A Widom–Rowlinson Jump Dynamics in the Continuum

We study the dynamics of an infinite system of point particles of two types. They perform random jumps in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} in the course of which particles of different types repel each other whereas those of the same type do not interact. The states of the system are probability measures on the corresponding configuration space. The main result is the construction of the global (in time) Markov evolution of such states by means of correlation functions. It is proved that for each initial sub-Poissonian state μ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _0$$\end{document}, the states evolve μ0↦μt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _0 \mapsto \mu _t$$\end{document} in such a way that μt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _t$$\end{document} is sub-Poissonian for all t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}. The mesoscopic (approximate) description of the evolution of states is also given. The stability of translation invariant stationary states is studied. In particular, we show that some of such states can be unstable with respect to space-dependent perturbations.


Posing
In this paper, we study the dynamics of an infinite system of point particles of two types placed in R d . The particles perform random jumps in the course of which particles of different types repel each other whereas those of the same type do not interact. We do not require that the B Yuri Kozitsky jkozi@hektor.umcs.lublin.pl Joanna Barańska asia13p@wp.pl 1 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland repulsion is of hard-core type. This model can be viewed as a dynamical version of the Widom-Rowlinson model [15] of equilibrium statistical mechanics -one of the few models of phase transitions in continuum particle systems, see the corresponding discussion in [7] where a similar birth-and-death model was introduced and studied. Note that the latter paper and this our work are the only publications where the dynamics of two-component infinite systems of interacting particles have been studied so far.
The phase space of our model is defined as follows. Let denote the set of all γ ⊂ R d that are locally finite, i.e., such that γ ∩ is a finite set whenever ⊂ R d is compact. Thus, is a configuration space as defined in [1,3,8,11]. In order to take into account the particle's type we use the Cartesian product 2 = × , see [5,7,9], the elements of which are denoted by γ = (γ 0 , γ 1 ). In a standard way, 2 is equipped with a σ -field of measurable subsets which allows one to deal with probability measures as states of the system. Among them one may distinguish Poissonian states in which the particles are independently distributed over R d . Sub-Poissonian states are characterized by a rather weak dependence between the particle's positions, see Definition 2.1 below. As was shown in [10], for infinite particle systems with birth-and-death dynamics the evolution of states exists and is such that they remain sub-Poissonian globally in time if the birth of the particles is in a sense controlled by their state-dependent death. For conservative dynamics, in which the particles do not appear or disappear and only change their positions, the interaction may in general change the sub-Poissonian character of the state in finite time (even cause an explosion), e.g., due to an infinite number of simultaneous correlated jumps. Thus, the conceptual outcome of the present study is that this is not the case for the considered model. Note that we do not impose any kind of restrictions on the model parameters, and the existence of the global in time evolution of states is proved to hold even if there may exist multiple equilibrium states (phase transitions), and hence no ergodicity can be expected. Our another aim in this paper is to study the dynamics of the considered model in the mesoscopic limit, which yields its though an approximate (mean-field like) but more detailed picture. We do this and show how this approximate picture and the description of the evolution of states are related to each other.

Presenting the Results
The Markov evolution of states of the system which we consider is described by the Kolmogorov equation where F t : 2 → R is an observable and the operator L specifies the model. It has the following form The evolution of states is supposed to be obtained by solving the Fokker-Planck equation 3) related to that in (1.1) by the duality 2 F t (γ )μ 0 (dγ ) = 2 F 0 (γ )μ t (dγ ). (1.4) As is usual for models of this kind, the direct meaning of (1.1) or (1.3) can only be given for states of finite systems, cf. [12]. In this case, the Banach space where the Cauchy problem in (1.3) is defined can be the space of signed measures with finite variation. For infinite systems, the evolution of states is constructed indirectly, by employing correlation functions and or the related Bogoliubov functionals, see [3,[6][7][8][9][10] and the references quoted therein. In this paper, in describing the evolution of states, see Theorem 3.5 below, we mostly follow the scheme elaborated in [10]. It consists in: (a) constructing the evolution of correlation functions k 0 → k t , t < T < +∞, based on the Cauchy problem in (3.1); (b) proving that each k t is the correlation function of a unique sub-Poissonian state μ t ; (c) constructing the continuation of thus obtained evolution k μ 0 = k 0 → k t = k μ t to all t > 0.
Step (b) is based on the use of the Denjoy-Carleman theorem [4]. In realizing step (c), we crucially use the result of (b). Our description of the mesoscopic limit is based on the scaling procedure described in Sect. 4. It is equivalent to the Lebowitz-Penrose scaling used in [7], and also to the Vlasov scaling used in [3,6]. In this procedure, passing to the mesoscopic level amounts to considering the system at different spatial scales parameterized by ε ∈ (0; 1] in such a way that ε = 1 corresponds to the micro-level, whereas the limit ε → 0 yields the meso-level description in which the corpuscular structure disappears and the system turns into a (two-component) medium characterized by a density function. The evolution of the latter is supposed to be found from the kinetic equation (3.15). In Theorem 3.8, we show that the kinetic equation has a unique global (in time) solution in the corresponding Banach space. In Theorem 3.9, we demonstrate that the micro-and meso-scopic descriptions are indeed connected by the scaling procedure in the sense of Definition 3.6. In Theorems 3.10 and 3.11, we describe the stability of translation invariant stationary solutions of the kinetic equation. In particular, we show that some of such solutions can be unstable with respect to space-dependent perturbations.
The rest of the paper has the following structure. In Sect. 2, we give necessary information on the analysis on configuration spaces and on the description of sub-Poissonian states on such spaces with the help of Bogoliubov functionals and correlation functions. We also describe in detail the model which we consider. In Sect. 3, we formulate the results mentioned above and prove Theorems 3.10 and 3.11. We also provide some comments; in particular, we relate our results with those of [7] describing a birth-and-death version of the Widom-Rowlinson dynamics in the continuum. Section 4 is dedicated to developing our main technical tool-Proposition 4.2. By means of it we realize step (a) in proving Theorem 3.5, see above. Steps (b) and (c) are based on Lemmas 5.1, 5.2, 5.4 and 5.5 proved in Sect. 5. Section 6 is dedicated to the proof of Theorems 3.8 and 3.10.

Two-Component Configuration Spaces
Here we present necessary information on the subject. A more detailed description can be found in, e.g., [5,7,9].
denote the sets of all Borel and all bounded Borel subsets of R d , respectively. The configuration space mentioned above is equipped with the vague topology and thus with the corresponding Borel σ -field B( ). It is known, see [11,Sect. 2.2], is generated by the counting maps γ → N (γ ) := |γ ∩ |, where | · | denotes cardinality. The elements of 2 := × are γ = (γ 0 , γ 1 ), i.e., the one-component configurations are always written with the subscript i = 0, 1. By B( 2 ) we denote the corresponding product σ -field. For i ∈ B(R d ), i = 0, 1, we denote = 0 × 1 and set is a sub-field of B( 2 ). Let p : 2 → 2 be the projection It is clearly measurable, and thus the sets Let P( 2 ) denote the set of all probability measures on ( 2 , B( 2 )). For a given μ ∈ P( 2 ), its projection on Let π be the standard homogeneous Poisson measure on ( , B( )) with density (intensity) = 1. Then the product measure π 2 := π ⊗ π is a probability measure on ( 2 , B( 2 )). By P π ( 2 ) we denote the set of all μ ∈ P( 2 ), for each of which the projections μ , with all possible = 0 × 1 , i ∈ B b (R d ), i = 0, 1, are absolutely continuous with respect to the corresponding projections of π 2 . It is known, see [5,Proposition 3.1], that for each μ ∈ P π ( 2 ) the following holds Since we are going to deal with elements of P π ( 2 ) only, from now on we assume that the configurations γ 0 and γ 1 are subsets of one and the same space R d .
Let 0 ⊂ be the set of all finite configurations. It is known that 0 ∈ B( ), see [11,Sect. 2.2]. Hence B( 0 ) := {A ⊂ 0 : A ∈ B( )} ia a sub-σ -field of B( ). At the same time, 0 can be equipped with the topology related to the Euclidean topology of R d , see [11,Sect. 2.1]. Let B( 0 ) be the corresponding Borel σ -field of subsets of 0 . Clearly, a function, The relationship between this measurability and the corresponding property of g : 0 ⊂ → R is clarified by Obata's result, see Lemma 1.1 and Proposition 1.3 in [14], by which B( 0 ) = B( 0 ). Thus, such a function g is B( )/B(R)-measurable if and only if there exist {g (n) : n ∈ N} with the above mentioned properties. For completeness, one adds to this family also g (0) = g(∅).
By standard arguments B bs ( 2 0 ) is a measure-defining class for measures on ( 2 0 , B( 2 0 )). The Lebesgue-Poisson measure λ on ( 2 0 , B( 2 0 )) is then defined by the following formula, see [5] and [9, page 130], . , x n 0 ; y 1 , . . . , y n 1 ) ×dx 1 · · · dx n 0 dy 1 · · · dy n 1 , which has to hold for all G ∈ B bs ( 2 0 ) with the usual convention regarding the cases n i = 0. The same can also be written as where both λ i are the copies of the standard Lebesgue-Poisson measure on the singlecomponent set 0 , see, e.g., [11]. In the sequel, both Lebesgue-Poisson measures on 2 0 and on 0 will be denoted by λ if no ambiguity may arise. For γ ∈ 2 , by writing η γ we mean that η i γ i , i = 0, 1, i.e., both η i are finite subsets of the corresponding γ i . For G ∈ B bs ( 2 0 ), we set see [5,9]. Note that the sums in (2.4) are finite and K G is a cylinder function on 2 . The latter means that it is B( 2 (G) )-measurable. Moreover, cf.

Correlation Functions
In the approach we follow, see [3,6,10], the evolution of states is constructed in the next way. Let Θ denote the set of all compactly supported continuous maps θ = (θ 0 , θ 1 ) : is measurable and bounded. Hence, for a state μ, one may define -the so called Bogoliubov functional corresponding to μ, considered as a map Θ → R. Definition 2.1 By P exp ( 2 ) we denote the set of sub-Poissonian states consisting of all those μ ∈ P π ( 2 ) for which B μ can be continued to an exponential type entire function of It can be shown that a given μ ∈ P π ( 2 ) is sub-Poissonian if and only if B μ can be written in the form, cf. (2.3), This, in particular, means that k μ is essentially bounded with respect to the Lebesgue-Poisson measure λ defined in (2.2). For the (heterogeneous) Poisson measure π , the Bogoliubov functional is where = ( 0 , 1 ) is the (two-component) density function. Then by (2.2) and (2.7) we have If one rewrites (2.6) in the form then the action of L on F as in (1.2) can be transformed to the action of L on k μ according to the following rule The main advantage of this is that k μ is a function of finite configurations. For μ ∈ P exp ( 2 ) and = ( 0 , 1 ), i ∈ B b (R d ), let μ be as in (2.1). Then μ is absolutely continuous with respect to the corresponding restriction λ of the measure defined in (2.2), and hence we may write Then the correlation function k μ and the Radon-Nikodym derivative R μ are related to each other by, cf. (2.3), Note that (2.13) relates R μ with the restriction of k μ to 2 . The fact that these are the restrictions of one and the same function k μ : 2 0 → R corresponds to the Kolmogorov consistency of the family {μ } . By (2.4), (2.1), and (2.12) we get 2 (K G)(γ )μ(dγ ) = G, k μ , holding for each G ∈ B bs ( 2 0 ) and μ ∈ P exp ( 2 ). Here for suitable G and k. Define By [11, Theorems 6.1 and 6.2 and Remark 6.3] one can prove that the following holds.

Proposition 2.2
Let a measurable function k : 2 0 → R have the following properties: with (c) holding for some C > 0 and λ-almost all η ∈ 2 0 . Then there exists a unique μ ∈ P exp ( 2 ) for which k is the correlation function.

The Model
The model we consider is specified by the operator L given in (1.2) where the coefficients are supposed to be of the following form By (1.2) and (2.11) one obtains the action of L in the following form. For x ∈ R d , we set 3), we introduce the maps where e is as in (2.8). Then This expression can be derived from the general form obtained in [9, Eqs. (4.4) and (4.5), page 142] by using the concrete form of the kernels given in (2.17). It can also be obtained directly from (1.2) and (2.11). Note that in (2.23) we use the convention x∈∅ = 0.

The Microscopic Level
As mentioned above, instead of directly studying the evolution of states by solving the problem in (1.3), we pass from μ 0 to the corresponding correlation function k μ 0 and then consider the problem where L is given in (2.23). For this problem, we prove the existence of a unique global solution k t which is the correlation function of a unique state μ t ∈ P exp ( 2 ). We begin by defining the problem (3.1) in the corresponding spaces of functions k : 2 0 → R. From the very representation (2.7), see also (2.2), it follows that μ ∈ P exp ( 2 ) implies holding for λ-almost all η ∈ 2 0 , some C > 0, and ϑ ∈ R. Keeping this in mind we set k ϑ = ess sup Then K ϑ := {k : 2 0 → R : k ϑ < ∞} is a Banach space with norm (3.2) and the usual linear operations. In fact, we are going to use the ascending scale of such spaces K ϑ , ϑ ∈ R, with the property where → denotes continuous embedding. Set, cf. (2.14) and (2.15), It is a subset of the cone By Proposition 2.2 it follows that each k ∈ K ϑ such that k(∅, ∅) = 1 is the correlation function of a unique μ ∈ P exp ( 2 ). Then we define As a sum of Banach spaces, the linear space K is equipped with the corresponding inductive topology which turns it into a locally convex space. For a given ϑ ∈ R, by (2.21)-(2.23) we define L ϑ as a linear operator in K ϑ with domain Proof For ϑ < ϑ, by (2.20), (2.21), (2.22), and (3.2) we have Now we apply the latter two estimates together with (2.18) in (2.23) and obtain By means of the inequality x exp(−σ x) ≤ 1/eσ , x, σ > 0, we get from (3.2) and (3.10) the following estimate which yields the proof.

Corollary 3.2
For each ϑ, ϑ ∈ R such that ϑ < ϑ, the expression in (2.23) defines a bounded linear operator L ϑϑ : K ϑ → K ϑ the norm of which can be estimated by means of (3.11).
In what follows, we consider two types of operators defined by the expression in (2.23): (a) unbounded operators (L ϑ , D(L ϑ )), ϑ ∈ R, with domains as in (3.7) and Lemma 3.1; (b) bounded operators L ϑϑ described in Corollary 3.2. These operators are related to each other in the following way: By means of the bounded operators L ϑϑ : K ϑ → K ϑ we define also a continuous linear operator L : K → K, see (3.6). In view of this, we consider the following two equations. The first one is d dt considered as an equation in a given Banach space K ϑ . The second equation is (3.1) with L given in (2.23) considered in the locally convex space K.
The main result of this subsection is contained in the following statement.

The Mesoscopic Level
As is commonly recognized, see [2,Chapter 8] and [13], the comprehensive understanding of the behavior of an infinite interacting particle system requires its multi-scale analysis. In the approach which we follow, see [3] (jump dynamics) and [7] (two-component system), passing from the micro-to the mesoscopic levels amounts to considering the system at different spatial scales parameterized by ε ∈ (0, 1] in such a way that ε = 1 corresponds to the micro-level, whereas the limit ε → 0 yields the meso-level description in which the corpuscular structure disappears and the system turns into a (two-component) medium characterized by a density function Then the evolution 0 → t , obtained from a kinetic equation, approximates (in the mean-field sense) the evolution of the system's states as it may be seen from the mesoscopic level.

The Kinetic Equation
Keeping in mind that the Poissonian state π is completely characterized by the density , see (2.9) and (2.10), we introduce the following notion, cf. [3, p. 1046] and [7, p. 70].
Our aim is to show that the evolution μ 0 → μ t obtained in Theorem 3.5 preserves the property just defined relative to the time dependent density t = ( 0,t , 1,t ), obtained from the following system of kinetic equations where * denotes convolution; e.g.,

Definition 3.7
By the global solution of the system of kinetic equations (3.15), subject to an initial condition, we understand a continuously differentiable map By the positive solution of (3.15) on the time interval [0, T ], 0 < T < ∞, we mean the corresponding restriction of this map.
The relationship between the micro-and mesoscopic descriptions is established by the following statement.

The Stationary Solutions
Stationary solutions i,t = i , t ≥ 0, of the system in (3.15) are supposed to solve the following system of equations It might be instructive to rewrite it in the form For each C i > 0, i = 0, 1, the system in (3.21) has constant solutions ψ i ≡ C i . Then the corresponding i are to be found from For given C 0 , C 1 > 0, let S( C 0 , C 1 ) be the set of all positive ( 0 , 1 ) ∈ L ∞ (R d → R 2 ) that satisfy (3.23). Let also S c ( C 0 , C 1 ) be the subset of S( C 0 , C 1 ) consisting of constant solutions i ≡ C i , i = 0, 1, with C i satisfying (3.24). The symmetric case of (3.24) with specific values of C i (as mentioned above) was studied in [7,Sect. 5]. Namely, for Here with some x 0 ∈ (0, 1). This solution is a stable node for a < e. For a > e, there exist three solutions: (a) The first two solutions are stable nodes and x 3 > 1.
The stability means the existence of a small neighborhood in S c ( C 0 , C 1 ) of the mentioned solution, which does not contain any other solution.
Let us now turn to the study of the stability of the constant solutions of (3.23) with respect to perturbations i = C i + i , i = 0, 1. By (3.23) and (3.24) we conclude that the perturbations ought to satisfy This means that there exists δ > 0 such that i ≡ C i , i = 0, 1, is the only solution in the set Proof Assume that 0 L ∞ > 0. By means of the inequality |e −α − 1| ≤ |α|e |α| we get from (3.26) holding for small enough δ in view of (3.27). This contradicts the assumption, and hence yields 0 = 0. The corresponding estimate for 1 L ∞ is obtained analogously.
. Then each solution of (3.26) is a fixed point of the nonlinear map : defined by the right-hand of (3.26). Note that this takes values in The zero solution of (3.26) gets unstable whenever there exist nonzero = ( 0 , 1 ) in the kernel of I − , where is the Fréchet derivative of at = (0, 0). By (3.26) we have Since contains convolutions, it can be partially diagonalized by means of the Fourier transformφ Note that bothφ i are uniformly continuous on R d and satisfy |φ i ( p)| ≤φ i (0) = φ i , that follows from their positivity. Moreover, |φ i ( p)| → 0 as | p| → +∞ (by the Riemann-Lebesgue lemma). Note also thatˆ i , i = 0, 1, exist since i are supposed to be integrable.

Theorem 3.11
Assume that the following holds, cf. (3.27), Then the constant solution i ≡ C i of (3.23), and hence of (3.20), is unstable with respect to the Proof In view of the mentioned continuity ofφ i and the Riemann-Lebesgue lemma, the condition in (3.29) implies the existence of p ∈ R d \ {0} such that The instability in question takes place whenever the equation = , cf. (3.28), has nonzero solutions in the considered space. By means of the Fourier transform it can be turned intô that has to hold for some p ∈ R \ {0}, which is certainly the case in view of (3.30).
Given C i , i = 0, 1, let = ( 0 , 1 ) solve (3.26). Then = (C 0 + 0 , C 1 + 1 ) solves (3.23) with C i as in (3.24) and hence lies in S( C 0 , C 1 ). Then Theorem 3.11 describes the instability of the solution ≡ (C 0 , C 1 ) in the latter set. For this reason, it is independent of the jump kernels a i . In order to study the corresponding instability in the set of all solutions of (3.20), one has to rewrite (3.20) in the form ( ) = 0 and then to show that the Fréchet derivative of at ≡ (C 0 , C 1 ), defined as a bounded linear self-map of , has nonzero in its kernel. By means of the arguments used in the proof of Theorem 3.11 one readily obtains that this is equivalent to, cf. (3.31), that has to hold for some nonzero p ∈ R d . Hereâ i ( p), i = 0, 1, are the Fourier transforms of the jump kernels, see (2.18). Thus, if both these kernels are such thatâ i ( p) <â i (0) = α i for all nonzero p, then the latter condition turns into that in (3.31).

The Microscopic Description
The existence of the global in time evolution stated in Theorem 3.5 is proved in the subsequent sections without any restrictions on the model parameters α i and φ i , i = 0, 1, see (2.18) and (2.19), respectively. That is, the global evolution exists, however, its ergodicity can hardly be expected. The analysis of the kinetic equation made in Theorem 3.11 points to the possibility of having a phase transition in the model, i.e., to the possibility of having multiple stationary states μ ∈ P exp ( 2 ). The only work on the Widom-Rowlinson dynamics of an infinite particle system is that in [7] where a birth-and-death (rather immigration-emigration) version was studied. In that version, the particles of two types appear and disappear at random; the appearance is subject to the repulsion from the particles of the other type. The system's evolution was described by means of the corresponding initial value problem for the Bogoliubov functional. Namely, for t < T , where T < ∞ is expressed via the model parameters, in [7, Theorem 1] there was constructed the evolution B μ 0 → B t , where B t : L 1 (R d → R 2 ) → R is an exponential type entire function and hence can be written down as, cf. (2.7), However, it was not shown that B t is the Bogoliubov functional, i.e., that k t above is the correlation function, of some state μ ∈ P exp ( 2 ). In the present work, for the jump version of the Widom-Rowlinson model we show (Theorem 3.5) that: (a) the evolution k μ 0 → k t , and hence also B μ 0 → B t , can be continued to all t > 0; (b) for each t > 0, B t is the Bogoliubov functional of a unique sub-Poissonian state μ t .

The Mesoscopic Description
In passing to the mesoscopic level of description, we use a scaling procedure described in Sect. 4 below. It is equivalent to the Lebowitz-Penrose scaling used in [7], and also to the Vlasov scaling used in [3,6]. Our Theorem 3.9 is analogous to [7, Theorem 2] proved for the birth-and-death version. Note that the convergence in (3.19) is uniform in t, whereas in the mentioned statement of [7] the convergence is point-wise. Now we turn to the stationary solutions of (3.15) which one obtains from the system in (3.20), or, equivalently, in (3.21). The latter may have nonconstant solutions ψ i , which then can be used to find the corresponding i from (3.22). These solutions may depend on the jump kernels a i . The set of all solutions of (3.20) contains the sets S( C 0 , C 1 ) for each pair C 0 , C 1 > 0. The corresponding solutions i are independent of the jump kernels. Moreover, S( C 0 , C 1 ) is exactly the set of solutions of the birth-and-death kinetic equation [7, Eq. (5.1)] corresponding to the specific values of C i . Thus, our Theorems 3.10 and 3.11 describe also the birth-and-death kinetic equation, which is an extension of the study in [7,Sect. 5].

The Rescaled Evolution
In this section, we construct the evolution q 0,ε → q t,ε , ε ∈ (0, 1], which then will be used for: (a) obtaining the evolution stated in Theorem 3.5 in the form k t = q t,1 ; (b) proving Theorem 3.9. To this end along with L defined in (2.23) we will use where L ε is obtained from L by multiplying both φ i by ε, and We refer the reader to [3,7] for more information on deriving operators as in (4.1). Denote, cf. (2.21), For ε ∈ (0, 1], let Q i y,ε be as in (2.22) with t i x replaced by t i x,ε given in (4.2). Then the action of L ε, is given by the right-hand side of (2.23) with both Q i y replaced by the corresponding Q i y,ε and τ i x replaced by τ i x,ε . Note that, cf. (2.19), For each ϑ ∈ R, k ∈ K ϑ , and ε ∈ (0, 1], by (4.4) both Q i y,ε k satisfy the estimates as in (3.8) and (3.9). Therefore, L ε, k satisfies (3.10), which allows one to introduce the corresponding linear operators L ε, , see also Corollary 3.2 and (3.12). Thus, along with (3.13) we will consider the problem Its solutions q t,ε ∈ D(L ϑ ) ⊂ K ϑ are defined analogously as in Definition 3.3. For ϑ, ϑ ∈ R such that ϑ < ϑ , we set, cf. (3.11), For a fixed ϑ ∈ R, T (ϑ , ϑ)) can be made as big as one wants by taking small enough ϑ. However, if ϑ is fixed, then sup ϑ >ϑ where δ(ϑ) is the unique positive solution of the equation The supremum in (4.7) is attained at Note also that δ(ϑ) → 0, and hence τ (ϑ) → 0, as ϑ → +∞.
Proof Take T < τ(ϑ 0 ) and then pick ϑ ∈ (ϑ 0 , ϑ 0 + δ(ϑ 0 )) such that T < T (ϑ , ϑ 0 ). Our aim is to construct the family defined by the series In (4.9), L(K ϑ 0 , K ϑ ) stands for the Banach space of bounded linear operators acting from K ϑ 0 to K ϑ equipped with the corresponding operator norm. In for n ∈ N. Now we take into account that ϑ l − ϑ l−1 = (ϑ − ϑ 0 )/n and that L ε, satisfies (3.11) for all ε ∈ (0, 1]. This yields the following estimate see (3.11) and (4.6). Then we apply (4.12) in (4.11) and conclude that the series in (4.10) converges in the operator norm, uniformly on [0, T ], to the operator-valued function In a similar way, we get Then (4.15) see Lemma 3.1, is a solution of (4.5) on the time interval [0, τ (ϑ 0 )) since T < τ(ϑ 0 ) has been taken in an arbitrary way and L ε, ϑϑ q t = L ε, ϑ q t whenever q t ∈ K ϑ , see (3.12). Let us prove that the solution given in (4.15) is unique. In view of the linearity, to this end it is enough to show that the problem in (4.5) with the zero initial condition has a unique solution. Assume that v t ∈ D(L ε, ϑ ) is one of such solutions. Then v t lies in K ϑ for each ϑ > ϑ, see (3.3). Fix any such ϑ and then take t < τ(ϑ 0 ) such that t < T (ϑ , ϑ). Then, cf. (3.12), where n ∈ N is an arbitrary number. Similarly as above we get from the latter Since n is an arbitrary number, this yields v s = 0 for all s ∈ [0, t]. The extension of this result to all t < τ(ϑ 0 ) can be done by repeating this procedure due times.

The Proof of Theorem 3.5
With the help of Proposition 4.2 we have already obtained the unique solution of (3.13) in the form where k μ 0 ∈ K ϑ 0 and ϑ ∈ (ϑ 0 , ϑ 0 + δ(ϑ 0 )) is taken such that t < T (ϑ 0 + δ(ϑ 0 ), ϑ). To prove Theorem 3.5 we first show (Lemma 5.1) that k t lies in the cone (3.4) and hence is a correlation function of a unique state μ t . Then, in Lemma 5.2, we construct an auxiliary evolution u 0 → u t , with which we compare the evolution k μ 0 → k t defined in (5.1). Thereby, we construct the extension of k μ t to all t > 0 as stated in the theorem.

The Identification Lemma
Our aim now is to show that the solution of (3.13) given in (5.1) has the property k t ∈ K ϑ , see (3.4). By this one can identify k t as k μ t for a unique state μ t . Recall that bounded operators L ϑϑ , ϑ < ϑ, were introduced in Corollary 3.2.
Note that (5.17) yields also that where G and k ,N t are as in (5.13) and cf. (5.10).

An Auxiliary Evolution
The evolution which we construct now will be used to continue the solution k t given in (5.1) to all t > 0 as stated in Theorem 3.5. The construction employs the operator obtained from L given in (2.23) by putting φ i = 0, i = 0, 1, and then dropping the second and fourth terms. Note thatL does not correspond to any Markov evolution as it describes (free) "half-jumps". Similarly as in (3.11), we get which allows us to introduce the operators (L ϑ , D(L ϑ )) andL ϑϑ ∈ L(K ϑ , K ϑ ) such that, cf. (3.12), ∀k ∈ ϑ L ϑϑ k =L ϑ k, ϑ < ϑ.
Proof Proceeding as in the proof of Proposition 4.2, by means of the estimate in (5.21) we prove the convergence of the series in (5.24). This allows also for proving (5.26), which yields the existence of the solution of (5.27) in the form given in (5.28). The uniqueness is proved analogously as in the case of Proposition 4.2. The stated positivity of u t follows from (5.24) and (5.22).

The Global Solution
As follows from Proposition 4.2 and Lemma 5.1, the unique solution of the problem (3.13) with k 0 ∈ K ϑ 0 lies in K ϑ for t ∈ (0, T (ϑ, ϑ 0 )). At the same time, for fixed ϑ 0 , T (ϑ, ϑ 0 ) is bounded, see (4.7). This means that the mentioned solution cannot be directly continued to all t > 0. In this subsection, by a comparison method we prove that, for t ∈ (0, T (ϑ, ϑ 0 )), k t satisfies (3.14) which is then used to get the continuation in question, cf. Corollary 5.3. Recall that the operators Q i y , i = 0, 1, were introduced in (2.22) and the cone K + ϑ was defined in (3.5).
Furthermore, by (5.33) we get We apply this in the last line of (5.34) and obtain which after the limiting transition as in (5.12) yields (5.32) for j = 1. For the same G, we setḠ = e(τ 0 y ; ·)G. Then by (2.21) and the second line in (5.35) we get which after the limiting transition as in (5.12) yields (5.32) for j = 2.
6 The Proof of Theorems 3.8 and 3.9

The Kinetic Equations
Here we prove Theorem 3.8. For a continuous function For a given T > 0, let C T stand for the Banach space of continuous functions Let also C + T denote the set of all positive ∈ C T , i.e., such that i,t (x) ≥ 0 for all i = 0, 1, t ∈ [0, T ], and Lebesgue-almost all x. By means of F i,t introduced in (6.1) we then define the map such that the values of F i ( ) are given in the right-hand sides of (6.1). By direct inspection one concludes that both F i,t ( ), i = 0, 1, are continuously differentiable in t, and the function as in (6.2) By (6.1) one readily gets that F : For ∈ Δ C , from the first equation in (6.1) one gets Similarly, F 1,t ( ) L ∞ ≤ Ce α 1 t , which proves (6.6). To solve (6.4) we apply the Banach contraction principle. To this end we pick T > 0 such that F is a contraction on (6.5). We do this as follows. For ,¯ ∈ Δ C , like in (6.7) we obtain ≤ e α 0 t −¯ T 1 − e −2α 0 t 1 − 2 3 C e 3α 0 t − 1 .
The corresponding estimate for F 1,t ( ) − F 1,t (¯ ) L ∞ (with e α 1 t ) can be obtained in the same way. Then according to (6.3) F is a contraction on Δ C whenever C > 0 and T satisfy Now we consider the problem (3.15) for (1) i,t = i,T +t , i = 0, 1, where is the solution just constructed. For this new problem, by (6.9) we have (1) i,0 L ∞ ≤ C 1 := e αT C, i = 0, 1.
Then we repeat the above construction and obtain the solution (1) on the time interval [0, T 1 ] with T 1 > 0 satisfying, cf. (6.8), By further repeating this construction we obtain i,t = i,T +T 1 +···+T n−1 +t , i = 0, 1, t ∈ [0, T n ], where the sequence {T n } n∈N is defined recursively by the condition e 3αT n = 1 + 1 C exp −α (T + T 1 + · · · + T n−1 ) , n ∈ N. (6.10) Thus, the global solution in question exists if the series n T n is divergent. Assume that this is not the case. Then the right-hand side of (6.10) is bounded from below by some b > 1, uniformly in n. This yields that T n ≥ log b/3α > 0, holding for all n ∈ N, which contradicts the summability of {T n } n∈N and thus completes the proof of Theorem 3.8.

The Scaling Limit
For each k and λ-almost all η ∈ 2 0 , we have that the following holds, cf.
y,0 k)(η 0 , η 1 )dy. (6.11) Like above, for each ϑ ∈ R and k ∈ K ϑ , both Q i y,0 k satisfy the estimates as in (3.8) and (3.9). Then for ϑ, ϑ ∈ R such that ϑ < ϑ, V k ϑ is bounded by the right-hand side of (3.11). This allows one to define the operators V ϑ and V ϑϑ analogous to L ϑ and L ϑϑ , respectively. For t being the solution as in Theorem 3.8, k π t satisfies d dt k π t = V ϑϑ k π t , t > 0, (6.12) where ϑ ∈ R is such that k π t ∈ K ϑ , see (3.18), and ϑ > ϑ is arbitrary. This can be checked by direct calculations based on (6.11) and (3.15). Moreover, if we set C = 0 ∞ , see (3.17), then k π t satisfies (3.14) with this C, which follows from (3.18). Thus, by Corollary 5.3 we conclude that k π t ∈ K ϑ(t) for all t > 0.