On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the $L^2$-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper. Stochastic Equations], which was proved for a special Glauber (Kawasaki, respectively) dynamics.


Introduction
This paper deals with two classes of equilibrium stochastic dynamics of infinite particle systems in continuum. Let Γ denote the space of all locally finite subsets of R d . Such a space is called the configuration space (of an infinite particle system in continuum). Elements of Γ are called configurations and each point of a configuration represents position of a particle.
One can naturally define a σ-algebra on Γ, and then a probability measure on Γ represents a random system of particles. A probability measure on Γ is often called a point process (see e.g. [9]). Configuration spaces and point processes are important tools of classical statistical mechanics of continuous systems. A central class of point processes which is studied there is the class of Gibbs measures. Typically one deals with Gibbs measures which correspond to a potential of pair interaction.
For a deep understanding of these dynamics, it is important to see how they are related to each other. For example, in the recent paper [10], it was shown that a typical diffusion dynamics can be derived through a diffusive scaling limit of a corresponding Kawasaki dynamics.
In [3], it was proved that a special Glauber dynamics can be derived through a scaling limit of Kawasaki dynamics. Furthermore, [3] conjectured that such a result holds, in fact, for a wide class of birth-and-death dynamics (dynamics of hopping particles, respectively), which are indexed by a parameter s ∈ [0, 1]. (Note that the result of [3] corrrespods to the choice of parameter s = 0.) The aim of this work is to show that the conjecture of [3] is indeed true, at least for parameters s ∈ [0, 1/2]. (In the case where s ∈ (1/2, 1], one needs to put additional, quite restrictive assumptions on the potential of pair interaction, and we will not treat this case in the present paper.) Thus, we show that the result of [3] is not a property of just one special Kawasaki (Glauber, respectively) dynamics, but rather represents a property which is common for many dynamics.
More specifically, we fix a class of cylinder functions on Γ, and prove that on this class of functions, the corresponding generators converge in the L 2 (Γ, µ)-space. Here, µ is a Gibbs measure in the low activity-high temperature regime, µ being invariant measure for all the processes under consideration. If one additionally knows that the class of cylinder functions is a core for the limiting generator, then our result implies weak convergence of finitedimensional distributions of the corresponding processes. Unfortunately, apart from a very special case [11], no result about a core for these generators is yet available.
The paper is organized as follows. In Section 2, we briefly discuss Gibbs measures in the low activity-high temperature regime, and the corresponding correlation and Ursell functions. In Section 3, we describe classes of birth-and-death processes and of dynamics of hopping particles. In Section 4, we formulate and prove the result about convergence of the generators.
The authors acknowledge numerous useful discussions with Dmirti Finkelshtein and Yuri Kondratiev.
2 Gibbs measures in the low activity-high temperature regime The configuration space over R d , d ∈ N, is defined by where |·| denotes the cardinality of a set. One can identify any γ ∈ Γ with the positive Radon measure x∈γ ε x ∈ M(R d ), where ε x is the Dirac measure with mass at x, x∈∅ ε x :=zero measure, and M(R d ) stands for the set of all positive Radon measures on the Borel σ-algebra B(R d ). The space Γ can be endowed with the relative topology as a subset of the space M(R d ) with the vague topology, i.e., the weakest topology on Γ with respect to which all maps are continuous. Here, C 0 (R d ) is the space of all continuous real-valued functions on R d with compact support. We will denote by B(Γ) the Borel σ-algebra on Γ.
Let µ be a probability measure on (Γ, B(Γ)). Assume that, for each n ∈ N, there exists a non-negative, measurable symmetric function k (n) µ on (R d ) n such that, for any measurable The functions k (n) µ are called correlation functions of the measure µ. If there exists a constant ξ > 0 such that then we say that the correlation functions k (n) µ satisfy the Ruelle bound. The following lemma gives a characterization of the correlation functions in terms of the Laplace transform of a given point process, see e.g. [10] Lemma 1 Let µ be a probability measure on (Γ, B(Γ)) which satisfies the Ruelle bound (1). Let f : R d → R be a measurable function which is bounded outside a compact set Λ ⊂ R d and such that e f − 1 ∈ L 1 (R d , dx). Then for µ-a.a. γ ∈ Γ, |f |, γ < ∞ and

Remark 1
Note that if f : R d → R is bounded outside a compact set Λ ⊂ R d and if, furthermore, f is bounded from above on the whole R d , then the condition Via a recursion formula, one can transform the correlation functions k (n) µ into the Ursell functions u (n) µ and vice versa, see e.g. [22]. Their relation is given by where Γ 0 := {γ ∈ Γ : |γ| < ∞}, for any η = {x 1 , . . . , x n } ∈ Γ 0 k µ (η) := k (n) µ (x 1 , . . . , x n ), u µ (η) := u (n) µ (x 1 , . . . , x n ), and the summation in (2) is over all partitions of the set η into nonempty mutually disjoint subsets η 1 , . . . , η j ⊂ η such that η 1 ∪ · · · ∪ η j = η, j ∈ N. Note that if the correlation functions (k then so are the Ursell functions (u For γ ∈ Γ and x ∈ R d \ γ, we define a relative energy of interaction between a particle at x and the configuration γ as follows: A probability measure µ on (Γ, B(Γ)) is called a (grand canonical) Gibbs measure corresponding to the pair potential φ and activity z > 0 if it satisfies the Georgii-Nguyen-Zessin identity ([17, Theorem 2]): for any measurable function F : Γ×R d → [0; +∞]. Here and below, for simplicity of notations, we just write x instead of {x}. We denote the set of all such measures µ by G(z, φ).
As a straightforward corollary of the Georgii-Nguyen-Zessin identity (3), we get the following equality: for any measurable function F : Let us formulate conditions on the pair potential φ.
(LA-HT) (Low activity-high temperature regime) We have: where B is as in (S).
The following classical theorem is due to Ruelle [21,22].
Theorem 1 Assume that (S) and (LAHT) are satisfied. Then there exists µ ∈ G(z, φ) which has the following properties: a) µ has correlation functions (k In what follows, we will assume that (S) and (LAHT) are satisfied, and we will keep the measure µ from Theorem 1 fixed.

Equilibrium birth-and-death (Glauber) dynamics and hopping particles' (Kawasaki) dynamics
In what follows, we will additionally assume that φ is bounded outside some ball in R d . Note that then (see e.g. [13]) for dx µ(dγ)-a.a. x ∈ R d and γ ∈ Γ and for µ-a.a. γ ∈ Γ and all x ∈ γ.
We fix a parameter s ∈ [0, 1/2]. We introduce the set We fix a bounded function a : . As we will see below, E G corresponds to a Glauber dynamics and E K corresponds to a Kawasaki dynamics. The next theorem follows from [13].
It follows from (5) that H G is (at least heuristically) the generator of a birth-and-death process, in which the factor exp[sE(x, γ \ x)] describes the rate at which particle x of the configuration γ dies, whereas the factor exp[(s − 1)E(x, γ)] describes the rate at which, given a configuration γ, a new particle is born at x. We see that particles tend to die in high energy regions, i.e., if E(x, γ \ x) is high, and they tend to be born al low energy regions, i.e., if E(x, γ) is low.
Next, by (6), H K is (again at least heuristically) the generator of a hopping particle dynamics, in which the factor describes the rate at which a particle x of configuration γ hops to y. We see that this rate is high if the relative energy of interaction between x and the rest of the configuration, γ \ x, is high, whereas the relative energy of interaction between y and γ \ x is low, i.e., particles tend to hop from high energy regions to low energy regions.

Scaling limit
In this section, we will show that the birth-and-death dynamics considered in Section 3 can be treated as a limiting dynamics of hopping particles. In other words, we will perform a scaling of Kawasaki dynamics which will lead to the Glauber dynamics. We will only discuss this convergence at the level of convergence of the generators on an appropriate set of cylinder functions. In fact, such a convergence implies weak convergence of finite-dimensional distributions of corresponding equilibrium processes if additionally the set of test functions forms a core for the limiting generator. However, in the general case, no core of this generator is yet known and this is an open, important problem, which we hope to return to in our future research. Our results will hold for all s ∈ [0, 1/2] (see Section 3). They will generalize Theorem 4.1 in [3], which was proved in the special case s = 0, and confirm the conjecture formulated in Section 6 of that paper.
So, let us consider the following scaling of the Kawasaki dynamics (for a fixed s ∈ [0, 1/2]). Recall that, for each bounded function a : R d → R such that a(x) ≥ 0, a ∈ L 1 (R d , dx), and a(−x) = a(x) for all x ∈ R d , we have constructed the corresponding generator of the Kawasaki dynamics. We now fix an arbitrary ε > 0 and define a function a ε : By the properties of the function a, we evidently have that the function a ε is also bounded, satisfies a ε (x) ≥ 0, for all x ∈ R d , a ε ∈ L 1 (R d , dx), and a ε (−x) = a ε (x) for all x ∈ R d . Hence, we can construct the Kawasaki generator which corresponds to the function a ε . It is convineant for us to denote this generator by (H ε , D(H ε )). We will also denote the generator of the Glauber dynamics by (H 0 , D(H 0 )). We first need the following lemma, whose proof is completely analogous to the proof of Lemma 4.1 in [3].
Lemma 2 For any ε ≥ 0 and any ϕ ∈ C 0 (R d ), the function F (γ) := e ϕ,γ belongs to D(H ε ) and the action of H ε on F is given by the right hand side of formula (6) for ε > 0 (with a replaced by a ε ), respectively by the right hand side of (5) for ε = 0.
We have So, we may rewrite the action of H ε for ε > 0 as follows: We can also rewrite Theorem 3 Let s ∈ [0, 1/2] be fixed. Assume that the pair potential φ and activity z > 0 satisfy conditions (S) and (LA-HT). Assume that φ is bounded outside some compact set in R d . Assume also that φ(x) → 0 as |x| → ∞.

Remark 4
In fact, condition (7) can be omitted, and instead one can use the fact that φ is an integrable function outside a compact set in R d (compare with [3]). However, in any reasonable application, the potential φ does satisfy condition (7).
Remark 5 Note that the integral on the right hand side of (8) is well defined and finite due to Lemma 1, see also Remark 1 Proof. We first need the following lemma, which generalizes Lemma 4.2 in [3].
Completely analogously, we get Using the Ruelle bound and Lemma 1, we conclude that the integral over Γ in the right hand side of equalities (17), (21)- (23), are bounded by a constant, which is indepeandent of ε. Therefore, by the dominated convergence theorem, to find the limit of (17), (21)-(23) as ε → 0 it suffices to find the point-wise limit of the functions appearing before the integral over Γ, as well as the limit of the integrals over Γ for fixed x (x 1 and x 2 respectively), y 1 and y 2 .