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Operator approach to Vlasov scaling for some models of spatial ecology

We consider Vlasov-type scaling for Markov evolution of birth-and-death type in continuum, which is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations. The existence of rescaled and limiting evolutions of correlation functions as well as convergence to the limiting evolution are shown. The obtained results enable to derive a non-linear Vlasov-type equation for the density of the limiting system.


Introduction
The Vlasov equation is a famous example of a kinetic equation which describes the dynamical behavior of a many-body system. In physics, it characterizes the Hamiltonian motion of an infinite particle system influenced by weak long-range forces in the mean field scaling limit. The detailed exposition of the Vlasov scaling for the Hamiltonian dynamics was given by W.Braun and K.Hepp [3] and later by R.L.Dobrushin [5] for more general deterministic dynamical systems. The limiting Vlasov-type equations for particle densities in both papers are considered in classes of integrable functions (or finite measures in the weak form). This corresponds, actually, to the situation of finite volume systems or systems with zero mean density in an infinite volume. The Vlasov equation for the integrable functions was investigated in details by V.V.Kozlov [16]. An excellent review about kinetic equations which describe dynamical multi-body systems was given by H.Spohn [22], [23]. Note that in the framework of interacting diffusions a similar problem is known as the McKean-Vlasov limit.
Motivated by the study of Vlasov scaling for some classes of stochastic evolutions in continuum for which the use of the mentioned above approaches breaks down (even in the finite volumes) we developed the general approach to study the Vlasov-type dynamics (see [8]). It is based on a proper scaling of the hierarchical equations for the evolution of correlation functions and can be interpreted in the terms of the rescaled Markov generators. Up to our knowledge, at the present time only this technique may give a possibility to control the convergence in the Vlasov limit in the case of non-integrable densities which is generic for infinite volume infinite particle systems. Saying about the evolutions, the kinetic equations of which can not be studied by the classical techniques described in [3] and [5], we have in mind, first of all, spatial birth-and-death Markov processes (e.g., continuous Glauber dynamics, spatial ecological models) and hopping particles Markov evolutions (e.g., Kawasaki dynamics in continuum). The main difficulty to carry out the approach proposed by W.Braun, K.Hepp [3] and R.L.Dobrushin [5] for such models is absence of the proper descriptions in terms of stochastic evolutional equations. Another problem concerns the possible variation of particles number in the evolution. The important point to note also is that an application of the technique proposed in [8] leads to a limiting hierarchy which posses a chaos preservation property.
The aim of this paper is to study the Vlasov scaling for the individual based model (IBM) in spatial ecology introduced by B.Bolker and S.Pacala [1,2], U.Dieckmann and R.Law [4] (BDLP model) using the scheme developed in [8]. A population in this model is represented by a configuration of motionless organisms (plants) located in an infinite habitat (an Euclidean space in our considerations). The unbounded habitat is taken to avoid boundary effects in the population evolution.
The evolution equation for the correlation functions of the BDLP model was studied in details in [9]. In [1,2], [4] this system was called the system of spatial moment equations for plant competition and, actually, this system itself was taking as a definition of the dynamics in the BDLP model. The mathematical structure of the correlation functions evolution equation is close to other well-known hierarchical systems in mathematical physics, e.g., BBGKY hierarchy for the Hamiltonian dynamics (see, e.g. [6]). As in all hierarchical chains of equations, we can not expect the explicit form of the solution, and even more, the existence problem for these equations is a highly delicate question.
According to the general scheme (see [8]), we state conditions on structural coefficients of the BDLP Markov generator, which give a weak convergence of the rescaled generator to the limiting generator of the related Vlasov hierarchy. Next, we may compute limiting Vlasov type equation for the BDLP model leaving the question about the strong convergence of the hierarchy solutions for a separate analysis. A control of the strong convergence of the rescaled hierarchy is, in general, a difficult technical problem. In particular, this problem remains be open for BBGKY hierarchy for the case of Hamiltonian dynamics as well as for Bogoliubov-Streltsova hierarchy corresponding to the gradient diffusion model. In the present paper we show the existence of the rescaled and limiting evolutions of correlation functions related to the Vlasov scaling of the BDLP model and the convergence to the limiting evolution. With this evolution for special class of initial conditions is related a non-linear equation for the density, which is called Vlasov equation for the considered stochastic dynamics.
Let us mention that a version of the BDLP model for the case of finite populations was studied in the paper [11]. In this work the authors developed a probabilistic representation for the finite BDLP process and applied this technique to analyze a mean-field limit in the spirit of classical Dobrushin or McKean-Vlasov schemes. They obtained an integro-differential equation for the limiting deterministic process corresponding to an integrable initial condition. The latter equation coincides with the Vlasov equation for the BDLP model derived below in our approach.
The present paper is organized in the following way. Section 2 is devoted to the general settings required for the description of the model which we study. In Subsection 3.1 we discuss the general Vlasov scaling approach for spatial continuos models. Subsection 3.2 is devoted to the abstract convergence result for semigroups in Banach spaces which will be crucial to prove the main statements of the paper presented in Subsection 3.3. The corresponding proofs are given in Subsection 3.4.
2 Basic fact and description of model

General facts and notations
Let B(R d ) be the family of all Borel sets in R d and B b (R d ) denotes the system of all bounded sets in B(R d ).
The space of n-point configuration is where |A| denotes the cardinality of the set A. The space Γ i.e. (R d ) n /S n , where S n is the permutation group over {1, . . . , n}. Hence one can introduce the corresponding topology and Borel σ-algebra, which we denote by O(Γ The space of finite configurations is equipped with the topology which has structure of disjoint union. Therefore, one can define the corresponding Borel σ-algebra B(Γ 0 ).
Λ . The Lebesgue-Poisson measure λ z on Γ 0 is defined as Here z > 0 is the so called activity parameter. The restriction of λ z to Γ Λ will be also denoted by λ z . The configuration space is equipped with the vague topology. It is a Polish space (see e.g. [14]). The corresponding Borel σ-algebra B(Γ) is defined as the smallest σ-algebra for which all mappings N Λ : Γ → N 0 , N Λ (γ) := |γ ∩ Λ| are measurable, i.e., One can also show that Γ is the projective limit of the spaces The Poisson measure π z on (Γ, B(Γ)) is given as the projective limit of the family of measures {π Λ z } Λ∈B b (R d ) , where π Λ z is the measure on Γ Λ defined by π Λ z := e −zm(Λ) λ z . We will use the following classes of functions: L 0 ls (Γ 0 ) is the set of all measurable functions on Γ 0 which have a local support, i.e. G ∈ L 0 ls (Γ 0 ) if there exists Λ ∈ B b (R d ) such that G ↾ Γ 0 \Γ Λ = 0; B bs (Γ 0 ) is the set of bounded measurable functions with bounded support, i.e. G ↾ Γ 0 \B = 0 for some bounded B ∈ B(Γ 0 ).
On Γ we consider the set of cylinder functions F cyl (Γ), i.e. the set of all measurable functions G on Γ, B(Γ)) which are measurable w.r.t. B Λ (Γ) for some Λ ∈ B b (R d ). These functions are characterized by the following relation: The following mapping between functions on Γ 0 , e.g. L 0 ls (Γ 0 ), and functions on Γ, e.g. F cyl (Γ), plays the key role in our further considerations: where G ∈ L 0 ls (Γ 0 ), see e.g. [13,17,18]. The summation in the latter expression is taken over all finite subconfigurations of γ, which is denoted by the symbol η ⋐ γ. The mapping K is linear, positivity preserving, and invertible, with Let M 1 fm (Γ) be the set of all probability measures µ on Γ, B(Γ) which have finite local moments of all orders, i.e. Γ |γ Λ | n µ(dγ) < +∞ for all The measure ρ µ := K * µ is called the correlation measure of µ. As shown in [13] for µ ∈ M 1 fm (Γ) and any Λ is absolutely continuous with respect to π Λ z for all Λ ∈ B Λ (R d ). In this case ρ µ := K * µ is absolutely continuous w.r.t λ z . We denote The functions are the correlation functions well known in statistical physics, see e.g. [20], [21].

Description of model
We consider the evolving in time system of interacting individuals (particles) in the space R d . The state of the system at the fixed moment of time t > 0 is described by the random configuration γ t from Γ. Heuristically, the mechanism of the evolution is given by a Markov generator, which has the following form (in other words, a − , a + are probability densities) and m, κ − , κ + > 0 are some positive constants. The pre-generator L describes the Bolker-Dieckmann-Law-Pacala BDLP model, which was introduced in [1,2,4]. During the corresponding stochastic evolution the birth of individuals occurs independently and the death is ruled not only by the global regulation (mortality) but also by the local regulation with the kernel κ − a − . This regulation may be described as a competition (e.g., for resources) between individuals in the population.
The evolution of the one dimensional distribution for such systems can be expressed in terms of their characteristics, e.g. the correlation functions (see (2.4)). The dynamics of correlation functions for the BDLP model was studied in [9]. The main result of this paper informally says the following: If the mortality m and the competition kernel κ − a − are large enough, then the dynamics of correlation functions, associated with the pre-generator (2.5), exists and preserves (sub-)Poissonian bound.
For the readers convenience we repeat below the relevant material from [9] without proofs.
LetL ± := K −1 L ± K be the K-image of L ± , which can be initially defined on functions from B bs (Γ 0 ). For arbitrary and fixed C > 0 we consider the operatorL :=L + +L − in the functional space Below, symbol · C stands for the norm of this space.
For any ω > 0 we define H(ω) to be the set of all densely defined closed operators T on L C , the resolvent set ρ(T ) of which contains the sector and for any ε > 0 where M ε does not depend on ζ.
The first non-trivial result, which is based on the perturbation theory, says that the operatorL with the domain To construct the corresponding evolution of correlation functions we note that the dual space ( where the isomorphism is provided by the isometry R C In fact, we have duality between Banach spaces L C and K C given by the following expression therefore,L * is the dual operator toL w.r.t. the duality (2.7). By [10], we have the precise form ofL * : Now we consider the adjoint semigroupT ′ (t) on (L C ) ′ and its imageT * (t) in K C . The latter one describes the evolution of correlation functions. Transferring the general results about adjoint semigroups (see, e.g., [7]) onto semigroupT * (t) we deduce that it will be weak*-continuous and weak*differentiable at 0. Moreover,L * will be the weak*-generator ofT * (t). Here and subsequently we mean "weak*-properties" w.r.t. the duality (2.7).

Description of Vlasov scaling
We begin with a general idea of the Vlasov-type scaling. It is of interest to construct some scaling L ε , ε > 0 of the generator L, such that the following scheme works. Suppose that we know the proper scaling of L and we are able to prove the existence of the semigroupT ε (t) with the generatorL ε := K −1 L ε K in the space L C for some C > 0. Let us consider the Cauchy problem corresponding to the adjoint operatorL * and take an initial function with the strong singularity in ε. Namely, where the function r 0 is independent of ε. The solution to this problem is described by the dual semigroupT * ε (t). The scaling L → L ε has to be chosen in such a way thatT * ε (t) preserves the order of the singularity: Another very important requirement for the proper scaling concerns the dynamics r 0 → r t . It should preserve Lebesgue-Poisson exponents: if r 0 (η) = e λ (ρ 0 , η) then r t (η) = e λ (ρ t , η) and there exists explicit (nonlinear, in general) differential equation for ρ t ∂ ∂t ρ t (x) = υ(ρ t (x)), (3.1) which will be called the Vlasov-type equation. Now let us explain the main technical steps to realize Vlasov-type scaling. Let us consider for any ε > 0 the following mapping (cf. (2.6)) on functions on Γ 0 (R ε r)(η) := ε |η| r(η).

(3.2)
This mapping is "self-dual" w.r.t. the duality (2.7), moreover, Therefore, we have to show that for any t ≥ 0 the operator family R εT * ε (t)R ε −1 , ε > 0 has limiting (in a proper sense) operator U(t) and In fact, we need that there exists an operatorV * (called Vlasov operator) such that exp {tR εL * Hence, heuristic way to produce the scaling L → L ε is to demand that if ρ t satisfies (3.1). The point-wise limit ofL * ε, ren will be natural candidate for V * . Having chosen the proper scaling we proceed to the following technical steps which give the rigorous meaning to the idea introduced above. Note that definition (3.4) impliesL ε, ren = R ε −1L ε R ε . We prove that "renormalized" operatorL ε, ren is a generator of a contraction semigroupT ε, ren (t) on L C . Next we show that this semigroup converges strongly to some semigroup T V (t) with the generatorV . This limiting semigroup leads us directly to the solution for the Vlasov-type equation. Below we show how to realize this scheme in details.

Approximation in Banach space
In this subsection we study general question about the strong convergence of semigroups in Banach spaces. The obtained results will be crucial in the realization of the Vlasov-type scaling for the BDLP model.
Let {U ε t , t ≥ 0} , ε ≥ 0 be a family of semigroups on a Banach space E. We set (L ε , D(L ε )) to be the generator of {U ε t , t ≥ 0} for each ε ≥ 0. Our purpose now is to describe the strong convergence of semigroups {U ε t , t ≥ 0} , ε ≥ 0 in terms of the corresponding generators as ε tends to 0. According to the classical result (see e.g. [12]), it is enough to show that there exists β > 0 and λ : Re λ > β such that where 1 1 is the identical operator. In this subsection we show how to prove (3.5) under the following assumptions on the family (L ε , D(L ε )), ε ≥ 0:
2. There exists β > 0 and λ: Re λ > β such that (a) λ belongs to the resolvent set of A 1 (ε) for any ε ≥ 0 and The strong convergence result for the family {U ε t , t ≥ 0} , ε ≥ 0 is established by our next theorem Theorem 3.1. Let (L ε , D(L ε )), ε ≥ 0 be the family of generators corresponding to the family of C 0 -semigroups {U ε t , t ≥ 0} , ε ≥ 0. Then, U ε t converges strongly to U 0 t as ε → 0 uniformly on each finite interval of time, provided assumptions (A) are satisfied.
Proof. The proof is completed by showing (3.5). For any ε ≥ 0 and λ from the resolvent set of A 1 (ε) we have Hence, Combining (3.6) with the assumption 2(c) of (A) we get the following representations for the resolvent From this formula, triangle inequality and assumptions 2(a), 2(b) and 2(d) of (A) we conclude the assertion of the theorem.

Main results
We check at once that the proper scaling for the BDLP pre-generator is the following one Next we consider the formal K-image of L ε and the corresponding renormalized operator on B bs (Γ 0 ): In the proposition below we calculate the precise form of the operatorL ε, ren for the BDLP model.
Proof. According to the definition, we haveL ε,ren = R ε −1L ε R ε , wherê As a result, and hence which completes the proof. Remark 3.3. It is easily seen that the operatorV := A 1 + A 2 will be the point-wise limit ofL ε, ren as ε tends to 0. Therefore, the adjoint operator tô V w.r.t. to the duality (2.7) (if it exists) can be considered as a candidate for the Vlasov operator in our model.
Below we give the rigorous meaning to the operatorL ε,ren . Let us define the set For any ε, m, κ − , C > 0 the operator with the domain D 1 is a generator of a contraction C 0 -semigroup on L C . Moreover, A 1 (ε) ∈ H (ω) for all ω ∈ 0; π 2 . Proof. See the proof of Proposition 4.2 in [9].
The next task is to show that for any ε > 0 the operator with the domain D 1 as well as the operator A 2 (0) := A 2 with the domain D 0 are relatively bounded w.r.t. the operator (A 1 (ε), D 1 ) and (A 1 , D 0 ), correspondingly. This is demonstrated in Propositions 3.6 and 3.7, which can be proved similarly to Lemmas 4.4 and 4.5 in [9].
Remark 3.8. Proposition 3.7 enables us to take D(B 2 ) = D 1 . As a result, Remark 3.5 shows that the domain of the operator A 2 (ε) will be D 0 ∩ D 1 = D 1 .
We are now in a position to show that the operator (L ε,ren , D 1 ) generates semigroup on L C . To this end we use the classical result about the perturbation of holomorphic semigroups (see, e.g. [12]). For the convenience of the reader we formulate below the main statement without proof: For any T ∈ H(ω), ω ∈ (0; π 2 ) and for any ǫ > 0 there exist positive constants α, δ such that if the operator A satisfies with a < δ, b < δ, then T + A is a generator of a holomorphic semigroup. In particular, if b = 0, then T + A ∈ H(ω − ǫ). Theorem 3.9. Let the functions a − , a + and the constants m, κ − , κ + , C > 0 satisfy Then, for any ε > 0 the operator (L ε,ren , D 1 ) is a generator of a holomorphic semigroupÛ t,ε , t ≥ 0 on L C .
Proof. Let ε > 0 be arbitrary and fixed. By definition, The direct application of the theorem about perturbation of holomorphic semigroups (see the formulation above the assertion of Theorem 3.9) to T = A 1 (ε) and A = A 2 (ε) gives now the desired claim. It is important to note that Proposition 3.4 enables us to consider δ equal to 1 2 in the formulation of the classical theorem introduced above. The appearance of the multiplicand 4 on the left-hand side of the both assumptions in assertion of Theorem 3.9 is motivated exactly by the latter fact. Then, the operatorV = A 1 + A 2 with the domain D 0 is a generator of a holomorphic semigroupÛ V t , t ≥ 0 on L C .
Proof. We use the same classical result as for Theorem 3.9 in the case: A 1 is a generator of holomorphic semigroup, A 2 is relatively bounded w.r.t. A 1 with the boundary less then 1 2 . Now we may repeat the same considerations as at the end of Section 2. Namely, transferring the general results about adjoint semigroups (see, e.g., [7]) onto semigroup (Û V t ) * in K C we deduce that it will be weak*-continuous and weak*-differentiable at 0. Moreover,V * will be the weak*-generator of T * (t). This means, in particular, that for any The explicit form ofV * follows from (2.10), namely, for any k ∈ D(V * ) As a result, we have that for any k 0 ∈ D(V * ) the function k t = (Û V t ) * k 0 provides a weak* solution of the following Cauchy problem In the next theorem we show that the limiting Vlasov dynamics has chaos preservation property, i.e. preserves the Lebesgue-Poisson exponents.
Theorem 3.11. Let conditions of Theorem 3.9 be satisfied and, additionally, C ≥ 4 16e−1 . Let ρ 0 ≥ 0 be a measurable nonnegative function on R d such that ess sup x∈R d ρ 0 (x) ≤ C. Then the Cauchy problem (3.15) with k 0 = e λ (ρ 0 ) has a weak* solution k t = e λ (ρ t ) ∈ K C , where ρ t is a unique nonnegative solution to the Cauchy problem
Proof. First of all, if (3.16) has a solution ρ t (x) ≥ 0 then and, therefore, u t L ∞ (R d ) . We denote by X + T the cone of all nonnegative functions from X T .
Let Φ be a mapping which assign to any v ∈ X T the solution u t of the linear Cauchy problem for a.a. x ∈ R d . Therefore, We have that v ∈ X + T implies Φv ≥ 0 as well as the estimate where we use the trivial inequality Therefore, Φv ∈ X + T . For simplicity of notations we denote for v ∈ X + Then, for any v, w ∈ X + We have where we used (3.19) and obvious inequalities |e −a −e −b | ≤ |a−b| for a, b ≥ 0; e −x x ≤ e −1 for x ≥ 0. Next, using another simple estimates for any a, b, p, q ≥ 0 and, using (3.12) and the inequalities above, one can continue if, e.g., 1 + C 4 ≤ 4Ce, that means C ≥ 4 16e−1 . As a result, by (3.11), Φ is a contraction mapping on the cone X + T . Taking, as usual, v (n) = Φ n v (0) , n ≥ 1 for v (0) ∈ X + T we obtain that {v (n) } ⊂ X + T is a fundamental sequence in X T which has, therefore, a unique limit point v ∈ X T . Since X + T is a closed cone we have that v ∈ X + T . Then, identically to the classical Banach fixed point theorem, v will be a fixed point of Φ on X T and a unique fixed point on X + T . Then, this v is the nonnegative solution of (3.16) on the interval [0; T ]. By the note above, v t (x) ≤ C. Changing initial value in (3.16) onto ρ t t=T (x) = v T (x) we may extend all our considerations on the time-interval [T ; 2T ] with the same estimate v t (x) ≤ C; and so on. As a a result, (3.16) has a unique global bounded non-negative solution ρ t (x) on R + .
The main result of the paper is formulated in the next theorem. Its proof will be given in Subsection 3.4.
Theorem 3.12. Under conditions of Theorem 3.9 the semigroupÛ t,ε converges strongly to the semigroupÛ V t as ε → 0 uniformly on any finite intervals of time.
In the following proposition we verify Assumption 2(a) of (A).
Next we check Assumption 2(b) of (A).
Proposition 3.14. Let λ > 0 be arbitrary and fixed. Then Proof. For any G ∈ L C and any ε > 0 This finishes the proof.
Assumption 2(c) of (A) is proved in the next Proposition.
We set In order to verify Assumption 2(d) of (A) we have to show that Q ε s −→ Q as ε → 0.
Suppose that we can show that Then, To check Q ε = (C ε + 1 1) −1 s −→ Q (3.24) we proceed as follows: Assuming (3.23) it is obvious now that convergence (3.24) is equivalent to sup ε>0 (C ε + 1 1) −1 < ∞, which is clear from The last bound we conclude from (3.22). As a result we shall have established Theorem 3.12 if we show (3.23). leads to the following formula Now, we are left with the task to show that But, for any G ∈ L C due to the Lebesgue's dominated convergence theorem.
Proof of Lemma 3.17. Since B 2 G C ≤ 1 4 B 1 G C , we have to show that εB 1 (A 1 (ε) − λ1 1) −1 G C → 0, as ε → 0. But, The last two lemmas conclude the proof of the main Theorem.
Remark 3.18. Under assumptions of Proposition 3.15 we get the following representation for the resolvents ofV andL ε,ren