Non-equilibrium stochastic dynamics in continuum: The free case

We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main probl...

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Дата:	2008
Автори:	Kondratiev, Y., Lytvynov, E., Röckner, M.
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Опубліковано:	Інститут фізики конденсованих систем НАН України 2008
Назва видання:	Condensed Matter Physics
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Цитувати:	Non-equilibrium stochastic dynamics in continuum: The free case / Y. Kondratiev, E. Lytvynov, M. Röckner // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 701-721. — Бібліогр.: 25 назв. — англ.

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spelling	irk-123456789-1195932017-06-08T03:03:44Z Non-equilibrium stochastic dynamics in continuum: The free case Kondratiev, Y. Lytvynov, E. Röckner, M. We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t > 0. We assume that X has infinite volume and, for each α ≥ 1, we consider the set θα of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from θα, will never leave θα, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in X), and free Kawasaki dynamics on the configuration space. We also show that if X = Rd, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics. Ми дослiджуємо проблему iдентифiкацiї вiдповiдного простору станiв для стохастичної динамiки вiльних частинок у континуумi з їх можливим народженням i знищенням. В цiй динамiцi рух окремої частинки описується за допомогою фiксованого маркiвського процесу M на рiмановому многовидi X. Головною проблемою тут є можливий колапс системи у наступному сенсi. Незважаючи на те, що початковий розподiл частинок є локально скiнченний, може iснувати в X така компактна множина, що з ймовiрнiстю 1 в момент часу t > 0 у цю множину потрапить безмежна кiлькiсть частинок. Ми вважаємо, що X має безмежний об’єм, а також, для кожного α ≥ 1, розглядаємо множину θα всiх безмежних конфiгурацiй в X, для яких число частинок в компактнiй множинi є обмежене добутком певної сталої i -го степеня об’єму цiєї множини. Ми знайшли цiлком загальнi умови на процес M, за яких вiдповiдний безмежно-частинковий процес, стартуючи з довiльної конфiгурацiї θα, нiколи не залишить θα, маючи при цьому cadlag (або, навiть, неперервнi) траєкторiї в ультра-слабкiй топологiї. Можливi такi застосування наших результатiв: броунiвський рух на конфiгурацiйнному просторi i вiльна динамiка Глаубера на конфiгурацiйному просторi (процес народження-знищення на X): вiльна динамiка Кавасакi на конфiгурацiйному просторi. Ми також показуємо, що у випадку X = Rd, для широкого класу стартових розкладiв (нерiвноважна) вiльна динамiка Глаубера є скейлiнговою границею (нерiвноважної) вiльної динамiки Кавасакi. 2008 Article Non-equilibrium stochastic dynamics in continuum: The free case / Y. Kondratiev, E. Lytvynov, M. Röckner // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 701-721. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 02.50.Ey, 02.50.Ga DOI:10.5488/CMP.11.4.701 http://dspace.nbuv.gov.ua/handle/123456789/119593 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t > 0. We assume that X has infinite volume and, for each α ≥ 1, we consider the set θα of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from θα, will never leave θα, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in X), and free Kawasaki dynamics on the configuration space. We also show that if X = Rd, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.
format	Article
author	Kondratiev, Y. Lytvynov, E. Röckner, M.
spellingShingle	Kondratiev, Y. Lytvynov, E. Röckner, M. Non-equilibrium stochastic dynamics in continuum: The free case Condensed Matter Physics
author_facet	Kondratiev, Y. Lytvynov, E. Röckner, M.
author_sort	Kondratiev, Y.
title	Non-equilibrium stochastic dynamics in continuum: The free case
title_short	Non-equilibrium stochastic dynamics in continuum: The free case
title_full	Non-equilibrium stochastic dynamics in continuum: The free case
title_fullStr	Non-equilibrium stochastic dynamics in continuum: The free case
title_full_unstemmed	Non-equilibrium stochastic dynamics in continuum: The free case
title_sort	non-equilibrium stochastic dynamics in continuum: the free case
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/119593
citation_txt	Non-equilibrium stochastic dynamics in continuum: The free case / Y. Kondratiev, E. Lytvynov, M. Röckner // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 701-721. — Бібліогр.: 25 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT kondratievy nonequilibriumstochasticdynamicsincontinuumthefreecase AT lytvynove nonequilibriumstochasticdynamicsincontinuumthefreecase AT rocknerm nonequilibriumstochasticdynamicsincontinuumthefreecase
first_indexed	2025-07-08T16:14:07Z
last_indexed	2025-07-08T16:14:07Z
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fulltext	Condensed Matter Physics 2008, Vol. 11, No 4(56), pp. 701–721 Non-equilibrium stochastic dynamics in continuum: The free case Y.Kondratiev1,2,3∗, E.Lytvynov4†, M.Röckner1,3‡ 1 Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D–33501 Bielefeld, Germany 2 Institute of Mathematics, Kiev, Ukraine 3 BiBoS, Univ. Bielefeld, Germany 4 University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K. Received January 31, 2008 We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t > 0. We assume that X has infinite volume and, for each α > 1, we consider the set Θα of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from Θα, will never leave Θα, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in X), and free Kawasaki dynamics on the configuration space. We also show that if X = Rd, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics. Key words: birth and death process, Brownian motion on the configuration space, continuous system, Glauber dynamics, independent infinite particle process, Kawasaki dynamics, Poisson measure PACS: 02.50.Ey, 02.50.Ga 1. Introduction In this paper, we study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X . A classical result by J.L. Doob [8] states that, if the initial distribution of free particles is Poissonian, then it will remain Poissonian at any moment of time t > 0, see also [7]. In [23,24], D. Surgails studied an independent infinite particle process as a Markov process whose generator is the second quantization of the generator of a Markov process in X . Following Surgailis’ papers, equilibrium independent infinite particle processes have been studied by many authors, see e.g. [22]. However, the problem of identification of the allowed initial configurations of the system was not addressed in these papers. Let us explain this problem, in more detail. When speaking about an infinite system of particles in continuum, we should consider such a system as an element of the configuration space Γ over X . This space is defined as the collection of all locally finite subsets of X . Now, consider, for example, a system of independent Brownian particles in Rd. Then, one can easily find a configuration γ ∈ Γ ∗E-mail: kondrat@mathematik.uni-bielefeld.de †E-mail: e.lytvynov@swansea.ac.uk ‡E-mail: roeckner@mathematik.uni-bielefeld.de c© Y.Kondratiev, E.Lytvynov, M.Röckner 701 Y.Kondratiev, E.Lytvynov, M.Röckner such that, if γ is the initial configuration of Brownian particles, then at some time t > 0, with probability one, the system will collapse, in the sense that there will be an infinite number of particles in a compact set, i.e., the system will not be a configuration anymore. Thus, generally speaking, the configuration space Γ appears to be too big and cannot serve as a state-space for the process. Thus, to study the non-equilibrium stochastic dynamics independent Markovian particles, one needs to identify a subset Θ of Γ such that the process starting at Θ will always remain in Θ with probability one. (We note, however, that, for a fixed independent infinite particle process, such a set Θ is not uniquely defined.) Next, if the underlying Markov process M has cadlag paths, then it is only natural to expect that the corresponding infinite particle process also has cadlag sample paths in Θ with respect to the vague topology. Again, this problem was not addressed in the above-mentioned papers. In our previous paper [13], we considered the case where M is a Brownian motion in a com- plete, connected, oriented, stochastically complete Riemannian manifold X of dimension > 2. We explicitly constructed a subset Γ∞ of the configuration space and proved that the corresponding infinite particle process can start at any γ ∈ Γ∞, will never leave Γ∞, and has continuous sam- ple paths in the vague topology (and even in a stronger one). In the case of a one-dimensional underlying manifold X , one cannot exclude collisions of particles, so that a modification of the construction of Γ∞ is necessary, see [13] for details. As we mentioned above, the aim of this paper is to consider the case of general Markovian particles. The interest in such particles, rather than just independent Brownian motions, is, in particular, connected with the study of the Glauber and Kawasaki dynamics on the configuration space, see [3,5,12,14]. For simpler notations, we assume that the underlying process M is symmetric. However, this condition can be easily omitted. Instead of trying to generalize (quite complicated) arguments of [13], we propose a new, simpler approach to the construction of a state-space of the system (which was Γ∞ in [13]), and to the proof that the process indeed has got the above discussed properties. So, we fix a system of closed balls B(r) of radius r ∈ N, centered at some x0 ∈ X , and for each α > 1, define Θα as the set of those infinite configurations for which the number of particles in each B(r) is bounded by a constant times the α-th power of the volume of B(r). The Θα’s form an isotone sequence of sets, and we also define Θ as the union of all the Θα’s. Note that the sets Θα are “big enough”, in the sense that already the smallest set Θ1 is of full Poisson measure with any intensity parameter z > 0. We find quite general conditions on a Markov process M which guarantee that the corresponding stochastic dynamics can start at each γ ∈ Θα (or even at each γ ∈ Θ), will never leave Θα (respectively Θ), and has cadlag (or, even, continuous) sample paths. We then consider the following examples of applicability of our results: • Brownian motion on the configuration space, i.e., the case where the underlying process M is a Brownian motion on X . Compared with [13], our result here also covers the case of manifolds which are not stochastically complete; • Free Glauber dynamics on the configuration space, or a birth-and-death process in X , com- pare with [5,12,14]; • Free Kawasaki dynamics on the configuration space, cf. [14]. We also show that, in the case where X = Rd, for a probability measure µ on Θ which is trans- lation invariant and has integrable Ursell functions, the (non-equilibrium) free Glauber dynamics having µ as initial distribution may be approximated by (non-equilibrium) Kawasaki dynamics. Note that, in the case of equilibrium dynamics, such an approximation can also be shown for interacting particles, see [10]. The paper is organized as follows. In section 2, we construct a Markov semigroup of kernels on the space Θα and prove that it corresponds to the second quantization of the (sub-)Markovian semigroup of the underlying process M . In section 3 we derive conditions which 702 Non-equilibrium stochastic dynamics in continuum guarantee that the corresponding infinite particle process can be realized as a Markov process on Θα with cadlag (respectively, continuous) sample paths. In section 4, we discuss the above menti- oned examples. Finally, in section 5, we prove the result on approximation of the Glauber dynamics by the Kawasaki dynamics. We would like to stress that this paper should be considered as a first step towards a construc- tion of a non-equilibrium dynamics of interacting particles, compare with [2,12,14,16], where the corresponding equilibrium processes were discussed. 2. Markov semigroup of kernels for the stochastic dynamics Let X be a complete, connected, oriented C∞ Riemannian manifold. Let B(X) denote the Borel σ-algebra on X . Let dx denote the volume measure on X , and we suppose that ∫ X dx = ∞. The configuration space Γ over X is defined as the set of all infinite subsets of X which are locally finite: Γ := {γ ⊂ X \| \|γ\| = ∞, \|γΛ\| < ∞ for each compact Λ ⊂ X}. Here, \| · \| denotes the cardinality of a set and γΛ := γ ∩ Λ. One can identify any γ ∈ Γ with the positive Radon measure ∑ x∈γ εx ∈ M(X). Here, εx denotes the Dirac measure with mass at x and M(X) stands for the set of all positive Radon measures on B(X). The space Γ can be endowed with the relative topology as a subset of the space M(X) with the vague topology, i.e., the weakest topology on Γ with respect to which all maps Γ 3 γ 7→ 〈ϕ, γ〉 := ∫ X ϕ(x) γ(dx) = ∑ x∈γ ϕ(x), ϕ ∈ C0(X), are continuous. Here, C0(X) denotes the set of all continuous functions on X with compact support. We shall denote the Borel σ-algebra on Γ by B(Γ). If Ξ is a subset of Γ, we shall denote by B(Ξ) the trace σ-algebra of B(Γ) on Ξ. Let us fix any x0 ∈ X and denote by B(r) := B(x0, r) the closed ball in X of radius r > 0, centered at x0. For each α > 1, we define Θα := { γ ∈ Γ \| ∃K ∈ N : ∀r ∈ N : \|γB(r)\| 6 K vol(B(r))α } , where vol(B(r)) denotes the volume of B(r). We evidently have that Θα1 ⊂ Θα2 if α2 > α1 > 1. Denote also Θ := ⋃ α>1 Θα . It easy to see that Θα ∈ B(Γ) for each α > 1, hence Θ ∈ B(Γ). For each z > 0, let πz denote the Poisson measure on (Γ,B(Γ)) with intensity measure z dx. This measure can be characterized by its Laplace transform ∫ Γ e〈ϕ,γ〉 πz(dγ) = exp (∫ X (eϕ(x) − 1) z dx ) , ϕ ∈ C0(X). (1) We refer to e.g. [15] for a detailed discussion of the construction of the Poisson measure on the configuration space. By e.g. [19], we have, for each z > 0, πz(Θ1) = 1 (in fact, by [19], the Poisson measure πz is concentrated on those configurations γ ∈ Γ for which limr→∞ \|γB(r)\|/ vol(B(r)) = z). Using multiple stochastic integrals with respect to the Poisson random measure, one constructs a unitary isomorphism Iz : F(L2(X, z dx)) → L2(Γ, πz), 703 Y.Kondratiev, E.Lytvynov, M.Röckner see e.g. [23]. Here, F(L2(X, z dx)) denotes the symmetric Fock space over L2(X, z dx), i.e., F(L2(X, z dx)) = ∞ ⊕ n=0 Fn(L2(X, z dx)), where Fn(L2(X, x dx)) := L2(X, z dx)�nn! , � standing for symmetric tensor product. Let us recall the following result of Surgailis [23]. Let A be a contraction in L2(X, z dx), and let Exp(A) denote the second quantization of A, i.e., Exp(A) is the contraction in F(L2(X, z dx)) given by Exp(A) � F0(L 2(X, z dx)) = 111, Exp(A)f⊗n = (Af)⊗n, f ∈ L2(X, z dx), n ∈ N. We shall keep the notation Exp(A) for the image of this operator under the isomorphism Iz . In the following, we shall restrict out attention to the case of a self-adjoint A (though the general case may be treated by an easy modification of the results below). Theorem 2.1 (Surgailis [23]) Let A be a self-adjoint contraction in L2(X, z dx). Then the op- erator Exp(A) is positivity preserving if and only if A is sub-Markov. In the latter case, Exp(A) is Markov. We recall that A being sub-Markov means that 0 6 Af 6 1 a.e. for each 0 6 f 6 1 a.e., f ∈ L2(X, z dx). If, additionally, Afn ↗ 1 a.e. for some sequence fn ↗ 1, fn ∈ L2(X, z dx), then A is called Markov. Let (Tt)t>0 be a Markov semigroup in L2(X, z dx), and let pt(x, ·), t > 0, x ∈ X , be a corre- sponding Markov semigroup of kernels. Consider the semigroup (Exp(Tt))t>0 in L2(Γ, πz), which is Markov by Theorem 2.1. Note that each operator Exp(Tt) is defined only πz-almost everywhere. We are now interested in an explicit point-wise realization of a Markov semigroup of kernels which would correspond to the semigroup (Exp(Tt))t>0. We consider the infinite product XN with the cylinder σ-algebra on it, denoted by C(XN). Let us recall the construction of a probability measure on Γ through a product measure on XN, see [13,25]. We define A ∈ C(XN) as the set of all elements (xn)∞n=1 ∈ XN such that xi 6= xj when i 6= j, and the sequence {xn} ∞ n=1 has no accumulation points in X . Let D := {(x, y) ∈ X2 : x = y}. Let νn, n ∈ N, be probability measures on (X,B(X)) such that νn ⊗ νm(D) = 0, n 6= m. Consider the product measure ν := ⊗∞ n=1 νn on (XN, C(XN)). Then, by the Borel–Cantelli lemma, ν(A) = 1 if and only if, for each r ∈ N, ∞ ∑ n=1 νn(B(r)) < ∞. In the latter case, we can consider ν as a probability measure on (A, C(A)), where C(A) denotes the trace σ-algebra of C(XN) on A. Define the mapping A 3 (xn)∞n=1 7→ E((xn)∞n=1) := ∞ ∑ n=1 εxn ∈ Γ, (2) 704 Non-equilibrium stochastic dynamics in continuum which is measurable. Thus, we can define a probability measure µ on (Γ,B(Γ)) as the image of ν under the mapping (2). Evidently, the measure µ is independent of the order of the νn’s. Assume now that pt(x, ·) ⊗ pt(y, ·)(D) = 0, x 6= y, t > 0, (3) and let γ ∈ Γ be such that ∑ x∈γ pt(x, B(r)) < ∞, t > 0, r ∈ N. (4) Then, for each t > 0, we define Pt(γ, ·) as the probability measure on Γ given through the product measure Pt((xn)∞n=1, ·) := ∞ ⊗ n=1 pt(xn, ·), where {xn} ∞ n=1 is an arbitrary numeration of the elements of γ. In what follows, we will always assume that the manifold X satisfies the following condition: there exist m ∈ N and C > 0 such that vol(B(βr)) 6 Cβm vol(B(r)), r > 0, β > 1. (5) By e.g. [6, Proposition 5.5.1], if X has non-negative Ricci curvature, then (5) is satisfied with C = 1 and m being equal to the dimension of X . Theorem 2.2 Let (pt)t>0 be a Markov semigroup of kernels on X satisfying (3) and let α > 1. Assume that ∃ε > 0 : ∀t ∈ (0, ε) ∀δ > 0 : ∞ ∑ n=1 sup x∈X pt(x, B(x, δn1/(αm))c) < ∞. (6) Then, each γ ∈ Θα satisfies (4), so that Pt(γ, ·) is a probability measure on Γ for each t > 0, and furthermore, (Pt)t>0 is a Markov semigroup of kernels on (Θα,B(Θα)). Additionally, for each z > 0 and t > 0 and F ∈ L2(Γ, πz), the function Θα 3 γ 7→ ∫ Θα F (ξ)Pt(γ, dξ) is a πz-version of the function Exp(Tt)F ∈ L2(Γ, πz). Remark 2.1 Under the assumptions of Theorem 2.2, if the condition (6) is satisfied for all α > 1, then (Pt)t>0 becomes a Markov semigroup of kernels on (Θ,B(Θ)). Proof of Theorem 2.2. For any x ∈ X , denote \|x\| := dist(x0, x), where dist denotes the Riemannian distance. Lemma 2.1 Let the conditions of Theorem 2.2 be satisfied. Then, for any γ ∈ Θα and t ∈ (0, ε), ∑ x∈γ pt(x, B(x, \|x\|/2)c) < ∞. Proof. Fix any γ ∈ Θα and choose any numeration {xn} ∞ n=1 of points of γ such that \|xn+1\| > \|xn\|, n ∈ N. Since γ ∈ Θα, there exists K ∈ N for which \|γB(r)\| 6 K vol(B(r))α, r ∈ N. (7) Define r(n) := max { i ∈ Z+ : i < (n/(KCα vol(B(1))α))1/(αm) } , n ∈ N, (8) 705 Y.Kondratiev, E.Lytvynov, M.Röckner where C is the constant from (5). By (5), (7), and (8), \|γB(r(n))\| 6 K vol(B(r(n)))α 6 KCαr(n)αm vol(B(1))α < n, n ∈ N. Therefore, xn 6∈ B(r(n)). Hence, by (8), \|xn\| > r(n) > (n/(KCα vol(B(1))α))1/(αm) − 1. Therefore, to prove the lemma, it suffices to show that ∞ ∑ n=1 pt(xn, B(xn, ((n/KCα vol(B(1))α)1/(αm) − 1)/2)c) < ∞, which evidently follows from (6). � Fix any (xn)∞n=1 ∈ E−1(Θα), where the mapping E is given by (2). Denote An := { (yk)∞k=1 ∈ XN : yn ∈ B(xn, \|xn\|/2) } , n ∈ N. (9) By Lemma 2.1, (9) and the Borel–Cantelli lemma, Pt ( (xn)∞n=1, lim inf n→∞ An ) = 1, t ∈ (0, ε). (10) Next, define A′ := lim inf n→∞ An ∩ {(yn)∞n=1 ∈ XN : yi 6= yj if i 6= j, i, j ∈ N)}. By (3) and (10), Pt((xn)∞n=1,A ′) = 1, t ∈ (0, ε). (11) We evidently have \|xn\| → ∞ as n → ∞, and therefore A′ ⊂ A. Hence, condition (4) is satisfied for γ = E((xn)∞n=1). Let us show that E−1(Θα) ⊂ A′. (12) Indeed, fix any (yn)∞n=1 ∈ A′ and define k ∈ Z+ as the number of those yn’s which do not belong to B(xn, \|xn\|/2). Then \|E((yn)∞n=1)B(r)\| 6 \|E((xn)∞n=1)B(2r)\| + k. (13) Then, by (5) and (13), we have, for each r ∈ N, \|E((yn)∞n=1)B(r)\| 6 K vol(B(2r))α + k 6 K(C2m vol(B(r)))α + k 6 K ′ vol(B(r))α for some K ′ ∈ N which is independent of r (note that vol(B(r)) → ∞ as r → ∞ since X has infinite volume). Hence E((yn)∞n=1) ∈ Θα. Thus, by (11) and (12), Pt((xn)∞n=1, E −1(Θα)) = 1, (xn)∞n=1 ∈ E−1(Θα), t ∈ (0, ε). (14) Then, it easily follows from (14) and the construction of Pt((xn)∞n=0, ·) that (Pt)t>0 is a Markov semigroup of kernels on E−1(Θα). Therefore, (Pt)t>0 is a Markov semigroup of kernels on Θα . The proof of the last statement of the theorem is quite analogous to the proof of [13, Theo- rem 5.1], so we only outline this. Let D := {ϕ ∈ C0(X) : −1 < ϕ 6 0}. (15) 706 Non-equilibrium stochastic dynamics in continuum Then, for any ϕ ∈ D, γ ∈ Θα, and t > 0, we easily get from the definition of Pt(γ, ·): ∫ Θα exp[〈log(1 + ϕ), ξ〉]Pt(γ, dξ) = ∏ x∈γ ∫ X (1 + ϕ(y)) pt(x, dy) = ∏ x∈γ (1 + (Ttϕ)(x)) = exp[〈log(1 + Ttϕ), γ〉]. (16) Next, it is well known (see e.g. [23, Corollary 2.1]) that, for any ϕ ∈ D, I−1 z exp[〈log(1 + ϕ), ·〉] = exp [ ∫ X ϕ(x) z dx ] ( (1/n!)ϕ⊗n )∞ n=0 . (17) Since ∫ X (Ttϕ)(x) z dx = ∫ X ϕ(x) z dx, it follows from (16), (17) and the definition of Exp(Tt) that the statement holds for F = exp[〈log(1 + ϕ), ·〉], ϕ ∈ D. Hence, analogously to the proof of [13, Theorem 5.1], we conclude the statement in the general case. � Let us now outline the case where the semigroup (Tt)t>0 is sub-Markov, but not Markov. We shall assume for simplicity that inf x∈X pt(x, X) > 0, t > 0. (18) Let X̂ := X ∪ {∆} be a one-point extension of X , and, as usual, consider (p̂t)t>0 as the extension of (pt)t>0 to a Markov semigroup of kernels on X̂. Let conditions (3) and (6) be satisfied. Consider the mapping X̂N 3 (xn)∞n=1 7→ Ê((xn)∞n=1) := ∞ ∑ n=1 111X(xn)εxn . Then P̂t((xn)∞n=1, ·) := ⊗∞ n=1 p̂t(xn, ·), t > 0, is a Markov semigroup of kernels on Ê−1(Θα) (notice that condition (18) guarantees that, for P̂t((xn)∞n=1, ·)-a.e. (yn)∞n=1 ∈ X̂N, an infinite number of yn’s belong to X). Set P̂t(γ, ·) to be the image of P̂t((xn)∞n=1, ·) under Ê , where γ = {xn} ∞ n=1 ∈ Θα, t > 0. Then, (P̂t)t>0 becomes a Markov semigroup of kernels on Θα. Next, we denote by πz,t the Poisson random measure over X with intensity measure (1 − pt(x, X)) z dx (notice that πz,t is concentrated on finite or infinite configurations in X , depending on whether the integral ∫ X (1−pt(x, X)) z dx is finite or infinite.) Define Pz,t(γ, ·) as the convolution of the measures P̂t(γ, ·) and πz,t, i.e., Pz,t(γ, A) = ∫ P̂t(γ, dξ1) ∫ πz,t(dξ2)111A(ξ1 + ξ2). (19) Note that Pz,t(γ, ·) is indeed concentrated on Γ, since for any fixed ξ1 ∈ Γ, the πz,t probability of those ξ2 which satisfy ξ1 ∩ ξ2 6= ∅ is equal to zero. Furthermore, we have Pz,t(γ, Θα) = 1 for each t > 0 and γ ∈ Θα. Indeed, for each t > 0, πz,t is either concentrated on finite configura- tions or πz,t(Θα) = 1, the latter being a consequence of the estimate 1 − pt(x, X) 6 1 and the support property of a Poisson measure [19]. Now, the equality Pz,t(γ, Θα) = 1 follows from the definition (19). Next, for any ϕ ∈ D (see (15)), γ ∈ Θα, and t > 0, we have ∫ Θα exp[log(1 + ϕ), ξ〉]Pz,t(γ, dξ) = ∫ exp[〈log(1 + ϕ), ξ1〉] P̂t(γ, dξ) ∫ exp[log(1 + ϕ), ξ2〉] πz,t(dξ2) = ( ∏ x∈γ ( 1 − pt(x, X) + ∫ X (1 + ϕ(y))pt(x, dy) )) exp [ ∫ X ϕ(x)(1 − pt(x, X)) z dx ] = exp [〈log(1 + Ttϕ), γ〉] exp [ ∫ X (ϕ(x) − (Ttϕ)(x)) z dx ] . (20) 707 Y.Kondratiev, E.Lytvynov, M.Röckner From here we conclude that Θα 3 γ 7→ ∫ Θα F (ξ)Pz,t(γ, dξ) is a πz-version of Exp(Tt)F . Finally, using (20), we have, for any ϕ ∈ D, t, s > 0, and γ ∈ Θα, ∫ Θα ∫ Θα exp[〈log(1 + ϕ), ξ2〉]Pz,s(ξ1, dξ2)Pz,t(γ, dξ1) = ∫ Θ exp[〈log(1 + ϕ), ξ〉]Pz,s+t(γ, dξ), from where it easily follows that (Pz,t)t>0 is a Markov semigroup of kernels on Θα. Remark 2.2 Note that, in the case where the semigroup (Tt)t>0 is Markov, the construction of the Markov semigroup of kernels (Pt)t>0 is independent of z, whereas in the case where (Tt)t>0 is sub-Markov, the (Pz,t)t>0 does depend on z. 3. Non-equilibrium independent infinite particle process Our next aim is to study the Markov process corresponding to the Markov semigroup of kernels (Pt)t>0, respectively (Pz,t)t>0 . For a metric space E, we denote by D([0,∞), E) the space of all cadlag functions from [0,∞) to E, i.e., right continuous functions on [0,∞) having left limits on (0,∞). We equip D([0,∞), E) with the cylinder σ-algebra C(D([0,∞), E)) constructed through the Borel σ-algebra B(E). In what follows, we will assume that a Markov process on X corresponding to the semigroup (Tt)>0 has cadlag paths. The latter, in particular, holds if the kernels (pt)t>0 determine a Feller semigroup (see e.g. [9, Chapter 2, Theorem 2.7]). We first consider the case where the semigroup (Tt)t>0 is Markov. For each x ∈ X , let P x denote the law of the Markov process (Xt)t>0 corresponding to (Tt)t>0 which starts at x. By our assumption, each P x is a probability measure on D([0,∞), X). We will also assume that P x ⊗ P y((X (1) t , X (2) t )t>0 \| ∃t > 0 : X (1) t = X (2) t ) = 0, x 6= y, (21) i.e., two independent Markov processes starting at x and y, x 6= y, will a.s. never meet. (If this condition is not satisfied, all the results below remain true, but for a corresponding space of multiple configurations.) Notice that condition (21) is stronger than (3). For each x ∈ X and r > 0, denote by τB(x,r)c the hitting time of B(x, r)c: D([0,∞), X) 3 ω 7→ τB(x,r)c(ω) := inf{t > 0 : ω(t) ∈ B(x, r)c}. Let α > 1 and assume: ∃ε > 0 : ∀δ > 0 : ∞ ∑ n=1 sup x∈X P x(τB(x,δn1/(αm))c > ε) < ∞. (22) Condition (22) is evidently stronger than (6). Consider the space D([0,∞), X)N equipped with the cylinder σ-algebra C(D([0,∞), X)N). Denote by Ωα,1 the set of those (ωn)∞n=1 ∈ D([0,∞), X)N which satisfy the following conditions: (i) for all t > 0, ωi(t) 6= ωj(t) if i 6= j; (ii) {ωn(0)}∞n=1 ∈ Θα ; (iii) there are only a finite number of ωn’s for which τB(xn,\|xn\|/2)c(ωn) 6 ε, where ε is as in (22). It is easy to see that Ωα,1 ∈ C(D([0,∞), X)N). Fix any (xn)∞n=1 ∈ E−1(Θα) and consider the product measure P (xn)∞n=1 := ∞ ⊗ n=1 P xn 708 Non-equilibrium stochastic dynamics in continuum on D([0,∞), X)N. Absolutely analogously to the proof of Theorem 2.2 we conclude from (21) and (22) that P (xn)∞n=1(Ωα,1) = 1. (23) Furthermore, it follows from the proof of Theorem 2.2 that, for each (ωn)∞n=1 ∈ Ωα,1, we have {ωn(t)}∞n=1 ∈ Θα for all t ∈ (0, ε]. For each k ∈ N, we now recurrently define Ωα,k+1 as the subset of Ωα,k consisting of those (ωn)∞n=1 which satisfy the following condition: there are only a finite number of ωn’s for which τB(ωn(εk),\|ωn(εk)\|/2)c(ωn(εk + ·)) 6 ε. Since under P x, Xt(ω) = ω(t), t > 0, is a time homogeneous Markov process starting at x, x ∈ X , we conclude, analogously to the above, that P (xn)∞n=1(Ωα,k) = 1, k ∈ N. Therefore, P (xn)∞n=1(Ωα) = 1, Ωα := ⋂ k∈N Ωα,k . Hence, we can consider P(xn)∞n=1 as a probability measure on Ωα equipped with the trace σ-algebra of C(D([0,∞), XN) on Ωα. Fix any (ωn)∞n=1 ∈ Ωα. Then, we evidently have {ωn(t)}∞n=1 ∈ Θα for all t > 0. Furthermore, for any compact Λ ⊂ X and for any T > 0, there are only a finite number of ωn’s which meet Λ during the time interval [0, T ]. Therefore, {ωn(·)}∞n=1 ∈ D([0,∞), Θα), where Θα is equipped with the relative topology as a subset of Γ. Thus, the following mapping is well-defined: Ωα 3 (ωn(·))∞n=1 7→ ∞ ∑ n=1 εωn(·) ∈ D([0,∞), Θα). (24) Furthermore, it easily follows from the definition of a cylinder σ-algebra that the mapping (24) is measurable. Thus, we can consider the image of the measure P(xn)∞n=1 under (24). This proba- bility measure on D([0,∞), Θα) will be denoted by P{xn}∞ n=1 (note that this measure is, indeed, independent of the numeration of points of {xn} ∞ n=1 ∈ Θα). Next, it is easy to see that the finite-dimensional distributions of Pγ , γ ∈ Θα, are given through the Markov semigroup of kernels Pt(γ, ·) on Θα (see Theorem 2.2). Hence, analogously to [13, Theorem 8.1], we get the following Theorem 3.1 Let (pt)>0 be a Markov semigroup of kernels on X and let (21) and (22) hold. Then there exists a time homogeneous Markov process M = (ΩΩΩ,F, (Ft)t>0, (θθθt)t>0, (P γ)γ∈Θ, (Xt)t>0) on the state space (Θα,B(Θα)) with cadlag paths and with transition probability function (Pt)t>0. Remark 3.1 In Theorem 3.1, M can be taken canonical, i.e., ΩΩΩ = D([0,∞), Θα), X(t)(ω) = ω(t) for t > 0 and ω ∈ ΩΩΩ, Ft = σ{Xs, 0 6 s 6 t} for t > 0, F = σ{Xt, t > 0}, (θθθtω)(s) = ω(s + t) for t, s > 0. Remark 3.2 From the proof of Theorem 3.1 we see that the Markov process M is a realization of the independent infinite particle process. Remark 3.3 If the underlying Markov process on X has continuous sample paths, then the Markov process M in Theorem 3.1 has continuous sample paths in Θα. Remark 3.4 Analogously to Remark 2.1, we note that if, under the assumptions of Theorem 3.1, condition (22) is satisfied for each α > 1, then Θα can be replaced by Θ in the statement of this theorem. 709 Y.Kondratiev, E.Lytvynov, M.Röckner Corollary 3.1 The statement of Theorem 3.1 remains true if, instead of (22), one demands that the following stronger condition be satisfied: ∃ε > 0 : ∀δ > 0 : ∞ ∑ n=1 sup t∈(0,ε] sup x∈X pt(x, B(x, δn1/(αm))c) < ∞. (25) Proof. For any x ∈ X , r > 0, and ε > 0, we have: P x(τB(x,r)c > ε) 6 2 sup t∈(0,ε] sup x∈X pt(x, B(x, r/2)c). (26) This estimate follows by a straightforward generalization of (the proof of) [18, Appendix A, Lemma 4] (see also [13, Lemma 8.1]) to the case of an arbitrary Markov process on X with cadlag paths. Now, by (26), condition (25) implies (22). � Let us consider the case where (pt)t>0 is sub-Markov. We will assume that, for each x ∈ X , pt(x, X) is continuously differentiable in t ∈ [0,∞) and there exists δ > 0 such that ∣ ∣ ∣ ∣ ∂ ∂t pt,x(X) ∣ ∣ ∣ ∣ 6 C, t ∈ [0, δ], x ∈ X, (27) for some C > 0. We set g(x) := − ∂ ∂t pt,x(X) ∣ ∣ ∣ t=0 , x ∈ X, (28) which is a bounded non-negative function. Using the semigroup property of (pt)t>0, we then con- clude from (27) and (28) that ∂ ∂t pt,x(X) = − ∫ X g(y)pt,x(dy), t > 0, x ∈ X. (29) We will now assume that (18), (21), and (22) hold. Analogously to the Markovian case, for each γ ∈ Θα, we define a probability measure P̂γ on D([0,∞), Θα) through the mapping Ω̂α 3 (ωn(·))∞n=1 7→ ∞ ∑ n=1 111X(ωn(·))εωn(·) ∈ D([0,∞), Θα), (30) where Ω̂α is a corresponding subset of D([0,∞), X̂)N (compare with (24)). Let Πz denote the Poisson random measure over X× [0,∞) with intensity measure g(x) z dx dt. Since the measure dx has no atoms in X , Πz is concentrated on the set of those configurations ξ which satisfy the following condition: for any different (x1, t1), (x2, t2) ∈ ξ, we have x1 6= x2. Any such configuration can be represented as the disjoint union ξ = ∞ ⋃ k=1 ξ(k), where each ξ(k) is a configuration in X×[k−1, k), and to each ξ(k) there corresponds a configuration γ(k) in X that is obtained by taking the X-components of the points from ξ(k). Denote Θα := Θα ∪ Γfin, where Γfin is the set of all finite configurations in X . We endow Θα with the vague topology. We note that, since the function g(x) is bounded, we have Πz-a.s. that γ(k) ∈ Θα. For each ξ(k), we now construct a probability measure Mξ(k) on D([0,∞,Θα). This measure is defined in the same way as the measure P̂γ(k) , but only, instead of (30), one uses the mapping (ωn(·))n>1 7→ ∑ n>1 111[tn,∞)(·)111X(ωn(· − tn))εωn(·−tn), 710 Non-equilibrium stochastic dynamics in continuum where ξ(k) = {(xn, tn)}n>1, so that γ(k) = {xn}n>1. For each γ ∈ Θα, we now define a probability measure Pγ z on D([0,∞), Θα) by setting, for each C ∈ C(D([0,∞), Θα), Pγ z (C) := ∫ P̂γ(dωωω(0)) ∫ Πz(dξ) ∫ ( ∞ ⊗ k=1 Mξ(k) ) (dωωω(1)(·), dωωω(2)(·), . . . )111C ( ∞ ∑ k=0 ωωω(k)(·) ) . We note that ∑∞ k=0 ωωω(k)(·) indeed a.s. belongs to D([0,∞), Θα), since for any K ∈ N and t < K, we have ∑∞ k=K ωωω(k)(t) = 0. Let us show that the finite-dimensional distributions of Pγ z are given through the Markov semigroup of kernels Pz,t(γ, ·) on Θα. For 0 = t0 < t1 < t2 < · · · < tn, n ∈ N, and any ϕ1, . . . , ϕn ∈ D (see (15)), we have ∫ Πz(dξ) ∫ ( ∞ ⊗ k=1 Mξ(k) ) (dωωω(1)(·), dωωω(2)(·), . . . ) n ∏ i=1 exp [ 〈 log(1 + ϕi), ∞ ∑ k=1 ωωω(k)(ti) 〉 ] = ∫ Πz(dξ) ∏ (x,τ)∈ξ ∫ P̂ x(dω(·)) n ∏ i=1 exp [ 〈log(1 + ϕi),111[t,∞)(ti)111X (ω(ti − τ))εω(ti−τ)〉 ] = exp [∫ X z dx g(x) ∫ ∞ 0 dτ ( − 1 + ∫ P̂ x(dω(·)) × n ∏ i=1 exp[〈log(1 + ϕi),111[t,∞)(ti)111X(ω(ti − τ))εω(ti−τ)〉] )] = exp [ n ∑ j=1 ∫ X z dx g(x) ∫ tj tj−1 dτ ( − 1 + ∫ P̂ x(dω(·)) × n ∏ i=1 exp[〈log(1 + ϕi),111X(ω(ti − τ))εω(ti−τ)〉] )] = exp [ n ∑ j=1 ∫ X z dx g(x) ∫ tj tj−1 dτ ( − 1 + (1 − ptj−τ (x, X)) + ∫ X ptj−τ (x, dy)(1 + ϕj(y)) × ∫ X P̂ y(dω(·)) n ∏ i=j+1 exp [ 〈log(1 + ϕi),111X(ω(ti − tj))εω(ti−tj)〉 ] )] . (31) Denote Fj(x) := −1 + (1 + ϕj(x)) ∫ P̂ x(dω(·)) n ∏ i=j+1 exp[〈log(1 + ϕi),111X(ω(ti − tj))εω(ti−tj)〉], where x ∈ X . Then we can proceed in (31) as follows: = exp [ n ∑ j=1 ∫ X z dx g(x) ∫ tj−tj−1 0 dτ ∫ X ptj−tj−1−τ (x, dy)Fj(y) = exp [ n ∑ j=1 ∫ tj−tj−1 0 dτ ∫ X z dx Fj(x) ∫ ptj−tj−1−τ (x, dy)g(y) ] = exp [ n ∑ j=1 ∫ X Fj(x)(1 − ptj−tj−1(x, X)) z dx ] , (32) where we used (29). 711 Y.Kondratiev, E.Lytvynov, M.Röckner On the other hand, by (19), ∫ Pz,t1(γ, dγ1) ∫ Pz,t2−t1(γ1, dγ2) · · · ∫ Pz,tn−tn−1(γn−1, dγn) n ∏ i=1 exp[〈log(1 + ϕi), γi〉] = ∫ P̂t1(γ, dγ1) ∫ P̂t2−t1(γ1, dγ2) · · · ∫ P̂tn−tn−1(γn−1, dγn) n ∏ i=1 exp[〈log(1 + ϕi), γi〉] n ∏ j=1 Aj , (33) where Aj = ∫ πz,tj−tj−1 (dθj) exp[〈log(1 + ϕj), θj〉] × ∫ P̂tj+1−tj (θj , dηj+1) · · · ∫ P̂tn−tn−1(ηn−1, dηn) n ∏ k=j+1 exp[〈log(1 + ϕk), ηk〉] = ∫ πz,tj−tj−1 (dθj) ∏ x∈θj ( (1 + ϕj(x)) ∫ p̂tj+1−tj (x, dy1) · · · ∫ p̂tn−tn−1(yn−1, dyn) × n ∏ k=j+1 exp[〈log(1 + ϕk),111X(yk)εyk 〉] ) = exp [ ∫ X Fj(x)(1 − ptj−tj−1 (x, X)) z dx ] . (34) By (31)–(34), we conclude that the finite-dimensional distributions of Pγ z are indeed given through the Markov semigroup of kernels Pz,t(γ, ·) Thus, analogously to Theorem 3.1, we obtain a time homogeneous Markov process Mz on the state space Θα (or Θ, provided (22) holds for all α > 1) with cadlag paths and with transition probability function (Pz,t)t>0. Remark 3.5 Analogously to Remark 2.2, we see that, in the case where (pt)t>0 is Markov, the process M from Theorem 3.1 has any Poisson measure πz, z > 0 as invariant measure, whereas, in the case of a sub-Markov (pt)t>0, only the measure πz is invariant for the process Mz . 4. Examples 4.1. Brownian motion on the configuration space Assume that (Tt)t>0 is the heat semigroup on X with generator 1 2∆X , the Laplace–Beltrami operator on X . We will denote by p(t, x, y) the corresponding heat kernel on X (see e.g. [6]). We recall that the corresponding Markov process on X is called Brownian motion on X . In the case where the manifold X is not stochastically complete, we will assume that condition (27) is satisfied. Theorem 4.1 Assume that the dimension of the manifold X is > 2. Assume that (pt)t>0 is either Markov, or (18) and (27) hold. Further assume that the heat kernel p(t, x, y) satisfies the Gaussian upper bound for small values of t: p(t, x, y) 6 Ct−n/2 exp [ − dist(x, y)2 Dt ] , t ∈ (0, ε], x, y ∈ X, (35) where n ∈ N, ε > 0 and C and D are positive constants. Then the corresponding independent infinite particle process exists as a Markov process MB on (Θ, C(Θ)) with either continuous paths if (pt)t>0 is Markov, or cadlag paths if (pt)t>0 is sub-Markov. 712 Non-equilibrium stochastic dynamics in continuum Remark 4.1 According to [1] and [13], the Markov process MB in Theorem 4.1 may be interpreted as a Brownian motion in the configuration space over X . Denote by FC∞ b (C∞ 0 (X), Θ) the set of all real-valued functions on Θ of the form F (γ) = gF (〈ϕ1, γ〉, . . . , 〈ϕN , γ〉), where gF ∈ C∞ b (R), ϕ1, . . . , ϕN ∈ C∞ 0 (X), N ∈ N. Assume first that (pt)t>0 is Markov. Then, the L2-generator of the process MB has the following representation on the set FC∞ b (C∞ 0 (X), Θ) (which is a core for this operator): (LBF )(γ) = 1 2 ∑ x∈γ ∆X x F (γ), where ∆X x F (γ) := ∆X y F (γ \ {y} ∪ {x}) ∣ ∣ y=x , see [1,13] for details. In the case where (pt)t>0 is sub-Markov, one can analogously show that the L2-generator of MB is given on the set FC∞ b (C∞ 0 (X), Θ) by (LBF )(γ) = 1 2 ∑ x∈γ ∆X y (F (γ \ {x} ∪ {y}) − F (γ \ {x}) ∣ ∣ y=x + ∫ X (F (γ ∪ {x}) − F (γ))g(x) dx. Proof of Theorem 4.1. Since the dimension of X is > 2, (21) is now satisfied, see e.g. (8.29) in [13]. Next, by [13, Lemma 8.2], (35) implies that there exists C > 0 such that sup t∈(0,ε] sup x∈X pt(x, B(x, r)c) 6 Ce−r, r > 0. It follows from here that (25) is satisfied. Now, the theorem follows from Corollary 3.1. � 4.2. Free Glauber dynamics on the configuration space Let a : X → [0,∞) be a bounded measurable function. Consider a Markov process on X with a finite lifetime that corresponds to the semigroup (Ttf)(x) = e−a(x)tf(x). Thus, the process stays at a starting point for some random time and then dies away, with pt(x, {x}) = pt(x, X) = e−a(x)t, x ∈ X, t > 0. The function g is now equal to a. Note also that pt(x, B(x, r)c) = 0 for each r > 0. Thus, all the above conditions are evidently satisfied and the corresponding infinite particle process exists as a Markov process MG on (Θ, C(Θ)) with cadlag paths. This process can be interpreted as a free Glauber dynamics on the configuration space, or a birth-and-death process in X , see [3,12,14] for details. The L2-generator of the process MG is given on its core FC∞ b (C∞ 0 (X), Θ) by (LGF )(γ) = ∑ x∈γ a(x)(F (γ \ {x}) − F (γ)) + ∫ X a(x)(F (γ ∪ {x} − F (γ)) dx, see [12,14]. 4.3. Free Kawasaki dynamics on the configuration space We now consider a Markov jump process on X . The generator of this process has the following representation on the set of all bounded functions on X : (Lf)(x) = ∫ X (f(y) − f(x))κ(x, y) dy, (36) and we assume that κ : X2 → [0,∞) 713 Y.Kondratiev, E.Lytvynov, M.Röckner is a measurable function satisfying κ(x, y) = κ(y, x), x, y ∈ X, (37) and λ := sup x∈X ∫ X κ(x, y) dy ∈ (0,∞). (38) Following [9], we can explicitly construct this Markov process as follows. For each x0 ∈ X , let {Y (k), k = 0, 1, 2, . . .} be a Markov chain in X starting at x0, with transition function µ(x, dy) = ( 1 − 1 λ ∫ X κ(x, y) dy ) εx(dy) + 1 λ κ(x, y) dy. Let (Zt)t>0 be an independent Poisson process with parameter λ. We now define the Markov process (Xt)t>0 starting at x0 by Xt := Y (Zt), t > 0. By [9], this process has generator (36). Theorem 4.2 Assume that (37) and (38) hold and assume that there exist C > 0 and α > m (m being as in (5)) such that sup x∈X ∫ B(x,r)c κ(x, y) dy 6 C rα , r > 0. (39) Then the corresponding independent infinite particle process exists as a Markov process MK on (Θα,B(Θα)) with cadlag paths. Remark 4.2 According to [14], the Markov process MK can be interpreted as a free Kawasaki dynamics on the configuration space. The L2-generator of the process MK has the following rep- resentation on FC∞ b (C∞ 0 (X), Θ): (LKF )(γ) = ∫ X γ(dx) ∫ X dy κ(x, y)(F (γ \ {x} ∪ {y})− F (γ)), see [14]. Proof. By construction, the process (Xt)t>0 has cadlag paths in X . Furthermore, as easily seen, condition (21) is now satisfied. In a standard way, the process (Xt)t> leads to a Markov semigroup (Tt)t>0 in L2(X, dx). Using (37) and the construction of the process, we see that each Tt is self- adjoint. By Theorem 3.1, it suffices to prove that condition (22) is satisfied. By using (39) and the construction of the process (Xt)t>0, we get, for any δ > 0. ∞ ∑ n=1 sup x∈X P x(τB(x,δn1/m)c > 1) = ∞ ∑ n=1 sup x∈X ∞ ∑ k=1 P x(τB(x,δn1/m)c > 1, Z1 = k) 6 ∞ ∑ n=1 sup x∈X ∞ ∑ k=1 P x(Z1 = k, ∃i ∈ {1, . . . , k} : dist(Y (i − 1), Y (i)) > δn1/m/k) = ∞ ∑ n=1 sup x∈X ∞ ∑ k=1 e−λ λk k! P x(∃i ∈ {1, . . . , k} : dist(Y (i − 1), Y (i)) > δn1/m/k) 6 ∞ ∑ n=1 ∞ ∑ k=1 e−λ λk k! k ∑ i=1 sup x∈X P x(dist(Y (i − 1), Y (i)) > δn1/m/k) 6 ∞ ∑ n=1 ∞ ∑ k=1 e−λ λkk k! λ sup x∈X ∫ B(x,δn1/m/k)c κ(x, y) dy 714 Non-equilibrium stochastic dynamics in continuum 6 ∞ ∑ n=1 ∞ ∑ k=1 e−λ λk−1 (k − 1)! C ( k δn1/m )α = e−λC δα ( ∞ ∑ n=1 1 nα/m )( ∞ ∑ k=1 λk−1kα (k − 1)! ) < ∞, since α > m. � 5. Free Glauber dynamics as a scaling limit of free Kawasaki dynamics Let µ be a probability measure on (Γ,B(Γ)). Assume that, for any n ∈ N, there exists a non- negative measurable symmetric function k (n) µ on Xn such that, for any measurable symmetric function f (n) : Xn → [0,∞], ∫ Γ ∑ {x1,...,xn}⊂γ f (n)(x1, . . . , xn) µ(dγ)= 1 n! ∫ Xn f (n)(x1, . . . , xn)k(n) µ (x1, . . . , xn) dx1 · · · dxn . (40) Then, the functions k (n) µ , n ∈ N, are referred to as the correlation functions of the measure µ. Via a recursion formula, one can transform the correlation functions k (n) µ into the Ursell func- tions u (n) µ and vice versa, see e.g. [21]. Their relation is given by kµ(η) = ∑ uµ(η1) · · ·uµ(ηj), η ∈ Γ0 , (41) where Γ0 := {η ⊂ X : 1 6 \|η\| < ∞}, for any η = {x1, . . . , xn} ∈ Γ0 kµ(η) := k(n) µ (x1, . . . , xn), uµ(η) := u(n) µ (x1, . . . , xn), and the summation in (41) is over all partitions of the set η into nonempty mutually disjoint subsets η1, . . . , ηj ⊂ η such that η1 ∪ · · · ∪ ηj = η, j ∈ N. Note that k (1) µ = u (1) µ . Let now X = Rd. We fix an arbitrary function ξ ∈ S(Rd) such that ξ(−x) = ξ(x) for all x ∈ Rd. Here, S(Rd) denotes the Schwartz space of rapidly decreasing, infinitely differentiable functions on R d. We define κ(x, y) := ξ(x − y), x, y ∈ R d. It can be easily checked that, by Theorem 4.2, the corresponding Kawasaki dynamics exists as a Markov process MK on (Θ,B(Θ)) with cadlag paths. Let µ be a probability measure on (Θ,B(Θ)) that satisfies the following conditions: (i) µ has correlation functions (k (n) µ )n∈N, and there exist 0 6 γ < 1 and C > 0 such that ∀n ∈ N, ∀(x1, . . . , xn) ∈ (Rd)n : k(n) µ (x1, . . . , xn) 6 (n!)γCn. (42) (ii) µ is translation invariant. In particular, the first correlation function k (1) µ is a constant. (iii) µ has decay of correlations in the following sense: for each n > 2 and 1 6 i 6 n u(n) µ (x1 ε , . . . , xi ε , xi+1, . . . , xn ) → 0 as ε → 0, (43) where the convergence is in the dx1 · · · dxn-measure on each compact set in Rd. For example, any double-potential Gibbs measure in the low activity-high temperature regime satisfies the above assumptions, see [4,17,21] 715 Y.Kondratiev, E.Lytvynov, M.Röckner Let us assume that the initial distribution of the Kawasaki dynamics is µ. We denote this stochastic process by MK µ . We scale this dynamics as follows. Instead of the function ξ used for the construction of MK µ , use the function ξε(x) := εdξ(εx), x ∈ R d, and denote the corresponding Kawasaki dynamics with initial distribution µ by MK µ, ε. We are interested in the limit of this dynamics as ε → 0. As in subsection 4.2, we construct the Glauber dynamics MG using a(x) := 〈ξ〉 and z = k (1) µ . Here and below, for any f ∈ L1(Rd, dx), we denote 〈f〉 := ∫ Rd f(x) dx. We assume that the initial distribution of the Glauber dynamics is µ and denote this stochastic process by MG µ . Below, we will use .. Γ to denote the space of multiple configurations over Rd equipped with the vague topology, see e.g. [11] for details. We have Θ ∈ B( .. Γ). Theorem 5.1 Under the above assumptions, consider MK µ,ε, ε > 0, and MG µ as stochastic pro- cesses taking values in .. Γ. Then, MK µ, ε → MG µ as ε → 0 in the sense of weak convergence of finite-dimensional distributions. Proof. Let (pt, ε)t>0 denote the semigroup in L2(Rd, dx) with generator (Lεf)(x) = ∫ Rd (f(y) − f(x))ξε(x − y) dy. Let F and F−1 denote the Fourier transform and its inverse, respectively, which we normalize so that they become unitary operators in L2(Rd → C, dx). As usual, we denote f̂ := Ff and f̌ := F−1f . We easily have: (pt, εf)(x) = e−t〈ξ〉f(x) + (Kt, εf)(x), (44) where (Kt, εf)(x) := ∫ Rd εdGt(ε(x − y))f(y) dy, t > 0. (45) Here Gt(x) := e−t〈ξ〉(exp[t(2π)d/2ξ̂] − 1)̌, x ∈ R d. (46) We note that, since ξ ∈ S(Rd), we have ξ̂ ∈ S(Rd), and therefore exp[t(2π)d/2ξ̂] − 1 ∈ S(Rd). Hence, for each t > 0, Gt ∈ S(Rd). Furthermore, since ξ(−x) = ξ(x), we get Gt(x) = Gt(−x). We fix any n ∈ N, 0 = t0 < t1 < t2 < · · · < tn, and ϕ0, ϕ1, . . . , ϕn ∈ C0(R d) with −1 < ϕi 6 0, i = 0, 1 . . . , n. We denote by (Pε t (γ, ·))t>0, γ∈Θ the transition semigroup of the ε-Kawasaki dynamics for ε > 0 and that of the Glauber dynamics for ε = 0. We also denote by (pε t (x, ·))t>0, x∈Rd the transition semigroup of the one-particle ε-dynamics, ε > 0. By (40), we have: ∫ Θ µ(dγ0) ∫ Θ Pε t1(γ0, dγ1) ∫ Θ Pε t2−t1(γ1, dγ2) × · · · × ∫ Θ Pε tn−tn−1 (γn−1, dγn) n ∏ i=0 exp[〈log(1 + ϕi), γi〉] = ∫ Θ µ(dγ) ∏ x∈γ (1 + ϕ0(x)) ∫ Rd pε t1(x, dx1) ∫ Rd pε t2−t1(x1, dx2) × · · · × ∫ Rd pε tn−tn−1 (xn−1, dxn) n ∏ i=1 (1 + ϕi(xi)) 716 Non-equilibrium stochastic dynamics in continuum = ∫ Θ µ(dγ) ∏ x∈γ (1 + gε(x)), = 1 + ∞ ∑ m=1 1 m! ∫ (Rd)m gε(x1) · · · g ε(xm)k(m) µ (x1, . . . , xm) dx1 · · · dxm . (47) Here, gε(x) := ϕ0(x) + fε(x) + ϕ0(x)fε(x) (48) with fε(x) : = ∑ 16i1<i2<···<ik≤n, k>1 ∫ Rd pε ti1 (x, dx1) × ∫ Rd pε ti2−ti1 (x1, dx2) × · · · × pε tik −tik−1 (xk−1, dxk)ϕi1 (x1) · · ·ϕik (xk). (49) We easily have from (48) and (49): sup ε>0 ∫ Rd \|gε(x)\| dx < ∞. Hence, by (42), in oder to find the limit of (47) as ε → 0, it suffices to find the limit of each term in the sum. By (44)–(46), (48), and (49), gε(x) = ϕ0(x) + (1 + ϕ0(x)) ∑ 16i1<i2<···<ik6n, k>1 (e−ti1 〈ξ〉 + Kti1 , ε)Mϕi1 × (e−(ti2−ti1 )〈ξ〉 + Kti2−ti1 , ε)Mϕi2 × · · · × (e−(tik −tik−1 )〈ξ〉 + Ktik −tik−1 , ε)ϕik , (50) where Mf denotes the operator of multiplication by a function f . By (41), for each m ∈ N, ∫ (Rd)m gε(x1) · · · g ε(xm)k(m) µ (x1, . . . , xm) dx1 · · ·dxm = ∑ {η1,...,ηj} j ∏ i=1 ∫ (Rd)\|ηi\| gε(x1) · · · gε(x\|ηi\|)u (\|ηi\|) µ (x1, . . . , x\|ηi\|) dx1 · · · dx\|ηi\| , (51) where the summation is over all partitions {η1, . . . , ηj}, j > 1, of the set {1, . . . , n} into nonempty, mutually disjoint subsets η1, . . . , ηj ⊂ η. We next have the following Lemma 5.1 Let the above assumptions be satisfied. Let k, n ∈ N, k 6 n, l1, . . . , lk ∈ N, t (j) i > 0, j = 1, . . . , li, i = 1, . . . , k. Let f (1) i , . . . , f (li) i ∈ C0(R d), i = 1, . . . , k. Let F : (Rd)n → R and f1, . . . , fk : Rd → R be measurable and bounded, and fk+1, . . . fn ∈ C0(R d). For any ε > 0, set Iε : = ∫ (Rd)n dx1 · · ·dxn F (x1, . . . , xn) k ∏ i=1 ( fi(xi) ∫ Rd dx (1) i εdG t (1) i (ε(xi − x (1) i ))f (1) i (x (1) i ) × ∫ Rd dx (2) i εdG t (2) i (ε(x (2) i − x (1) i ))f (2) i (x (2) i ) × · · · × ∫ Rd dx (li) i εdG t (li) i (ε(x (li−1) i − x (li) i ))f (li) i (x (li) i ) ) n ∏ j=k+1 fj(xj). (52) We then have: 717 Y.Kondratiev, E.Lytvynov, M.Röckner (i) If at least one li > 2, then Iε → 0 as ε → 0. (ii) If n > 2, l1 = · · · = lk = 1 and F = u (n) µ , then Iε → 0 as ε → 0. (iii) If l1 = · · · = lk = 1, F = 1, and at least one fi ∈ C0(R d). Then. Iε → 0 as ε → 0. (iv) If l1 = · · · = lk = 1, f1 = · · · = fk = 1, and F = 1, then for each ε > 0, Iε = ( k ∏ i=1 (1 − e−t (1) i 〈ξ〉)〈f (1) i 〉 ) × n ∏ j=k+1 〈fj〉. Proof. In the right hand side of (52), make the following change of variables x′ i = ε(xi), (x (j) i )′ = εx (j) i , j = 1, . . . , li − 1, i = 1, . . . , n. Then by the dominated convergence theorem and (43), we get the statements (i)–(iii). In the same way, in the case of (iv), we get Iε = ( k ∏ i=1 〈G t (1) i 〉〈f (1) i 〉 ) × n ∏ j=k+1 〈fj〉. By (44), we have, for any t > 0: 〈Gt〉 = 1 − e−t〈ξ〉, from where the lemma follows. � By (41), (50), (51), and Lemma 5.1, and taking into account that each Ursell function u (n) µ is bounded, we have: ∫ (Rd)m gε(x1) · · · g ε(xm)k(m) µ (x1, . . . , xm) dx1 · · · dxm → m ∑ l=0 ( m l )∫ (Rd)l l ∏ j=1  ϕ0(xj) + (1 + ϕ0(xj)) ∑ 16i1<i2<···<ik6n, k>1 e−tik 〈ξ〉(ϕi1 · · ·ϕik )(xj)   × k(l) µ (x1, . . . , xl) dx1 · · · dxl   ∑ 16i1<i2<···<ik6n, k>1 k(1) µ (1 − e−ti1 〈ξ〉)e−(tik −ti1 )〈ξ〉〈ϕi1 · · ·ϕik 〉   m−l . (53) Therefore, by (40), the left hand side of (47) converges to exp   ∑ 16i1<i2<···<ik6n, k>1 k(1) µ (1 − e−ti1 〈ξ〉)e−(tik −ti1 )〈ξ〉〈ϕi1 · · ·ϕik 〉   × ∫ Θ exp[〈log(1 + ϕ0), γ〉] ∏ x∈γ  1 + ∑ 16i1<i2<···<ik6n, k>1 e−tik 〈ξ〉(ϕi1 · · ·ϕik )(x)   µ(dγ) (54) as ε → 0. We next have the following 718 Non-equilibrium stochastic dynamics in continuum Lemma 5.2 For each γ ∈ Θ and any 0 < t1 < t2, · · · < tn, n ∈ N, ∫ Θ P0 t1(γ, dγ1) ∫ Θ P0 t2−t1(γ1, dγ2) × · · · × ∫ Θ P0 tn−tn−1 (γn−1, dγn) n ∏ i=1 exp[〈log(1 + ϕi), γi〉] = exp   ∑ 16i1<i2<···<ik6n, k>1 k(1) µ (1 − e−ti1 〈ξ〉)e−(tik −ti1 )〈ξ〉〈ϕi1 · · ·ϕik 〉   × ∏ x∈γ  1 + ∑ 16i1<i2<···<ik6n, k>1 e−tik 〈ξ〉(ϕi1 · · ·ϕik )(x)   . (55) Proof. By section 2 and subsection 4.2, for n = 1, we have: ∫ Θ exp[〈log(1 + ϕ1), γ1〉]P 0 t1(γ, dγ1) = exp [ k(1) µ (1 − e−t1〈ξ〉)〈ϕ1〉 ] ∏ x∈γ (1 + e−t1〈ξ〉ϕ1(x)), which is (55) in this case. Now, assume that (55) holds for n ∈ N, and let us prove it for n + 1. We then have: ∫ Θ P0 t1(γ, dγ1) ∫ Θ P0 t2−t1(γ1, dγ2) × · · · × ∫ Θ P0 tn+1−tn (γn, dγn+1) n+1 ∏ i=1 exp[〈log(1 + ϕi), γi〉] = ∫ Θ P0 t1(γ, dγ1) exp[log(1 + ϕ1), γ1〉] × ∏ x∈γ1  1 + ∑ 26i1<···<ik6n+1, k>1 e−(tik −t1)〈ξ〉(ϕi1 · · ·ϕik )(x)   × exp   ∑ 26i1<···<ik6n+1, k>1 k(1) µ (1 − e−(ti1−t1)〈ξ〉)e−(tik −ti1 )〈ξ〉〈ϕi1 · · ·ϕik 〉   = ∏ x∈γ  e−t1〈ξ〉(1 + ϕ1(x))  1 + ∑ 26i1<···<ik6n+1, k>1 e−(tik −t1)〈ξ〉(ϕi1 · · ·ϕik )(x)  + 1 − e−t1〈ξ〉   × ∫ Θ π (1+e−t1〈ξ〉)k (1) µ (dγ1) ∏ x∈γ1 (1 + ϕ1(x))  1 + ∑ 26i1<···<ik6n+1, k>1 e−(tik −t1)〈ξ〉(ϕi1 · · ·ϕik )(x)   × exp   ∑ 26i1<···<ik6n+1, k>1 k(1) µ (1 − e−(ti1−t1)〈ξ〉)e−(tik −ti1 )〈ξ〉〈ϕi1 · · ·ϕik 〉   = ∏ x∈γ  1 + ∑ 16i1<i2<···<ik6n+1, k>1 e−tik 〈ξ〉(ϕi1 · · ·ϕik )(x)   exp  (1 − e−t1〈ξ〉)k(1) µ × 〈 −1 + (1 + ϕ1)  1 + ∑ 2≤i1<···<ik6n+1, k>1 e−(tik −t1)〈ξ〉(ϕi1 · · ·ϕik )(x)   〉 + ∑ 26i1<···<ik6n+1, k>1 k(1) µ (1 − e−(ti1−t1)〈ξ〉)e−(tik −ti1 )〈ξ〉〈ϕi1 · · ·ϕik 〉   719 Y.Kondratiev, E.Lytvynov, M.Röckner = ∏ x∈γ  1 + ∑ 16i1<i2<···<ik6n+1, k>1 e−tik 〈ξ〉(ϕi1 · · ·ϕik )(x)   × exp   ∑ 16i1<i2<···<ik≤n+1, k>1 k(1) µ (1 − e−ti1 〈ξ〉)e−(tik −ti1 )〈ξ〉〈ϕi1 · · ·ϕik 〉   . Thus, by induction, the lemma is proved. � Let 0 6 t0 < t1 < · · · < tn, and let µε t0,t1,...,tn , ε > 0, denote the joint distribution of the process MK µ, ε at times t0, t1, . . . , tn for ε > 0, and respectively that of the process MG µ for ε = 0. By (54) and Lemma 5.2, for any ϕ0, ϕ1 . . . , ϕn ∈ C0(R d), ϕi 6 0, i = 0, 1, . . . , n, ∫ .. Γn+1 n ∏ i=0 exp[〈ϕi, γi〉] dµε t0,t1,...,tn (γ0, γ1, . . . , γn) → ∫ .. Γn+1 n ∏ i=0 exp[〈ϕi, γi〉] dµε t0,t1,...,tn (γ0, γ1, . . . , γn) (56) as ε → 0. By (56), we, in particular, get, for any t > 0 and any ϕ ∈ C0(R d), ϕ 6 0, ∫ .. Γ exp[〈ϕ, γ〉] dµε t (γ) → ∫ .. Γ exp[〈ϕ, γ〉] dµ0 t (γ) as ε → 0. Hence, by [11, Theorem 4.2], µε t → µ0 t weakly in M( .. Γ) as ε → 0. Here, M( .. Γ) denotes the space of probability measures on .. Γ, see e.g. [20] for details. Therefore, the set {µε t \| 0 < ε 6 1} is tight in M( .. Γ) . This implies that, for any 0 6 t0 < t1 < · · · < tn, the set {µε t0,t1,...,tn \| 0 < ε 6 1} is tight in M( .. Γn+1). Hence, by (56), µε t0,t1,...,tn → µ0 t0,t1,...,tn weakly in M( .. Γn+1) as ε → 0. Thus, the theorem is proved. � Acknowledgements The authors acknowledge the financial support of the SFB 701 “ Spectral structures and topo- logical methods in mathematics”. References 1. Alberverio S., Kondratiev Yu.G., Röckner M., J. Func. Anal., 1998, 154, 444–500. 2. Alberverio S., Kondratiev Yu.G., Röckner M., J. Func. Anal., 1998, 157, 242–291. 3. Bertini L., Cancrini N., Cesi F., Ann. Inst. H. Poincaré Probab. Statist., 2002, 38, 91–108. 4. Brox T. Gibbsgleichgewichtsfluktuationen für einige Potentiallimiten, PhD thesis. Universität Heidel- berg, 1980. 5. Boudou A.-S., Caputo P., Dai Pra P., Posta G., J. Funct. Anal., 2006, 232, 222–258. 6. Davies E.B. Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge, 1989. 7. Dobrushin R.L. Ukrain. Mat. Zh., 1956, 8, 127–134 (Russian). 8. Doob J.L. Stochastic Processes. John Wiley & Sons, New York/London, 1953. 9. Ethier S.N., Kurtz T.G. Markov processes. Characterization and Convergence. John Wiley, New York, 1986. 10. Finkelshtein D.L., Kondratiev Yu.G., Lytvynov E., Random Oper. Stoch. Equ., 2007, 15, 105–126. 11. Kallenberg O. Random Measures. Academic Press, London/New York/ San Francisco, 1976. 12. Kondratiev Yu.G., Lytvynov E., Ann. Inst. H. Poincaré Probab. Statist., 2005, 41, 685–702. 13. Kondratiev Yu.G., Lytvynov E., Röckner M., Publ. Res. Inst. Math. Sci., 2002, 39, 1–48. 14. Kondratiev Yu.G., Lytvynov E., Röckner M., Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2007, 10, 185–209. 15. Kingman J.F.C. Poisson Processes. Oxford University Press, New York, 1993. 16. Ma Z.-M., Röckner M., Osaka J. Math., 2000, 37, 273–314. 17. Minlos R.A., Funktsional’nyj Analiz i Ego Prilozhenija, 1967, 1, No. 2, 60–73. 720 Non-equilibrium stochastic dynamics in continuum 18. Nelson E., J. Math. Phys., 1964, 5, 332–343. 19. Nguyen X.X., Zessin H., Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1976, 37, 191–200. 20. Parthasarathy K.R. Probability Measures on Metric Spaces. Academic Press, New York/London, 1967. 21. Ruelle D. Statistical Mechanics. Rigorous Results. Benjamins, 1969. 22. Shiga T., Takanashi Y., Publ. RIMS, Kyoto Univ., 1974, 9, 505–516. 23. Surgailis D., Probab. Math. Statist., 1984, 3, 217–239. 24. Surgailis D., On Poisson multiple stochastic integrals and associated equilibrium Markov processes. – In: “Theory and application of random fields (Bangalore, 1982)”, pp. 233–248, Lecture Notes in Control and Inform. Sci., Vol. 49, Springer, Berlin, 1983. 25. Vershik A.M., Gelfand I.M., Graev M.I., Russian Math. Surveys, 1975, 30, No. 6, 1–50. Нерiвноважна стохастична динамiка в континуумi: випадок вiльної системи Ю.Кондратьєв1,2,3, Є.Литвинов4, М.Рьокнер1,3 1 Факультет математики, Унiверситет м. Бiлефельд, Нiмеччина 2 Iнститут математики, Київ, Україна 3 Дослiдницький центр BiBoS, Унiверситет м. Бiлефельд, Нiмеччина 4 Факультет математики, Унiверситет м. Суонсi, Суонсi, Великобританiя Отримано 31 сiчня 2008 р. Ми дослiджуємо проблему iдентифiкацiї вiдповiдного простору станiв для стохастичної динамiки вiльних частинок у континуумi з їх можливим народженням i знищенням. В цiй динамiцi рух окремої частинки описується за допомогою фiксованого маркiвського процесу M на рiмановому многовидi X. Головною проблемою тут є можливий колапс системи у наступному сенсi. Незважаючи на те, що початковий розподiл частинок є локально скiнченний, може iснувати в X така компактна множина, що з ймовiрнiстю 1 в момент часу t > 0 у цю множину потрапить безмежна кiлькiсть частинок. Ми вважаємо, що X має безмежний об’єм, а також, для кожного α > 1, розглядаємо множину Θα всiх безмежних конфiгурацiй в X, для яких число частинок в компактнiй множинi є обмежене добутком певної сталої i α-го степеня об’єму цiєї множини. Ми знайшли цiлком загальнi умови на процес M , за яких вiдповiдний безмежно-частинковий процес, стартуючи з довiльної конфiгурацiї Θα, нiколи не залишить Θα, маючи при цьому cadlag (або, навiть, неперервнi) траєкторiї в ультра-слабкiй тополо- гiї. Можливi такi застосування наших результатiв: броунiвський рух на конфiгурацiйнному просторi i вiльна динамiка Глаубера на конфiгурацiйному просторi (процес народження-знищення на X): вiль- на динамiка Кавасакi на конфiгурацiйному просторi. Ми також показуємо, що у випадку X = Rd, для широкого класу стартових розкладiв (нерiвноважна) вiльна динамiка Глаубера є скейлiнговою границею (нерiвноважної) вiльної динамiки Кавасакi. Ключовi слова: процеси народження-знищення, броунiвський рух у конфiгурацiйному просторi, неперервнi системи, динамiка Глаубера, динамiка Кавасакi, мiра Пуассона PACS: 02.50.Ey, 02.50.Ga 721 722

Non-equilibrium stochastic dynamics in continuum: The free case

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