Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics

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1 Equilibrium Glauber ynamics of continuous particle systems as a scaling limit of Kawasaki ynamics Dmitri L. Finkelshtein Institute of Mathematics, National Acaemy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kiev 01601, Ukaine. e.mail: finkelshtein@gmail.com Yuri G. Konratiev Fakultät für Mathematik, Universität Bielefel, Postfach , D Bielefel, Germany; BiBoS, Univ. Bielefel, Germany; Kiev-Mohyla Acaemy, Kiev, Ukraine. konrat@mathematik.uni-bielefel.e Eugene W. Lytvynov Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K. e.lytvynov@swansea.ac.uk Abstract A Kawasaki ynamics in continuum is a ynamics of an infinite system of interacting particles in which ranomly hop over the space. In this paper, we eal with an equilibrium Kawasaki ynamics which has a Gibbs measure µ as invariant measure. We stuy a scaling limit of such a ynamics, where the scaling is of Kac type. Informally, we expect that, in the limit, only jumps of infinite length will survive, i.e., we expect to arrive at a Glauber ynamics in continuum a birth-an-eath process in. We prove that, in the low activity-high temperature regime, the generators of the Kawasaki ynamics converge to the generator of a Glauber ynamics. The convergence is on the set of exponential functions, in the L 2 µ- norm. Furthermore, aitionally assuming that the potential of pair interaction is positive, we prove the weak convergence of the finite-imensional istributions of the processes. MSC: 60K35, 60J75, 60J80, 82C21, 82C22 Keywors: Continuous system; Gibbs measure; Glauber ynamics; Kawasaki ynamics; Scaling limit 1 Introuction A Kawasaki ynamics in continuum is a ynamics of an infinite system of interacting particles in which ranomly hop over the space. In this paper, we eal with an equilibrium Kawasaki ynamics which has a Gibbs measure µ as invariant measure. 1

2 About µ we assume that it correspons to an activity parameter z > 0 an a potential of pair interaction φ. The generator of the Kawasaki ynamics is given, on an appropriate set of cyliner functions, by F γ = y ax y exp φu y x γ u γ\x F γ \ x y F γ, γ. 1.1 ere, enotes the configuration space over, i.e., the space of all locally finite subsets of, an, for simplicity of notations, we just write x instea of {x}. About the function a in 1.1 we assume that it is non-negative, integrable an symmetric with respect to the origin. The factor ax y exp [ ] u γ\x φu y in 1.1 escribes the rate with which, given a configuration γ, a particle x γ jumps to y. Uner very mil assumptions on the Gibbs measure µ, it was prove in [10] that there inee exists a Markov process on with calag paths whose generator is given by 1.1. We assume that the initial istribution of this ynamics is µ, an perform the following scaling of this ynamics. For each ε > 0, we consier the equilibrium Kawasaki ynamics whose generator is given by formula 1.1 in which a is replace by the function a ε := ε aε. 1.2 We enote this generator by ε, an stuy the limit of the corresponing ynamics as ε 0 Kac-type limit. Informally, we expect that, in the limit, only jumps of infinite length will survive, i.e., jumps from a point to infinity an from infinity to a point. Thus, we expect to arrive at a Glauber ynamics in continuum, i.e., a birth-an-eath process in, cf. [9, 10]. In fact, heuristic calculations show that the limiting Glauber ynamics has the generator where k 1 0 F γ = α x γf γ \ x F γ [ α z x exp ] φu x F γ x F γ, 1.3 u γ α = z 1 k µ 1 ax x, 1.4 µ being the first correlation function of the measure µ. Thus, α escribes the rate with which a particle x γ ies, whereas αz exp [ ] u γ φu x escribes the rate with which, given a configuration γ, a new particle is born at x \γ. The existence 2

3 of a Markov process on with calag paths, whose generator is given by 1.3, was prove in [9] see also [10]. The main results of this paper are as follows: For any stable potential φ in the low activity-high temperature regime, the generators ε converge to the generator 0. The convergence is on the set of exponential functions, in the L 2, µ-norm. For any positive potential φ in the low activity-high temperature regime, the finite-imensional istributions of the Kawasaki ynamics with generator ε an initial istribution µ weakly converge to the finite-imensional istributions of the Glauber ynamics with generator 0 an initial istribution µ. To prove the first main result, we essentially use the uelle boun on the correlation functions of the measure µ, as well as the integrability of the Ursell cluster functions of µ, prove by Brox [3]. To erive from here the convergence of the finite-imensional istributions of the ynamics, we aitionally nee that the set of finite sums of exponential functions forms a core for the generator of the limiting ynamics, 0. For this, we use a result from [9] on a core for 0, which hols uner the assumptions of positivity of the potential φ. We note that the generator of the Kawasaki ynamics is inepenent of the activity parameter z > 0. ence, at least heuristically, the Kawasaki ynamics has a continuum of symmetrizing Gibbs measures, inexe by the activity z > 0. On the other han, the limiting Glauber ynamics has only one of these measures as the symmetrizing one. Thus, the result of the scaling essentially epens on the initial istribution of the ynamics. In the case of no interaction between particles, φ = 0, it is also possible to prove the Kawasaki to Glauber convergence for a non-equilibrium ynamics whose initial istribution has Ursell functions ecaying at infinity, see [11] for etails. The paper is organize as follows. In Section 2, we recall some known facts about Gibbs measures on the configuration space. In Section 3, we recall a rigorous construction of the equilibrium Kawasaki an Glauber ynamics. Our two main results are prove in Sections 4 an 5, respectively. Finally, in Section 6, we make remarks on the results obtaine, an iscuss some relate open problems. 2 Gibbs measures in the low activity-high temperature regime The configuration space over, N, is efine by := {γ : γ Λ < for each compact Λ }, 3

4 where enotes the carinality of a set an γ Λ := γ Λ. One can ientify any γ with the positive aon measure x γ ε x M, where ε x is the Dirac measure with mass at x, x ε x :=zero measure, an M stans for the set of all positive aon measures on the Borel σ-algebra B. The space can be enowe with the relative topology as a subset of the space M with the vague topology, i.e., the weakest topology on with respect to which all maps γ f, γ := fxγx = fx, f C 0, x γ are continuous. ere, C 0 is the space of all continuous real-value functions on with compact support. We will enote by B the Borel σ-algebra on. We note that enowe with the vague topology is a Polish space, see e.g. [14]. A pair potential is a Borel-measurable function φ : {+ } such that φ x = φx for all x \ {0}. For γ an x \ γ, we efine a relative energy of interaction between a particle at x an the configuration γ as follows: φx y, if φx y < +, Ex, γ := y γ y γ +, otherwise. A probability measure µ on, B is calle a gran canonical Gibbs measure corresponing to the pair potential φ an activity z > 0 if it satisfies the Georgii Nguyen Zessin ientity [17, Theorem 2], see also [12, Theorem 2.2.4]: µγ γxf γ, x = µγ z x exp [ Ex, γ] F γ x, x 2.1 for any measurable function F : [0; + ]. We enote the set of all such measures µ by Gz, φ. Let us formulate conitions on the pair potential φ. S Stability There exists B 0 such that, for any γ, γ <, φx y B γ. {x,y} γ In particular, conition S implies that φx 2B, x. P Positivity We have φx 0, x. The conition P is stronger than S. More precisely, if P hols, then we can choose B = 0 in S. 4

5 LA-T Low activity-high temperature regime We have: e φx 1 z x < 2e 1+2B 1, where B is as in S. In particular, if P hols, then LA-T means: e φx 1 z x < 2e 1. Let µ Gz, φ. Assume that, for any n N, there exists a non-negative, measurable symmetric function k µ n on n such that, for any measurable symmetric function f n : n [0, + ] f n, : γ n : µγ = 1 n! f n x 1,..., x n k n n µ x 1,..., x n x 1 x n. ere f n, : γ n : := {x 1,...,x n} γ f n x 1,..., x n. The functions k n µ are calle correlation functions of the measure µ. If there exists a constant ξ > 0 such that x 1,..., x n n : k n µ x 1,..., x n ξ n, 2.2 then we say that the correlation functions k µ n satisfy the uelle boun. Uner the conitions S an LA-T, there exists a Gibbs measure µ Gz, φ which has correlation functions satisfying the uelle boun, see e.g. [20]. This measure µ is constructe as a weak limit of finite volume Gibbs measures with empty bounary conition, see [16] for etails. We will call this measure the Gibbs measure corresponing to z, φ an the construction with empty bounary conition. In what follows, we will always assume that S an LA-T are satisfie an the Gibbs measure µ as iscusse above is fixe. We note that, if the conition P is satisfie, then this measure µ is unique in the set Gz, φ, see [20] an [13, Theorem 6.2]. We also note that the relative energy Ex, γ is finite x µγ-a.e. on. Via a recursion formula, one can transform the correlation functions k µ n into the Ursell functions u n µ an vice versa, see e.g. [20]. Their relation is given by k µ η = u µ η 1 u µ η j, η 0, η, 2.3 where 0 := {γ : γ < }, 5

6 for any η = {x 1,..., x n } 0 k µ η := k n µ x 1,..., x n, u µ η := u n µ x 1,..., x n, an the summation in 2.3 is over all partitions of the set η into nonempty mutually isjoint subsets η 1,..., η j η such that η 1 η j = η, j N. For example, k µ 1 x = u 1 µ x, k µ 2 x 1, x 2 = u 2 µ x 1, x 2 + u 1 µ x 1 u 1 µ x 2. For our fixe Gibbs measure µ, both the correlation functions an the Ursell functions of µ are translation invariant. In particular, the first correlation function k µ 1 is a constant, which we enote by k µ 1. Furthermore, for any n N, where U n+1 µ L 1 n, x1 x n, 2.4 U n+1 µ x 1,..., x n := u n+1 µ x 1,..., x n, 0, x 1,..., x n n, 2.5 see [3, Theorem 4.5]. As a straightforwar corollary of the Georgii Nguyen Zessin ientity 2.1, we get the following equality: µγ γx 1 γx 2 F γ, x 1, x 2 = µγ z x 1 z x 2 exp [ Ex 1, γ Ex 2, γ φx 1 x 2 ] F γ {x 1, x 2 }, x 1, x 2 + µγ z x exp [ Ex, γ] F γ x, x, x 2.6 for any measurable function F : [0, + ]. Let f : be such that e f 1 L 1, x. Then, using the representation e e f,γ = 1 + f 1 n, : γ : n, we get e f,γ µγ = 1 + n=1 1 n! n=1 n e f 1 n x1,..., x n k n µ x 1,..., x n x 1 x n. 2.7 ence, by using the uelle boun, we conclue that e f, L 1, µ. Furthermore, if e 2f 1 L 1, x, then e f, L 2, µ. 6

7 3 Kawasaki an Glauber ynamics We introuce the set FC b C0, of all functions of the form γ F γ = g F ϕ 1, γ,..., ϕ N, γ, where N N, ϕ 1,..., ϕ N C 0, an g F C b, where C b enotes the set of all continuous boune functions on N. For each function F :, γ, an x, y, we enote D x F γ := F γ \ x F γ, D + xy F γ := F γ \ x y F γ. We fix a function a : [0, + such that a x = ax, x, an a L 1, x. We efine bilinear forms E ε F, G := 1 µγ γx y a ε x y 2 exp [ Ey, γ \ x] Dxy + F γ Dxy + G γ, ε > 0, E 0 F, G := α µγ γx Dx F γ Dx G γ, where F, G FC b C0,, a ε is efine by 1.2, an α is given by 1.4. The next theorem follows from [9, Proposition 3.1 an Theorem 3.1] an [10, Proposition 4.3]. Theorem 3.1. i For each ε 0, the bilinear form E ε, FC b C0, is closable on L 2, µ an its closure will be enote by E ε, Dom E ε. ii Denote by ε, Dom ε, ε 0, the generator of E ε, Dom E ε. Then FC b C0, ε 0 Dom ε, an for any F FC b C0, ε F γ = γx y a ε x y exp [ Ey, γ \ x] Dxy + F γ, ε > 0, F γ = α γx Dx F γ α z x exp [ Ex, γ] D x + F γ. 3.2 iii For each ε 0, there exists a conservative unt process M ε = Ω ε, F ε, F ε t t 0, Θ ε t t 0, X ε t t 0, P ε γ γ 7

8 on see e.g. [15, p. 92] which is properly associate with E ε, Dom E ε, i.e., for all µ-versions of F L 2, µ an all t > 0 the function γ p ε tf γ := F X ε tp ε γ is an E ε -quasi-continuous version of exp [ t ε ] F. M ε is up to µ-equivalence unique cf. [15, Chap. IV, Sect. 6]. In particular, M ε has µ as invariant measure. emark 3.1. In Theorem 3.1, M ε can be taken canonical, i.e., Ω ε is the set D[0, +, of all calag functions ω : [0, + i.e., ω is right continuous on [0, + an has left limits on 0, +, X ε tω = ωt, t 0, ω Ω ε, F ε t t 0 together with F ε is the correponing minimum complete amissible family cf. [6, Section 4.1] an Θ ε t, t 0, are the corresponing natural time shifts. Ω 4 Convergence of the generators We will now stuy the limiting behavior of the generators of the Kawasaki ynamics, ε, as ε 0. We start with the following Lemma 4.1. For any ε 0 an any ϕ C 0, the function F γ := e ϕ,γ belongs to Dom ε an the action of ε on F is given by formula 3.1 for ε > 0 an by 3.2 for ε = 0. Proof. We first note that since e 2ϕ 1 L 1, x, we have e ϕ, L 2, µ. Assume that ε > 0. For each n N, we efine g n C b by g n u = { e u, u n, e n, u > n. 4.1 Then g n ϕ, FC b C0,. Since g n ϕ, γ e ϕ,γ, γ, by the majorize convergence theorem, we have g n ϕ, e ϕ, in L 2, µ as n. 8

9 Next, by 2.6, 2.7, an the uelle boun, µγ γx y a ε x y exp [ Ey, γ \ x + ϕ, γ ] e ϕx+ϕy 1 = z x 1 y 1 z x 2 y 2 a ε x 1 y 1 a ε x 2 y 2 e ϕx 1 e ϕy1 1 + e ϕx1 1 e ϕx 2 e ϕy2 1 + e ϕx2 1 exp [ φx 1 x 2 φy 1 x 2 φy 2 x 1 ] [ µγ exp ] φu x 1 + φu x 2 + φu y 1 + φu y 2 u γ + z x y 1 y 2 a ε x y 1 a ε x y 2 e ϕx e ϕy1 1 + e ϕx 1 e ϕx e ϕy2 1 + e ϕx 1 [ µγ exp ] φu x + φu y 1 + φu y 2 u γ <. ere, we use the following estimate: for any x, y 1, y 2 e φu x φu y 1 φu y 2 1 u = e φu x φu y 1 e φu y e φu x e φu y e φu x 1 u e 4B + e 2B + 1 e φu 1 u <, where B is as in S, an an analogous estimate for the function e φu x 1 φu x 2 φu y 1 φu y 2 1. By using 4.1, we get, for any γ, x γ, an y : g n ϕ, γ \ x y g n ϕ, γ = g n ϕ, γ ϕx + ϕy g n ϕ, γ exp [max { ϕ, γ ϕx + ϕy, ϕ, γ }] ϕx + ϕy exp [ ϕ, γ + ϕx + ϕy ] ϕx + ϕy, 9

10 an, hence, for any γ, γx y a ε x y exp [ Ey, γ \ x]dxy + g n ϕ, γ γx y a ε x y exp [ Ey, γ \ x + ϕ, γ + ϕx + ϕy ] ϕx + ϕy. 4.3 Analogously to 4.2, we conclue that the right han sie of 4.3, as a function of γ, belongs to L 2, µ. Therefore, by the majorize convergence theorem, γx y a ε x y exp [ Ey, γ \ x]dxy + g n ϕ, γ γx y a ε x y exp [ Ey, γ \ x]dxy + e ϕ,γ in L 2, µ as n. From here, the statement of the lemma follows in the case ε > 0. The case ε = 0 can be treate analogously. We may rewrite ε = ε + + ε, ε 0, where ε F γ = γx Dx F γ y e Ey,γ\x a ε x y, + ε F γ = y γx e Ey,γ\x a ε x y[f γ \ x y F γ \ x], 0 F γ = α γx Dx F γ, + 0 F γ = α y exp [ Ey, γ] D y + F γ. Theorem 4.1. Assume that the pair potential φ an activity z > 0 satisfy the conitions S an LA-T. Let µ be the Gibbs measure from Gz, φ which correspons to the construction with empty bounary conition. Then, for any ϕ C 0, so that ± ε e ϕ, ± 0 e ϕ, in L 2, µ as ε 0, ε e ϕ, 0 e ϕ, in L 2, µ as ε 0, Proof. We fix any ϕ C 0 an enote F γ := e ϕ,γ. We nee to prove that ± ε F 2 γ µγ ± 0 F 2 γ µγ as ε 0, 4.4 ± ε F γ 0 ± F γ µγ ± 0 F 2 γ µγ as ε

11 Using 2.1 an 2.6, we get 0 F 2 γ µγ an = α 2 z x e ϕx 1 2 µγ exp [ Ex, γ + 2ϕ, γ ] + α 2 z x z x e ϕx 1 e ϕx e ϕx 1 e ϕx e φx x µγ exp [ Ex, γ Ex, γ + 2ϕ, γ ] F 2 γ µγ = α 2 z x z x e ϕx 1 e ϕx 1 µγ exp [ Ex, γ Ex, γ + 2ϕ, γ ]. 4.7 Using the same arguments an changing the variables, we have ε F 2 γ µγ = z x y y e ϕx 1 2 aε x ya ε x y µγ exp [ Ex, γ Ey, γ Ey, γ + 2ϕ, γ ] + z x z x y y e ϕx 1 e ϕx e ϕx 1 e ϕx a ε x ya ε x y e φx x e φx y e φy x µγ exp [ Ex, γ Ex, γ Ey, γ Ey, γ + 2ϕ, γ ] = z x y y e ϕx 1 2 ayay [ y y µγ exp Ex, γ E ε + x, γ E ε + x, γ + z x z x y y e ϕx 1 e ϕx e ϕx 1 e ϕx ayay e φx x e φ y ε +x x e φ y ε +x x ] + 2ϕ, γ 11

12 [ µγ exp Ex, γ Ex, γ y ] y E ε + x, γ E ε + x, γ + 2ϕ, γ. 4.8 Analogously, + ε F 2 γ µγ = z x y y e ϕx 1 [ µγ exp E x y ε, γ + z x z x y e ϕ y y +x ε 1 ayay y y Ex, γ E + x, γ ε y e ϕy 1 e ϕ x ε +y e ϕy 1 y x ε y ] + 2ϕ, γ e ϕ x ε +y axax e φ x x +y y ε e φ x ε +y y φ e [ x µγ exp E ε + y, γ ] x E ε + y, γ Ey, γ Ey, γ + 2ϕ, γ. 4.9 In the same way, ε F γ 0 F γ µγ = α z x y e ϕx 1 2 ay [ µγ exp Ex, γ E + α z x z x y ε + x, γ ] + 2ϕ, γ e ϕx 1 e ϕx y e ϕx 1 e ϕx aye φx x e φ y ε +x x [ y µγ exp Ex, γ Ex, γ E ε + x, γ an finally, + ε F γ 0 + F γ µγ ] + 2ϕ, γ, 4.10 = α z x y y e ϕy 1 e ϕy 1 e ϕ x +y ε e φ x ε +y y ax 12

13 [ x ] µγ exp E ε + y, γ Ey, γ Ey, γ + 2ϕ, γ Since the functions e ϕ 1, e φ 1 are boune an integrable on, from the uelle boun an 2.7 we conclue that all the integrals over in the right han sies of the equalities are boune by constants. Therefore, by the majorize convergence theorem, to fin the limit of these expressions as ε 0, it suffices to fin the limit of the corresponing integrals over for fixe variables x, x, y, y. Therefore, ue to 4.6 an 4.7, formulas 4.4, 4.5 will immeiately follow from the following Lemma 4.2. Let a function ψ : be such that e ψ 1 is boune an integrable. Suppose that x, y, x, y an x y. Then [ x y ] exp E ε + x, γ E ε + y, γ + ψ, γ µγ as ε 0. k 1 µ z 2 2 exp [ ψ, γ ] µγ, 4.12 [ x ] exp E ε + x, γ + ψ, γ µγ k1 µ exp [ ψ, γ ] µγ z 4.13 Proof. By 2.7, [ x y ] exp E ε + x, γ E ε + y, γ + ψ, γ µγ = 1 + n=1 1 n! exp [ φ xε φ yε + ψ ] 1 n u 1,..., u n n k n µ u 1,..., u n u 1 u n, 4.14 where xε := x + ε x, yε := y + ε y. Using the uelle boun, S, an LA-T, we conclue that, in orer to fin the limit of the right han sie of 4.14 as ε 0, it suffices to fin the limit of each term C ε n : = exp [ φ xε φ yε + ψ ] 1 n u 1,..., u n n = n 1 +n 2 +n 3 =n n n 1 n 2 n 3 k µ n u 1,..., u n u 1 u n f n 1 n 1,ε f n 2 2,ε f n 3 3,ε u1,..., u n 13 k n µ u 1,..., u n u 1 u n, 4.15

14 where f 1,ε u := exp [ ψu] 1 exp [ φu xε φu yε], f 2,ε u := exp [ φu xε] 1 exp [ φu yε], f 3,ε u := exp [ φu yε] 1, u. Using 2.3, we see that f n 1 n 1,ε f n 2 2,ε f n 3 3,ε u1,..., u n k µ n u 1,..., u n u 1 u n = f n 1 n 1,ε f n 2 2,ε f n 3 3,ε u1,..., u n u µ η 1 u µ η j u 1 u n, where the summation is over all partitions of η = {u 1,..., u n }. We now have to istinguish the three following cases. Case 1: Each element η i of the partition is either a subset of {u 1,..., u n1 }, or a subset of {u n1 +1,..., u n1 +n 2 }, or a subset of {u n1 +n 2 +1,..., u n }. By using the translation invariance of the Ursell functions, we get that the corresponing term is equal to f n 1 n 1,ε g n 2 2,ε g n 3 3,ε u1,..., u n u µ η 1 u µ η j u 1 u n, 4.16 where [ g 2,ε u := exp [ φu] 1 exp φ u + x y + x y ], ε g 3,ε u := exp [ φu] 1, u. Since e φ 1 L 1, x, for each δ > 0 there exists r > 0 such that I x r, φx δ x < δ, 4.17 where I enotes the inicator function. ence, by the majorize convergence theorem, since x y, 4.16 converges to exp [ψ ] 1 n1 n exp [ φ ] 1 n 2+n 3 u 1,..., u n u µ η 1... u µ η j u 1 u n. Case 2: There is an element of the partition which has non-empty intersections with both sets {u 1,..., u n1 } an {u n1 +1,..., u n }. 14

15 By using 2.4, 2.5, 4.17, an the majorize convergence theorem, we conclue that the term converges to zero as ε 0. Case 3: Case 2 is not satisfie, but there is an element η l of the partition which has non-empty intersections with both sets {u n1 +1,..., u n1 +n 2 } an {u n1 +n 2 +1,..., u n }. Shift all the variables entering η l by yε. Now, analogously to Case 2, the term converges to zero as ε 0. Thus, again using 2.3, for each n N, C n ε n 1 +n 2 +n 3 =n n 1 n 2 n 3 n n 1 n 2 n 3 exp [ψ ] 1 n 1 u 1,..., u n1 k n 1 µ u 1,..., u n1 u 1 u n1 exp [ φ ] 1 n 2 u n1 +1,..., u n1 +n 2 k n 2 µ u n1 +1,..., u n1 +n 2 u n1 +1 u n1 +n 2 exp [ φ ] 1 n 3 u n1 +n 2 +1,..., u n k n 3 µ u n1 +n 2 +1,..., u n u n1 +n 2 +1 u n. Therefore, the right han sie of 4.14 converges to [ exp ] 2 φu µγ µγ exp [ ψ, γ ]. u γ Let f C 0, f 0, f 0. Then k µ 1 fx x = = µγ γxfx [ µγ z x exp ] φu x u γ = z x fx = z x fx µγ exp µγ exp [ [ fx u γ φu x u γ φu ]. ] 15

16 ence, exp [ u γ φu ] µγ = k1 µ z, which proves The proof of 4.13 is analogous. 5 Convergence of the processes For each ε 0, we take the canonical version of the process M ε from Theorem 3.1, an efine a stochastic process Y ε = Y ε t t 0 whose law is the probability measure on D[0, +, given by Q ε := P ε γ µγ. By virtue of Theorem 3.1, the process Y ε has µ as invariant measure. Theorem 5.1. Let P an LA-T, be satisfie. Then the finite-imensional istributions of the process Y ε weakly converge to the finite-imensional istributions of Y 0 as ε 0. Proof. Fix any 0 t 1 < t 2 < < t n, n N. For ε 0, enote by µ ε t 1,...,t n the finiteimensional istribution of the process Y ε at times t 1,..., t n, which is a probability measure on n. Since is a Polish space, by [18, Chapter II, Theorem 3.2], the measure µ is tight on. Since all the marginal istributions of the measure µ ε t 1,...,t n are µ, we therefore conclue that the set {µ ε t 1,...,t n ε > 0} is pre-compact in the space M n of the probability measures on n with respect to the weak topology, see e.g. [18, Chapter II, Section 6]. ence, by [5, Chapter 3, Theorem 3.17], the statement of the theorem will follow from Theorem 4.1 if we show that the set of all finite linear combinations of the functions of the form e ϕ,, ϕ C 0, constitutes a core of 0, Dom 0. Uner the assumptions of the theorem, it follows from the proof of [9, Theorem 4.1] that the set of all finite sums of the functions of the form n i=1 f i,, f i C 0, i = 1,..., n, an constants forms a core of 0, Dom 0. Therefore, by the polarization ientity see e.g. [1, Chapter 2, formula 2.7] the set of all finite linear combinations of the functions of the form f, n, f C 0, n = 0, 1, 2,..., forms a core of 0, Dom 0. ence, to prove the theorem, it suffices to show that, for each n N an f C 0, the function f, n can be approximate by finite linear combinations of functions e ϕ,, ϕ C 0, in the graph norm of the operator 0, Dom 0. This statement will follows from the two following lemmas. Lemma 5.1. For any ϕ, ψ C 0 an n N, ψ, n e ϕ, Dom 0. Proof. The proof of this lemma is absolutely analogous to the proof of Lemma

17 Lemma 5.2. For any ϕ C 0, t, an n = 0, 1, 2,..., ϕ, n 1 e t+u ϕ, e t ϕ, ϕ, n+1 e t ϕ, as u 0, 5.1 u where convergence is in the sense of the graph norm of the operator 0, Dom 0. Proof. Using the estimate 1 e u ϕ,γ 1 e ϕ,γ ϕ, γ, u 1, ϕ C 0, u an the majorize convergence theorem, we easily get the convergence 5.1 in L 2, µ. Next, we have the estimate, for u 1 γx Dx ϕ, n 1 e t+u ϕ, e t ϕ, γ u [ = γx ϕ, γ \ x n ϕ, γ n 1 e t+u ϕ,γ\x e t ϕ,γ\x u + ϕ, γ n e t ϕ,γ\x e t ϕ,γ 1 e u ϕ,γ\x 1 u γx [ n 1 + ϕ, γ n e t ϕ,γ 1 u e u ϕ,γ\x e u ϕ,γ ] ϕ, γ k ϕx n k e t +1 ϕ,γ ϕ, γ k=0 + ϕ, γ n e t ϕ,γ t ϕx e ϕ,γ ϕ, γ ] + ϕ, γ n e t ϕ,γ e ϕ,γ ϕx Therefore, by the majorize convergence theorem γx Dx ϕ, n 1 e t+u ϕ, e t ϕ, γ u γx Dx ϕ, n+1 e t ϕ, γ as u 0 in L 2, µ. Finally, noticing that exp [ Ex, γ] 1 ue to P, we conclue, analogously to the above, that z x exp [ Ex, γ] D x + ϕ, n 1 e t+u ϕ, e t ϕ, γ u z x exp [ Ex, γ] D x + ϕ, n+1 e t ϕ, γ as u 0 in L 2, µ. From here, the statement of the lemma follows. 17.

18 Using the inuction in n = 0, 1, 2,..., we conclue from Lemma 5.2 that any function of the form f, n+1 e t f,, f C 0, may be approximate by finite linear combinations of the functions e ϕ,, ϕ C 0, in the graph norm of 0, Dom 0. Letting t = 0, we get the neee statement. 6 Concluing remarks an open problems It is known cf. [19] that any Gibbs measure of uelle type, corresponing to a superstable potential φ see [21], satisfies the generalize uelle boun: [ k µ x 1,..., x n C n exp ] φx i x j i<j n for some C > 0 inepenent of n N compare with the usual uelle boun 2.2. Using 6.1 an harmonic analysis on the configuration space cf. [8], it is possible to erive the convergence of the generators, as in Theorem 4.1, uner the following assumption on a ecay of correlations of the measure µ: For each n N an for x 1 x n -a.e. x 1,..., x n n, u n+1 µ x 1,..., x n, y 0, ε u n+2 µ x 1,..., x n, yε, y ε 0 as ε 0, 6.2 where the convergence is in the y y -measure on each compact set in 2. By 2.4, 2.5, this assumption is satisfie in the low activity-high temperature regime. It is sill an open problem whether other Gibbs measures of uelle type satisfy 6.2. We believe that 6.2 inee hols for any Gibbs measure of uelle type which is a pure phase. Note that, if this were so, we woul erive the convergence of the processes, as in Theorem 5.1, assuming aitionally P. Next, let us consier the following generalization of the Kawasaki an Glauber ynamics. For any s [0, 1], let us efine bilinear forms Eε s F, G := 1 µγ γx y c s 2 εx, y, γ \ x Dxy + F γ Dxy + G γ, ε > 0, E0F, s G := α µγ γxc s 0x, γ \ x Dx F γ Dx G γ, where F, G FC b C0, an c s εx, y, γ := a ε x y exp [sex, γ 1 sey, γ], c s 0x, γ := exp [sex, γ]. 18

19 In particular, for s = 0, E 0 ε = E ε an E 0 0 = E 0. By [10], each of these bilinear forms leas to an equilibrium Markov processes on, which is a Kawasaki ynamics for ε > 0, an a Glauber ynamics for ε = 0. Uner the same assumptions as in Theorem 4.1, it can prove that, for any F γ := e ϕ,γ with ϕ C 0, E s ε F, F E s 0F, F as ε 0. The iea of proof is the same as before, we only use the secon statement of Lemma 4.2. Moreover, we expect that uner an aitional assumption if s 1/2, 1] an analog of Theorem 4.1 is also true in this case. owever, to erive from here the convergence of the corresponing processes, as in Theorem 5.1, we are still missing a theorem on a core for the generator of the closure of the bilinear form E0. s Another version of our results may be applie to the ynamics consiere in [2, Section 5]. There, the Kawasaki ynamics corresponing to the following bilinear form was stuie: E N Λ F, G := 1 2 Λ N Λ µ N Λ γ γx y Dxy + F γ Dxy + G γ. 6.3 Λ Λ ere, Λ is a measurable boune omain in, by Λ we enote the volume of Λ, N Λ is the set of all N-point subsets of Λ an µn Λ is the canonical Gibbs measure on N Λ corresponing to the potential φ see [2] for etail, an note that we have chosen empty bounary conition. Now, let Λ, N an N ρ = const the so-calle N/V limit. Then, by Λ [7], the measures µ N Λ have a limiting point in the weak topology of probability measures on. This limiting point is a gran canonical Gibbs measure µ corresponing to the potential φ. A heuristic calculation shows that the Kawasaki ynamics corresponing to 6.3 converges to the Glauber ynamics with equilibrium measure µ. In this way, we obtain an N/V -approximation of the Glauber ynamics on. It was shown in [2] that, uner P, the generator of 6.3 has a spectral gap which is 3N e φx x, Λ provie that the above value is positive. ence, we shoul expect that the generator of the limiting Glauber ynamics has a spectral gap which is 1 3ρ 1 e φx x. This can be compare with the lower boun of the spectral gap 1 z 1 e φx x, 19

20 obtaine in [9]. There is still an open problem whether an equilibrium Glauber ynamics in infinite volume can have a spectral gap if the pair potential φ has a negative part. We hope that the finite-volume approximations as iscusse above shoul give an insight into this problem. It shoul also be possible to approximate the Glauber ynamics in infinite volume by Glauber ynamics in a finite volume which have a gran canonical Gibbs measure as invariant measure. There is another interesting open problem in this irection: approximation of the Kawasaki ynamics in infinite volume by finite-volume Kawasaki ynamics. Consier e.g. the bilinear form E 1 as in Section 3. This form can be approximate by the forms E Λ F, F = 1 µ Λ γ γx y ax y exp [ Ey, γ \ x] Dxy + F γ 2. 2 Λ Λ Λ ere, Λ is the set of all finite subsets of Λ an µ Λ is the gran canonical Gibbs measure on Λ corresponing to φ an empty bounary conition. The problems are: 1 prove that the generator of E Λ has a spectral gap, 2 estimate how quickly it shrinks as Λ. Note that problems of such type have been stuie in the lattice case, see e.g. [4]. Acknowlegements The authors acknowlege the financial support of the SFB 701 Spectral structures an topological methos in mathematics, Bielefel University. eferences [1] Yu. M. Berezansky an Yu. G. Konratiev, Spectral Methos in Infinite Dimensional Analysis, Kluwer Aca. Publ., Dorrecht/Boston/Lonon, [2] A.-S. Bouou, P. Caputo, P. Dai Pra, G. Posta, Spectral gap estimates for interacting particle systems via a Bochner-type ientity, Preprint, 2005, available at [3] T. Brox, Gibbsgleichgewichtsfluktuationen für einige Potentiallimiten, PhD thesis, Universität eielberg, [4] N. Cancrini an F. Martinelli, On the spectral gap of Kawasaki ynamics uner a mixing conition revisite, J. Math. Phys , [5] E. B. Davies, One-Parameter Semigroups, Acaemic Press, Lonon,

21 [6] M. Fukushima, Dirichlet Forms an Symmetric Markov Processes, North- ollan, Amsteram/New York, [7] M. Grotahaus, Yu. G. Konratiev, an M. öckner, N/V-limit for stochastic ynamics in continuous particle systems, to appear in Probab. Theory elate Fiels. [8] Yu. G. Konratiev an T. Kuna, armonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. elat. Top , [9] Yu. G. Konratiev an E. Lytvynov, Glauber ynamics of continuous particle systems, Ann. Inst.. Poincaré Probab. Statist , [10] Yu. G. Konratiev, E. Lytvynov, an M. öckner, Equilibrium Glauber an Kawasaki ynamics of continuous particle systems, Preprint, 2005, available at [11] Yu. G. Konratiev, E. Lytvynov, an M. öckner, Non-equilibrium stochastic ynamics in continuum: The free case, in preparation. [12] T. Kuna, Stuies in Configuration Space Analysis an Applications, Ph.D. thesis, Bonn University, [13] T. Kuna, Properties of marke Gibbs measures in high temperature regime, Methos Funct. Anal. Topology , no. 3, [14] K. Matthes, J. Kerstan, an J. Mecke, Infinitely Divisible Point Processes, John Wiley & Sons, Chichester/New York/Brisbane, [15] Z.-M. Ma an M. öckner, An Introuction to the Theory of Non-Symmetric Dirichlet Forms, Springer-Verlag, Berlin, [16]. A. Minlos, Gibbs limit istribution, Funktsional nyj Analiz i Ego Prilozhenija, , no. 2, [17] X.X. Nguyen an. Zessin, Integral an ifferentiable characterizations of the Gibbs process, Math. Nachr , [18] K.. Parthasarathy, Probability Measures on Metric Spaces, Acaemic Press, New York/Lonon, [19] A. L. ebenko, A new proof of uelle s superstability bouns, J. Statist. Phys , [20] D. uelle, Statistical Mechanics. igorous esults, Benjamins, New York/Amsteram,

22 [21] D. uelle, Superstable interaction in classical statistical mechanics, Comm. Math. Phys ,