Glauber Dynamics in Continuum: A Constructive Approach to Evolution of States

The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.


Introduction
In the statistical theory of large systems [2], the system states are described as probability measures on the corresponding phase space rather than pointwise, which is typical for the standard theory of dynamical systems. For such large systems, in order to obtain the description independent of the system size one employs the models where the system is infinite and distributed over a noncompact manifold with positive density. A particular case constitute models of interacting point particles distributed over R d , which are widely used in mathematical physics, ecology, sociology, etc, see [3,6,[8][9][10][13][14][15]. Here the states are probability measures on the space of the particle configurations Γ ≡ Γ(R d ) := {γ ⊂ R d : |γ ∩ K| < ∞ for any compact K ⊂ R d }, (1.1) where |A| denotes the cardinality of A. The system is characterized by a collection of appropriate functions F : Γ → R, called observables. For a state µ, the quantity is called the mean value of observable F in state µ. Then the system evolution is described as the evolution of observables obtained from the Kolmogorov equation (1.2) where the 'generator' L is specified within the choice of the model. The evolution of states is obtained from the Fokker-Planck equation Note that L ought to be Markov in order that the solutions of (1.3) be probability measures. One of the possibilities here is to describe the evolution pathwise-by constructing a stochastic Markov process X µ0 t , corresponding to the 'generator' L and to the initial state µ 0 . Then the state µ t is just the distribution law of X µ0 t . However, for a number of important models this way encounters serious problems and hence is rather unrealistic. Moreover, the mere existence of the process tells not too much about the properties of the system evolution. Thus, the main idea which we realize in this work is to describe the system evolution as the Markov evolution of states µ 0 → µ t , not necessarily based on the pathwise description, and accompanied with a more detailed study of its properties. In a sense, our approach is suggested by classical works on the Hamiltonian dynamics where the system evolution is described as the evolution of the corresponding correlation functions obtained by solving the equation In the Hamiltonian dynamics, the analog of (1.5) is the BBGKY hierarchy. As mentioned in [2], kinetic equations of the Hamiltonian dynamics allow one to describe the evolution approximately but in more detail and in simpler terms.
Such kinetic equations can be obtained from equations like (1.5), provided all necessary information about their solutions is available, see the corresponding discussion in [2, paragraph 6].
In the present article, we describe the Markov evolution of states in terms of the correlation functions on both microscopic, obtained from (1.5), and mesoscopic levels. The latter will be done by means of a nonlinear (kinetic) equation obtained from (1.5) in the Vlasov scaling limit. As in [8,15], our object is the Glauber dynamics described by the 'generator' (1.2) having the form Here the first term describes the particle death with constant rate, whereas the second one is the birth term with activity κ > 0 and an interaction potential φ ≥ 0, which is supposed to obey a natural integrability condition only. In contrast to [8,15], where κ and φ were subject to a certain constraint, here we obtain the evolution k 0 → k t for all κ and φ, which, however, is restricted to a limited time interval [0, T * ). Instead of the semigroup techniques used in [8,15], we apply Picard-like approximations and a method suggested in [4, pp. 94, 95], which allows for constructing classical solutions in a scale of Banach spaces 1 {K α } α∈I⊂R , K α ′ ⊂ K α ′′ for α ′′ < α ′ . Namely, in Theorem 3.6 we show that, for any α 0 ∈ R and any α < α 0 , there exists T (α 0 , α) > 0 such that, for any t ∈ [0, T (α 0 , α)), there exists α t ∈ (α, α 0 ) such that the problem (1.5) with k 0 ∈ K α0 has a classical solution k t ∈ K αt being the correlation function of a certain µ t . The latter fact is obtained by means of the corresponding result of [3]. This yields the evolution µ 0 → µ t . In addition, in Theorem 3.9 we show that, for k 0 (η) ≤ κ |η| , the solution obeys k t (η) ≤ κ |η| and hence can be continued in time to the whole R + . These are the main results of Section 3. In Section 4, we perform the Vlasov scaling and obtain the Vlasov hierarchy-a linear evolution equation (d/dt)r t = L V r t , which we study in the same scale of Banach spaces where the correlation functions evolve. Its main peculiarity is the fact that if r 0 is the correlation function of a nonhomogeneous Poisson measure π ̺0 with density ̺ 0 , then the solution r t is the correlation function for π ̺t with ̺ t satisfying a nonlinear nonlocal equation, see Lemma 4.2 and Theorem 4.4. Finally, in Theorem 4.5 we show that the rescaled correlation functions converge in the scaling limit to the corresponding r t . In Section 5, we briefly summarize and compare with each other the results of Sections 3 and 4.
2 The basic notions and the model

The notions
All the details of the framework used in this paper can be found in [6,[8][9][10][11]14]. We consider an infinite system of point particles moving in R d , d ≥ 1. By B(R d ) and B b (R d ) we denote the set of all Borel and the set of all bounded Borel subsets of R d , respectively. For X ∈ B(R d ), the set of n-particle configurations in X is where | · | denotes cardinality. Γ (n) X can be identified with the symmetrization of {(x 1 , . . . , x n ) ∈ X n : x i = x j , for j = j}, which allows one to introduce the corresponding topology and hence the Borel σ-algebra B(Γ We equip it with the topology of the disjoint union and hence with the Borel σ-algebra B(Γ 0,X ). For X = R d , we write Γ (n) and Γ 0 meaning Γ (n) X and Γ 0,X , respectively. The restriction of the Lebesgue product measure dx 1 dx 2 · · · dx n to (Γ (n) , B(Γ (n) )) is denoted by m (n) . Then the Lebesgue-Poisson measure on For any X ∈ B(R d ), the restriction of λ to Γ 0,X will also be denoted by λ.
The set of all configurations in R d is We equip it with the vague topology-the weakest topology in which all the maps are continuous. Here C 0 (R d ) stands for the set of all continuous f : R d → R, which have compact supports. The vague topology on Γ admits a metrization, which turns it into a complete and separable metric (Polish) space, see e.g. [12]. By B(Γ) we denote the corresponding Borel σ-algebra. It turns out that the measurable space (Γ, B(Γ)) is the projective limit of the family Then the Poisson measure π on (Γ, B(Γ)) is the projective limit of the family which is supposed to be finite. Then π ̺ is defined by its projections For a measurable f : R d → R and η ∈ Γ 0 , the Lebesgue-Poisson exponent is Clearly e(f, ·) ∈ L 1 (Γ 0 , dλ) for any f ∈ L 1 (R d ), and for some Λ ∈ B b (R d ) and N ∈ N. By B bs (Γ 0 ) we denote the set of all bounded measurable functions G : Γ 0 → R, which have bounded supports. That is, each such G is the zero function on Γ 0 \ M for some bounded M . Noteworthy, any measurable G : Γ 0 → R is in fact a sequence of measurable symmetric functions G (n) : (R d ) n → R. For Λ ∈ B b (R d ) and γ ∈ Γ, by γ Λ we denote γ ∩ Λ; thus, γ Λ ∈ Γ 0,Λ . A measurable function F : Γ → R is called a cylinder function if there exist Λ ∈ B b (R d ) and a measurable G : Γ 0,Λ → R such that F (γ) = G(γ Λ ) for all γ ∈ Γ. By F cyl (Γ) we denote the set of all cylinder functions. For γ ∈ Γ, by writing η ⋐ γ we mean that η ⊂ γ and η is finite, i.e., η ∈ Γ 0 . For G ∈ B bs (Γ 0 ), we set It is known that K is linear and positivity preserving, and maps B bs (Γ 0 ) into F cyl (Γ) (see e.g. [11]). By M 1 fm (Γ) we denote the set of all probability measures on (Γ, B(Γ)) which have finite local moments, that is, for which Γ |γ Λ | n µ(dγ) < ∞ for all n ∈ N and Λ ∈ B b (R d ). (2.10) A measure ρ on (Γ 0 , B(Γ 0 )) is said to be locally finite if ρ(M ) < ∞ for every bounded M ⊂ Γ 0 . By M lf (Γ 0 ) we denote the set of all such measures. For is said to be locally absolutely continuous with respect to the Poisson measure π if, for every Λ ∈ B b (R d ), µ Λ := µ • p −1 Λ is absolutely continuous with respect to π Λ , see (2.3).
Let M ⊂ Γ 0 be bounded, and let I M be its indicator function on Γ 0 . Then I M is in B bs (Γ 0 ) and hence one can apply (2.9). For µ ∈ M 1 fm (Γ), let which uniquely determines a measure ρ µ ∈ M lf (Γ 0 )). It is called the correlation measure for µ. This defines the map K * : M 1 fm (Γ) → M lf (Γ 0 )) such that K * µ = ρ µ . In particular, K * π = λ. It is known that, see [11,Proposition 4.14], ρ µ is absolutely continuous with respect to λ if µ is locally absolutely continuous with respect to π. In this case, we have that for any Λ The Radon-Nikodym derivative k µ is called the correlation function corresponding to the measure µ. Finally, we mention the following integration rule, c.f.
which holds for any appropriate function H.

The model
Let φ : R d → R + := [0, +∞) be such that φ(x) = φ(−x) and be integrable in the following sense with the possibility that E φ (x, γ) = +∞ for some γ. In the model we consider, the dynamics of the observables is defined by the 'generator' (1.6) with φ just mentioned and κ > 0 being the birth activity parameter. The action of the Thus, the sum and the integral in (1.6) are finite.
Following the general scheme developed in [13] one constructs the evolution of the quasi-observables, which are functions on Γ 0 . This evolution is obtained as a solution to the following Cauchy problem where e(t x , ·) is defined in (2.6), and Clearly, the action ofL on G ∈ B bs (Γ 0 ) is well-defined. Its extension to wider classes of G will be done in a while. For a measurable function k : Γ 0 → R and G ∈ B bs (Γ 0 ), we define This pairing can be extended to the corresponding classes of G and k. Then (2.16) and (2.19) lead to the following (dual) Cauchy problem The action of L ∆ is obtained by means of (2.13) from and from (2.19) and (2.17). It thus has the form Of course, the case of a special interest in (2.20) is where k 0 is the correlation function of a certain µ 0 ∈ M 1 fm (Γ), see (2.12). However, the mere existence of the solution k t does not guarantee that this k t is a correlation function.
In [13][14][15], the solution G 0 → G t of (2.16), for all t ≥ 0 and 'small' κ and c φ , was obtained in a certain Banach space by means of the construction of a C 0 -semigroup based on perturbation methods. Then the evolution of the correlation functions k 0 → k t was obtained in the weak sense, in which k t is defined by k 0 via the relation Regarding the problems (2.16) and (2.20), in the present article we realize the following program: • Show that (2.16) has a unique classical solution for all κ > 0 and c φ , which we do in Theorem 3.1 for t belonging to a bounded interval.
• Show that the solution of (2.16) exists for all t ≥ 0 if κc φ < 1/e, which we do in Theorem 3.2.
• Show that (2.20) has a unique classical solution k t for all κ > 0 and c φ , being the correlation function of a certain µ t ∈ M 1 fm (Γ), which yields the evolution of states µ 0 → µ t . We do this in Theorem 3.6 for t belonging to a bounded interval.
These results give the microscopic evolution of states corresponding to (1.6).
A similar program concerning the mesoscopic evolution is formulated and realized in Section 4 below.
3 The microscopic description

The evolution of quasi-observables
First we study the problem (2.16), (2.17). For α ∈ R, we consider the Banach space We will seek the solution of (2.16), (2.17) as the limit of {G The latter can be iterated to give For any α ′ , α ′′ such that α ′ < α ′′ , we have where we have used the following obvious estimate Furthermore, where we have used also (2.7) and (2.13). By means of the latter estimate and (3.5) we finally get From (3.6) we see thatL can be defined as a bounded linear operatorL : Given α 0 ∈ R, α > α 0 , m ∈ N, and l = 0, . . . , m, we take α l = α 0 + lǫ, ǫ = (α − α 0 )/m. Then by (3.7) we get Put .
Note that Hence, we can set Our main result concerning the problem (2.16), (2.17) is the following statement.
The statement just proven describes systems with any κ and c φ . It is, however, possible to get more if one imposes appropriate restrictions on these parameters. Namely, the dynamics in this case is described by a C 0 -semigroup S(t) : G α → G α , where the space is the same as in (3.1). Set and also Theorem 3.2. Assume that For such φ, let α φ = ln c φ . Then, for every G 0 ∈ H α φ , the problem (2.16), The above theorem is true not only for α = α φ but also for α ∈ (α φ − δ, α φ + δ) for some δ > 0, see the proof below. (b) The condition (3.14) is well-known, see [16,Chapter 4]. It provides the convergence of cluster expansions for the gas of classical particles with the pair-wise repulsion U (x, y) = φ(x − y).
In the proof of Theorem 3.2 we use two statements. The first one is an adaptation of [1,Corollary 5.16].
Proposition 3.5. Let A and B ± be as above. Suppose that there exist real α and α ′ , α ′ < α, such that for some positive C and ε. Then the closure ofL + in G α generates a positive C 0 -semigroup in this space.

The evolution of correlation functions and states
In this subsection, the problem (2.20), (2.21) will be studied in the Banach space, c.f. (3.1), For the sake of convenience, here we use the same notation · α as in (3.2), which, however, should not cause any ambiguity as it will always be clear from the context which norm is meant.
Recall that M 1 fm (Γ) stands for the set of probability measures on Γ obeying (2.10). Given α ∈ R, by M α we denote the set of all those µ ∈ M 1 fm (Γ) whose correlation functions, defined in (2.12), belong to K α . The main result of this section is contained in the following statement.
Proof. First we prove the existence of the solution k t in the Banach space K αt and then show that it is a correlation function.
Thus, for α ′ and α ′′ as in (3.6), we get from (2.21) Similarly as in producing (3.6) we then get from the latter Hence, L ∆ can be defined as a bounded linear operator L ∆ : Now we fix t ∈ (0, T (α 0 , α)), and forα ∈ (α, α 0 ) and a given m ∈ N, set α l = α 0 − lǫ, ǫ = (α 0 −α)/m. Then, by (3.27), see (3.9). Pick δ > 0 such that t + δ < T (α 0 , α). Then for we have from (3.28) Applying this estimate in (3.24) we obtain the convergence of the sequence {k (n) t } n∈N0 in K αt , which yields the existence and uniqueness of the solution k t similarly to that in the proof of Theorem 3.1. Now let us show that the solution just constructed is such that, for any t ∈ (0, T (α 0 , α)), there exists µ t ∈ M 1 fm (Γ) such that, c.f. (2.12), if k 0 ∈ K α0 is the correlation function of the corresponding µ 0 ∈ M 1 fm (Γ). Under the condition (2.14), there exists a proper S ⊂ Γ and an S-valued process with sample paths in the Skorokhod space D S (R + ) associated with L, see [3,Theorem 2.13]. This yields the evolution µ 0 → µ t , where µ t is the law of the process, and hence the evolution of the corresponding correlation functions k µ0 → k µt , which satisfy (2.20). By the uniqueness just established, k t = k µt for all t ∈ [0, T (α 0 , α)), which yields (3.30) and hence completes the proof.
Remark 3.7. Theorem 3.6 establishes the evolution k 0 → k t which takes places in the scale of spaces K αt . It is clear from the proof that all such spaces are contained in K α , that is, the mentioned theorem can be formulated similarly as Theorem 3.1.
Another our remark addresses the regularity of the solutions k t . Instead of (3.22) let us consider where this time Remark 3.8. Let α 0 , α, and T (α 0 , α) be as in Theorem 3.6. Suppose in addition that the function φ is continuous. Then the problem (2.20) with k 0 ∈ K α0 has a unique classical solution k t ∈ K αt with t ∈ [0, T (α 0 , α)), where α t is the same as in Theorem 3.6. Now we show that the evolution of k t obtained in Theorems 3.1 and 3.6 can be continued in time to the whole R + . Recall that the space K α was defined in (3.22) and κ > 0 is the birth activity parameter, see (1.6). Set Theorem 3.9. The solution of the problem (2.20) with k 0 ∈ K κ can be continued to any positive t. Moreover, for every t ≥ 0, the solution k t is also in K κ .
The proof of Theorem 3.9 is based on the following estimate.
Lemma 3.10. Suppose that k 0 ∈ K κ . For α < α κ , let k t be the solution described by Theorem 3.6. Then, for all t ∈ [0, T (α κ , α)), we have that k t is also in K κ .
Proof. For G ∈ B bs (Γ 0 ) and k t as in Theorem 3.6, we set where µ t is the corresponding state. According to (1.6), we have that By (2.9), Here all sums are finite since G ∈ B bs (Γ 0 ). Applying these equalities in (3.35) together with the following x∈γ ξ⋐γ\x Assume now that G is positive as an element of L 1 (Γ 0 , dλ). As φ in (2.15) is also positive, the second line in (3.36) can be estimated In obtaining the latter equality we have used (2.13). Applying this estimate in (3.36) we arrive at which holds for λ-almost all η ∈ Γ 0 . By standard methods we get from this As a correlation function, k t (η) ≥ 0 for λ-almost all η. For a fixed t and |η| = n, assume that k s (ξ) ≤ κ |ξ| for all s ∈ [0, t] and all |ξ| = n − 1. As we also assume k 0 (η) ≤ κ |η| , by (3.38) k s (η) ≤ κ |η| , also for all s ∈ [0, t]. Thus, k t ∈ K κ .

The mesoscopic description
The mesoscopic description of the dynamics of our model will be conducted in the Vlasov scaling framework, see [5] where the detailed presentation of this approach and the most updated related bibliography can be found. Our program in this section is as follows: • Derive the Vlasov hierarchy as the (Vlasov) scaling limit of (2.20) and prove the existence of its solutions, see (4.12) and Proposition 4.1.
• Then derive the Vlasov equation from the Vlasov hierarchy and prove the existence of its solutions, see (4.15) and Theorem 4.4.
• Prove the convergence of the rescaled correlation functions to the solutions of the Vlasov hierarchy, see Theorem 4.5.

The Vlasov hierarchy
In the Vlasov scaling limit, which is achieved by letting ε → 0, the particle density is supposed to diverge whereas the interaction gets weak and of long range. Thus, we assume that the correlation function k (ε) of the particle system we consider depends of the scaling parameter and diverges in such a way that the renormalized function has a finite limit, which we denote by r. The evolution of yet not rescaled functions is described by the equation (2.20) in which the generator L ∆ ε contains εφ and ε −1 κ in place of φ and κ, respectively. Thus, the evolution of the rescaled functions (4.1) is described by the equation By (2.21), we thus have Note that, for small enough ε, R −1 ε k might not be a correlation function, even if k is, see (4.3).
Note that the passage from (4.4) to (4.10) was only 'informal', so we have no information how 'close' is r t to k (ε) t,ren . Another observation is that (4.12) has a very special solution, which we obtain now.
Proof. First of all we note that e(̺, ·) ∈ K α0 if and only if ̺ ∈ ∆ α0 , see (3.23). Now setr t = e(̺ t , ·) with ̺ t solving (4.15). Thisr t solves (4.12), which can easily be checked by computing d/dt and employing (4.15). In view of the uniqueness as in Proposition 4.1, we then haver t = r t on the time interval where both solutions exist. Let us show now that the problem (4.15) does have the solution we need (c.f. [7,Theorem 3.3]). In a standard way, this problem can be transformed into the following integral equation and ≥ 0 for all n ∈ N 0 . Thus, we have to show that ̺ (n) t (x) ≤ e −α0 , at least for some t > 0. By the induction over n, we see that this holds, for all t > 0, if κ ≤ e −α0 . t } n∈N0 is a Cauchy sequence in L ∞ (R d ), assuming ̺ (n) s ∈ ∆ α0 , for all n ∈ N 0 and s ≤ t. From (4.19), using elementary inequality Now we take T > 0 such that q(T ) < 1. Then the latter estimate yields Therefore, the sequence {̺ (n) t } n∈N0 converges in L ∞ (R d ), uniformly on [0, T ]. Thus, its limit is the unique classical solution of (4.15). Since this limit is still in ∆ + α0 , the evolution can be continued. Taking into account Lemma 4.2 we come to the following conclusion.

The scaling limit ε → 0
Our final task in this work is to show that the solution of (4.2) k (ε) t converges in K α0 uniformly in on [0, T ], T < T (α 0 , α), to the solution of (4.12), see Proposition 4.1. Here we should impose an additional condition on the potential φ, which, however, seems quite natural. Recall that in this section we suppose φ ∈ L 1 (R d ).
Proof. Given n ∈ N, let k (ε) t,n and r t,n be defined as in (3.24) with L ε,ren and L V , respectively. Like in the proof of Theorem 3.6, one can show that the sequences of k (ε) t,n and r t,n converge in K α to k (ε) t and r t , respectively, uniformly on every [0, T ], T < T (α 0 , α). Then, for δ > 0, one finds n ∈ N such that, for all t ∈ [0, T ], t,ren α + r t,n − r t α < δ/2.
where, see (3.27), Here we used the following representation  Thus, we have to show that L ε,ren − L V α0α → 0, as ε → 0, (4.25) which will allow us to make the first summand in the right-hand side of (4.23) also smaller than δ/2 and thereby to complete the proof. Subtracting (4.10) from (4.4) we get For t > 0, the function e −t − 1 + t takes positive values only; hence Ψ(t) := (e −t − 1 + t)/t 2 , t > 0, is positive and bounded, say by C > 0. Then by means of the following elementary analog of (4.24) b 1 · · · b n − a 1 · · · a n ≤ n i=1 we obtain e −α0|ξ| e(φ(x − ·), ξ) This yields and thereby (4.25).

Concluding remarks
Regarding the evolution of quasi-observables, in Theorem 3.1 we have proven its existence in G α if G 0 ∈ G α0 , for any α 0 and any α > α 0 , and for all values of the model parameters c φ and κ, however, on a bounded time interval. Note that the bound T (α, α 0 ) is small for big c φ κ, see (3.10). Note also that there exists the scale of spaces G αt ⊂ G α such that G t ∈ G αt for t ∈ [0, T (α, α 0 )), similarly to Theorem 3.6. For c φ κ < 1/e, the evolution G 0 → G t is described by a C 0 -semigroup, and hence has no time bounds, see Theorem 3.2.
Turn now to the evolution of states and correlation functions. The main peculiarity of Theorem 3.6 is that, in contrast to the results of [13][14][15], here we (a) impose no restrictions on c φ and κ; (b) describe the evolution directly, not as a weak evolution via (2.22). The price is the time restriction, similar as in Theorem 3.1. Again, we can start in K α0 with any α 0 ∈ R, and obtain that k t ∈ K αt ⊂ K α , also for any α < α 0 . The time bound T (α 0 , α) depends on the choice of α 0 and α. If the initial states is dominated by the Poisson measure with intensity κ, that is, if k 0 (η) ≤ κ |η| , then the solution described by Theorem 3.6 has also the property k t (η) ≤ κ |η| , and hence can be continued in time ad infinitum, see Theorem 3.9. Of course, in this case α 0 should obey (4.20). The main aim of using the Vlasov hierarchy (4.12) is obtaining the scaling limit of the rescaled correlation functions k (ε) t,ren . For any α 0 ∈ R and r 0 ∈ K α0 , this hierarchy has a unique classical solution r t in any K α , α < α 0 , with t ∈ [0, T (α 0 , α)), see Proposition 4.1. Here, however, for general r 0 we have no tools for continuing r t , like we did in Theorem 3.9 where we used the connection of L ∆ with L given by (1.6), since neither Markov operator corresponds to L V . But if r 0 is Poissonian, i.e., r 0 = e(̺ 0 , ·), then (4.12) has the solution r t = e(̺ t , ·) with infinite time lives in 'sufficiently large' K α0 , see Theorem 4.4. The latter means that the Poissonian correlation function k(η) = κ |η| belongs to this K α0 , see (4.20).
Note also that in the recent paper [7] it was shown the existence and strong convergence in the Vlasov scaling for the classical solution in one space K α but again under the condition c φ κ < 1/e.