 , that, by (4.46) and (4.47), yields (4.45). This completes the proof.
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