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 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 $\alpha\ge1$, we consider the set $\Theta_\alpha$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $\alpha$-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 $\Theta_\alpha$, will never leave $\Theta_\alpha$, 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 $X=\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.


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. After Surgailis' papers, equilibrium independent infinite particle processes have been studied by many authors, see e.g. [22]. However, the problem of identification of 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 R d . Then, one can easily find a configuration γ ∈ Γ 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, for study of the non-equilibrium stochastic dynamics of 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 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 complete, 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 sample 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 Mar-kovian particles. The interest in such particles, rather than just independent Brownian motions, is, in particular, connected with study of the Glauber and Kawasaki dynamics on the configuration space, see [3,12,14,5].
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 the above discussed properties.
So, we fix a system of closed balls B(r) of radius r ∈ N, centered at some x 0 ∈ X, and for each α ≥ 1, define Θ α 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 Θ α 's form an isotone sequence of sets, and we also define Θ as the union of all the Θ α 's. Note that the sets Θ α are "big enough", in the sense that already the smallest set Θ 1 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 ∈ Θ α (or even at each γ ∈ Θ), will never leave Θ α (respectively Θ), and has cadlag (or, even, continuous) sample paths. We then consider the following examples of application 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, compare with [12,14,5]; • Free Kawasaki dynamics on the configuration space, cf. [14].
We also show that, in the case where X = R d , for a probability measure µ on Θ which is translation invariant and has integrable Ursell functions, the (non-equilibrium) free Glauber dynamics having µ as initial distribution may be approximated by (nonequilibrium) 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 Θ α 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 guarantee that the corresponding infinite particle process can be realized as a Markov process on Θ α with cadlag (respectively, continuous) sample paths. In Section 4, we discuss the above mentioned 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 construction of a non-equilibrium dynamics of interacting particles, compare with [2,16,12,14], where the corresponding equilibrium processes were discussed.

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: Here, | · | denotes the cardinality of a set and γ Λ := γ ∩ Λ. One can identify any γ ∈ Γ with the positive Radon measure x∈γ ε x ∈ M(X). Here, ε x 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 are continuous. Here, C 0 (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 x 0 ∈ X and denote by B(r) := B(x 0 , r) the closed ball in X of radius r > 0, centered at x 0 . For each α ≥ 1, we define where vol(B(r)) denotes the volume of B(r). We evidently have that It easy to see that Θ α ∈ B(Γ) for each α ≥ 1, hence Θ ∈ B(Γ).
For each z > 0, let π z 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 π z (dγ) = exp 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, π z (Θ 1 ) = 1 (in fact, by [19], the Poisson measure π z is concentrated on those configurations γ ∈ Γ for which lim r→∞ |γ B(r) |/ vol(B(r)) = z).
Using multiple stochastic integrals with respect to the Poisson random measure, one constructs a unitary isomorphism I z : F (L 2 (X, z dx)) → L 2 (Γ, π z ), see e.g. [23]. Here, F (L 2 (X, z dx)) denotes the symmetric Fock space over L 2 (X, z dx), i.e., ⊙ standing for symmetric tensor product. Let us recall the following result of Surgailis [23]. Let A be a contraction in L 2 (X, z dx), and let Exp(A) denote the second quantization of A, i.e., Exp(A) is the contraction in F (L 2 (X, z dx)) given by Exp(A) ↾ F 0 (L 2 (X, z dx)) = 1 1 1, Exp(A)f ⊗n = (Af ) ⊗n , f ∈ L 2 (X, z dx), n ∈ N. We shall keep the notation Exp(A) for the image of this operator under the isomorphism I z .
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). We recall that A being sub-Markov means that 0 ≤ Af ≤ 1 a.e. for each 0 ≤ f ≤ 1 a.e., f ∈ L 2 (X, z dx). If, additionally, Af n ր 1 a.e. for some sequence f n ր 1, f n ∈ L 2 (X, z dx), then A is called Markov.
Let (T t ) t≥0 be a Markov semigroup in L 2 (X, z dx), and let p t (x, ·), t ≥ 0, x ∈ X, be a corresponding Markov semigroup of kernels. Consider the semigroup (Exp(T t )) t≥0 in L 2 (Γ, π z ), which is Markov by Theorem 2.1. Note that each operator Exp(T t ) is defined only π z -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(T t )) t≥0 .
We consider the infinite product X N with the cylinder σ-algebra on it, denoted by C(X N ). Let us recall the construction of a probability measure on Γ through a product measure on X N , see [25,13].
We define A ∈ C(X N ) as the set of all elements (x n ) ∞ n=1 ∈ X N such that x i = x j when i = j, and the sequence {x n } ∞ n=1 has no accumulation points in X. Let D := {(x, y) ∈ X 2 : x = y}.
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(X N ) on A. Define the mapping which is measurable. Thus, we can define a probability measure µ on (Γ, B(Γ)) as the image of ν under the mapping (2.2). Evidently, the measure µ is independent of the order of the ν n 's. Assume now that and let γ ∈ Γ be such that Then, for each t ≥ 0, we define P t (γ, ·) as the probability measure on Γ given through the product measure where {x n } ∞ 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)) ≤ Cβ m vol(B(r)), r > 0, β ≥ 1. (2.5) By e.g. [6, Proposition 5.5.1], if X has non-negative Ricci curvature, then (2.5) is satisfied with C = 1 and m being equal to the dimension of X.
Proof of Theorem 2.2. For any x ∈ X, denote |x| := dist(x 0 , x), where dist denotes the Riemannian distance.
where the mapping E is given by (2.2). Denote By Lemma 2.1, (2.9) and the Borel-Cantelli lemma, Next, define

By (2.3) and (2.10),
We evidently have |x n | → ∞ as n → ∞, Indeed, fix any (y n ) ∞ n=1 ∈ A ′ and define k ∈ Z + as the number of those y n 's which do not belong to B(x n , |x n |/2). Then Then, by (2.5) and (2.13), we have, for each r ∈ N, for some K ′ ∈ N which is idependent of r (note that vol(B(r)) → ∞ as r → ∞ since X has infinite volume). Hence E((y n ) ∞ n=1 ) ∈ Θ α . Thus, by (2.11) and (2.12), (2.14) Then, it easily follows from (2.14) and the construction of The proof of the last statement of the theorem is quite analogous to the proof of [13, Theorem 5.1], so we only outline it. Let Then, for any ϕ ∈ D, γ ∈ Θ α , and t ≥ 0, we easily get from the definition of P t (γ, ·): Next, it is well known (see e.g. [23, Corollary 2.1]) that, for any ϕ ∈ D, Let us now outline the case where the semigroup (T t ) t≥0 is sub-Markov, but not Markov. We shall assume for simplicity that LetX := X ∪ {∆} be a one-point extension of X, and, as usual, consider (p t ) t≥0 as the extension of (p t ) t≥0 to a Markov semigroup of kernels onX. Let conditions (2.3) and (2.6) be satisfied. Consider the mappinĝ , t ≥ 0, is a Markov semigroup of kernels on E −1 (Θ α ) (notice that condition (2.18) guarantees that, forP t ((x n ) ∞ n=1 , ·)-a.e. (y n ) ∞ n=1 ∈ X N , an infinite number of y n 's belong to X).
Then, (P t ) t≥0 becomes a Markov semigroup of kernels on Θ α . Next, we denote by π z,t the Poisson random measure over X with intensity measure (1 − p t (x, X)) z dx (notice that π z,t is concentrated on finite or infinite configurations in X, depending on whether the integral X (1 − p t (x, X)) z dx is finite or infinite.) Define P z,t (γ, ·) as the convolution of the measuresP t (γ, ·) and π z,t , i.e., Note that P z,t (γ, ·) is indeed concentrated on Γ, since for any fixed ξ 1 ∈ Γ, the π z,t probability of those ξ 2 which satisfy ξ 1 ∩ ξ 2 = ∅ is equal to zero. Furthermore, we have P z,t (γ, Θ α ) = 1 for each t ≥ 0 and γ ∈ Θ α . Indeed, for each t > 0, π z,t is either concentrated on finite configurations or π z,t (Θ α ) = 1, the latter being a consequence of the estimate 1 − p t (x, X) ≤ 1 and the support property of a Poisson measure [19]. Now, the equality P z,t (γ, Θ α ) = 1 follows from the definition (2.19). Next, for any ϕ ∈ D (see (2.15)), γ ∈ Θ α , and t ≥ 0, we have Finally, using (2.20), we have, for any ϕ ∈ D, t, s > 0, and γ ∈ Θ α , from where it easily follows that (P z,t ) t≥0 is a Markov semigroup of kernels on Θ α .
Remark 2.2 Note that, in the case where the semigroup (T t ) t≥0 is Markov, the construction of the Markov semigroup of kernels (P t ) t≥0 is independent of z, whereas in the case where (T t ) t≥0 is sub-Markov, the (P z,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 (P t ) t≥0 , respectively (P z,t ) t≥0 . In what follows, we will assume that a Markov process on X corressponding to the semigroup (T t ) ≥0 has cadlag paths. The latter, in particular, holds if the kernels (p t ) t≥0 determine a Feller semigroup (see e.g. [9, Chapter 2, Theorem 2.7]).
We first consider the case where the semigroup (T t ) t≥0 is Markov. For each x ∈ X, let P x denote the distribution of the Markov process (X t ) t≥0 corresponding to (T t ) 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 i.e., two independent Markov process starting at x and y, x = 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 (3.1) is stronger than (2.3).
Since under P x , X t (ω) = ω(t), t ≥ 0, is a time homogeneous Markov process starting at x, x ∈ X, we conclude, analogously to the above, that Therefore, Hence, we can consider P (xn) ∞ n=1 as a probability measure on Ω α equipped with the trace σ-algebra of C(D([0, ∞), X N ) on Ω α .
Fix any (ω n ) ∞ n=1 ∈ Ω α . Then, we evidently have {ω n (t)} ∞ n=1 ∈ Θ α for all t ≥ 0. Furthermore, for any compact Λ ⊂ X and for any T > 0, there are only a finite number of ω n 's which meet Λ during the time interval [0, T ]. Therefore, {ω n (·)} ∞ n=1 ∈ D([0, ∞), Θ α ), where Θ α is equipped with the relative topology as a subset of Γ. Thus, the following mapping is well-defined: Furthermore, it easily follows from the definition of a cylinder σ-algebra that the mapping (3.4) is measurable. Thus, we can consider the image of the measure P (xn) ∞ n=1 under (3.4). This probability measure on D([0, ∞), Θ α ) will be denoted by P {xn} ∞ n=1 (note that this measure is, indeed, independent of the numeration of points of {x n } ∞ n=1 ∈ Θ α ). Next, it is easy to see that the finite-dimensional distributions of P γ , γ ∈ Θ α , are given through the Markov semigroup of kernels P t (γ, ·) on Θ α (see Theorem 2.2). From here, analogously to [13, Theorem 8.1], we get the following 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.   Proof. For any x ∈ X, r > 0, and ǫ > 0, we have: Let us consider the case where (p t ) t≥0 is sub-Markov. We will assume that, for each x ∈ X, p t (x, X) is continuously differentiable in t ∈ [0, ∞) and there exists δ > 0 such that for some C > 0. We set which is a bounded non-negative function. Using the semigroup property of (p t ) t≥0 , we then conclude from (3.7) and (3.8) that ∂ ∂t p t,x (X) = − X g(y)p t,x (dy), t > 0, x ∈ X. (3.9) We will now assume that (2.18), (3.1), and (3.2) hold. Analogously to the Markovian case, for each γ ∈ Θ α , we define a probability measureP γ on D([0, ∞), Θ α ) through the mappingΩ whereΩ α is a corresponding subset of D([0, ∞),X) N (compare with (3.4)). Let Π z 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, Π z is concentrated on the set of those configurations ξ which satisfy the following condition: for any different (x 1 , t 1 ), (x 2 , t 2 ) ∈ ξ, we have x 1 = x 2 . Any such configuration can be represented as the disjoint union 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 Θ α := Θ α ∪ Γ fin , where Γ fin denotes the set of all finite configurations in X. We endow Θ α with the vague topology. We note that, since the function g(x) is bounded, we have Π z -a.s. that γ (k) ∈ Θ α .

Brownian motion on the configuration space
Assume that (T t ) 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 (3.7) is satisfied.
Theorem 4.1 Assume that the dimension of the manifold X is ≥ 2. Assume that (p t ) t≥0 is either Markov, or (2.18) and (3.7) hold. Further assume that the heat kernel p(t, x, y) satisfies the Gaussian upper bound for small values of t: where n ∈ N, ǫ > 0 and C and D are positive constants. Then the corresponding independent infinite particle process exists as a Markov process M B on (Θ, C(Θ)) with either continuous paths if (p t ) t≥0 is Markov, or cadlag paths if (p t ) t≥0 is sub-Markov.
Remark 4.1 According to [1] and [13], the Markov process M B in Theorem 4.1 may be interpreted as a Brownian motion in the configuration space over X.
Denote by F C ∞ b (C ∞ 0 (X), Θ) the set of all real-valued functions on Θ of the form Assume first that (p t ) t≥0 is Markov. Then, the L 2 -generator of the process M B has the following representation on the set F C ∞ b (C ∞ 0 (X), Θ) (which is a core for this operator): [1,13] for details. In the case where (p t ) t≥0 is sub-Markov, one can analogously show that the L 2 -generator of M B is given on the set F C ∞ b (C ∞ 0 (X), Θ) by Proof of Theorem 4.1. Since the dimension of X is ≥ 2, (3.1) is now satisfied, see e.g. (8.29) in [13]. Next, by [13, Lemma 8.2], (4.1) implies that there exists C > 0 such that sup It follows from here that (3.5) is satisfied. Now, the theorem follows from Corollary 3.1.

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 life time that corresponds to the semigroup (T t f )(x) = e −a(x)t f (x). Thus, the process stays at a starting point for some random time and then dies, with x ∈ X, t > 0.
The function g is now equal to a. Note also that p t (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 M G 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 L 2 -generator of the process M G is given on its core see [12,14].

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: and we assume that κ : is a measurable function satisfying κ(x, y) = κ(y, x), x, y ∈ X, Following [9], we can explicitly construct this Markov process as follows. For each x 0 ∈ X, let {Y (k), k = 0, 1, 2, . . . } be a Markov chain in X starting at x 0 , with transition function Let (Z t ) t≥0 be an independent Poisson process with parameter λ. We now define the Markov process (X t ) t≥0 starting at x 0 by By [9], this process has generator (4.2). Then the corresponding independent infinite particle process exists as a Markov process M K on (Θ α , B(Θ α )) with cadlag paths.
Remark 4.2 According to [14], the Markov process M K can be interpreted as a free Kawasaki dynamics on the configuration space. The L 2 -generator of the process M K has the following representation on F C ∞ b (C ∞ 0 (X), Θ): see [14].
Proof. By construction, the process (X t ) t≥0 has cadlag paths in X. Furthermore, as easily seen, condition (3.1) is now satisfied. In a standard way, the process (X t ) t≥ leads to a Markov semigroup (T t ) t≥0 in L 2 (X, dx). Using (4.3) and the construction of the process, we see that each T t is self-adjoint. By Theorem 3.1, it suffices to prove that condition (3.2) is satisfied. By using (4.5) and the construction of the process (X t ) t≥0 , we get, for any δ > 0.

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 X n such that, for any measurable symmetric function f (n) : Then, the functions k µ and vice versa, see e.g. [21]. Their relation is given by where Γ 0 := {η ⊂ X : 1 ≤ |η| < ∞}, and the summation in (5.2) 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 k µ . Let now X = R d . We fix an arbitrary function ξ ∈ S(R d ) such that ξ(−x) = ξ(x) for all x ∈ R d . Here, S(R d ) 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 M K 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 ≤ γ < 1 and C > 0 such that (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 ≤ i ≤ n where the convergence is in the dx 1 · · · dx n -measure on each compact set in R d . For example, any double-potential Gibbs measure in the low activity-high temperature regime satisfies the above assumptions, see [17,21,4] Let us assume that the initial distribution of the Kawasaki dynamics is µ. We denote this stochastic process by M K µ . We scale this dynamics as follows. Instead of the function ξ used for the construction of M K µ , use the function and denote the corresponding Kawasaki dynamics with initial distribution µ by M K µ, ε . We are interested in the limit of this dynamics as ε → 0.
As in subsec. 4.2, we construct the Glauber dynamics M G using a(x) := ξ and z = k (1) µ . Here and below, for any f ∈ L 1 (R d , dx), we denote We assume that the initial distribution of the Glauber dynamics is µ and denote this stochastic process by M G µ . Below, we will use ..
Γ to denote the space of multiple configurations over R d equipped with the vague topology, see e.g. [11] for details. We have Θ ∈ B( .. Proof. Let (p t, ε ) t≥0 denote the semigroup in L 2 (R d , dx) with generator
Then by the majorized convergence theorem and (5.4), we get the statements (i)-(iii).