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Vlasov scaling for the Glauber dynamics in continuum 1 1 0 Dmitri Finkelshtein∗ Yuri Kondratiev† Oleksandr Kutoviy‡ 2 n a J Abstract 2 2 We consider Vlasov-type scaling for the Glauber dynamics in continuum with a positive integrable potential, and construct rescaled and ] h limiting evolutions of correlation functions. Convergence to the limit- p ing evolution for the positive density system in infinite volume is shown. - Chaos preservation propertyof thisevolution givesapossibility toderive h a non-linear Vlasov-type equation for the particle density of the limiting t a system. m [ 1 Introduction 3 v Kinetic equations are a useful approximation for the description of dynamical 2 processes in multi-body systems, see, e.g., the reviews by H.Spohn [32], [33]. 6 Among them, the Vlasov equation has important role in physics (in particular, 7 4 physics of plasma). It describes the Hamiltonian motion of an infinite particle . system in the mean field scaling limit when the influence of weak long-range 2 0 forces is taken into account. The convergence of the Vlasov scaling limit was 0 shown rigorously by W.Braun and K.Hepp [1] (for the Hamiltonian dynamics) 1 and by R.L.Dobrushin [3] (for more general deterministic dynamical systems). : v However, the resulting Vlasov-type equations for particle densities are consid- i eredinclassesofintegrablefunctions (or,in the weakform, offinite measures). X This, in fact, restricts us to the case of finite volume systems or systems with r a zeromeandensity in aninfinite volume. Detailedanalysis ofVlasov-typeequa- tions for integrable functions is presented in the recent paper by V.V.Kozlov [25]. In [9], we proposed a general approach to study the Vlasov-type scaling for some classesofstochastic evolutionsin the continuum,in particular,for spatial birth-and-death Markov processes. The approaches mentioned above are not ∗Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine ([email protected]). †Fakultät fu¨r Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany ([email protected]) ‡Fakultät fu¨r Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany ([email protected]). 1 applicable to these dynamics (even in a finite volume) due to essential reasons (see [9] for details). One of them is a possible variation of the particle number during the evolution. More essentially is that for these processes the possi- bility of their descriptions in terms of proper stochastic evolutional equations for particle motion is, generally speaking, absent. There are only few works concerning general spatial birth-and-death evolutions, see [30], [16], [13], [14], [29], [31]. However, the conditions for the existence (in different senses) of the evolutions considered therein are quite far from the general form. Therefore,welookedforanalternativeapproachtothe derivationofkinetic Vlasov-type equations from stochastic dynamics. The correct Vlasov limit can be easily guessed from the BBGKY hierarchy for the Hamiltonian system, see, e.g.,[32]. Suchaheuristicderivationdoesnotassumetheintegrabilitycondition forthedensity,butuntilnow,itcouldnotbemaderigorouslyduetothelackof detailedinformationaboutthepropertiesofsolutionstotheBBGKYhierarchy. Our approach is based on this observation applied in a new dynamical framework. Notethatwealreadyknowthatmanystochasticevolutionsincontinuum admit effective descriptions in terms of hierarchical equations for correlation functions which generalize the BBGKY hierarchyfrom Hamiltonian to Markov setting, see, e.g., [12] and the references therein. Even more, these hierarchical equations are often the only available technical tools for a construction of considered dynamics [20], [21], [8]. Developingthispointofview,ourschemefortheVlasovscalingofstochastic dynamics is based on the proper scaling of the hierarchical equations. This schemehasalsoaclearinterpretationinthetermsofscaledMarkovgenerators. An application of the considered scaling leads to the limiting hierarchy which posses a chaos preservation property. Namely, if we start from a Poissonian (non-homogeneous) initial state of the system, then during the time evolution this propertywill be preserved. Moreover,a specialstructure of the interaction in the resulting virtual Vlasov system gives a non-linear evolutional equation for the density of the evolving Poissonstate. The control of the convergence of Vlasov scalings for the considered hier- archies is a quite difficult technical problem which should be analyzed for any particular model separately. In the present paper, we solve this problem for the Glauber dynamics in continuum. These dynamics have given reversible states which are grand canonical Gibbs measures. The corresponding equilib- riumdynamicswhichpreservetheinitialGibbs stateinthetime evolutionwere considered in, e.g., [22], [23], [24], [11]. Note that, in applications, the time evolution of initial state is the subject of the primary interest. Therefore, we understand the considered stochastic (non-equilibrium) dynamics as the evolution of initial distributions for the system. Actually, the correspondingMarkov process (provided it exists) itself gives a general technical equipment to study this problem. Moreover,using the techniques developedin [13], it is possible to constructthis Markovprocessasasolutionofastochasticdifferentialequation. Unfortunately, this approach does not give any information about the properties of the corresponding correlation functions which we need for the study of Vlasov scaling as was mentioned above. 2 However, we note that the transition from the micro-state evolution corresponding to the given initial configuration to the macro-state dynamics is the well developed concept in the theory of infinite particle systems. This point of view appeared initially in the framework of the Hamiltonian dynamics of classical gases, see, e.g., [4]. Again, the lack of the general Markov processes techniques for the consideredsystems makes it necessaryto developalternative approaches to study the state evolutions in the Glauber dynamics. Such ap- proacheswerealizedin[20],[21],[10],[7]. Thedescriptionofthetimeevolutions formeasuresonconfigurationspacesintermsofaninfinitesystemofevolutional equationsforthecorrespondingcorrelationfunctionswasusedthere. Thelatter systemis aGlauber evolution’sanalogofthe famous BBGKY-hierarchyfor the Hamiltonian dynamics. Here we extend the approximation approach proposed in [10], [7] to the VlasovscalingfortheGlauberdynamicsincontinuum. Weconstructandstudy semigroupscorrespondingtoproperlyrescaledMarkovgeneratoroftheGlauber dynamics(Propositions3.8and3.11). Weprovefortheintegrableandbounded potential the convergence of these semigroups to the limiting semigroup which describeVlasovevolution(Theorem3.12). WederivethecorrespondingVlasov- type equationfromthis evolution(Theorem3.14). Note thatthe stationaryso- lutionofthisequationwillsatisfiedthewell-knownKirkwood–Monroeequation in the freezing theory (Remark 3.15). 2 Glauber dynamics in continuum 2.1 Basic facts and notation Let B(Rd) be the family of all Borel sets in Rd, d ≥ 1; B (Rd) denotes the b system of all bounded sets in B(Rd). The configuration space over space Rd consists of all locally finite subsets (configurations) of Rd, namely, Γ=ΓRd := γ ⊂Rd |γΛ|<∞, for all Λ∈Bb(Rd) . (2.1) n (cid:12) o (cid:12) Here γΛ := γ ∩Λ, and |·| means(cid:12) the cardinality of a finite set. The space Γ is equipped with the vague topology, i.e., the minimal topology for which all mappings Γ ∋ γ 7→ f(x) ∈ R are continuous for any continuous function x∈γ f onRd withcompactsupport; notethatthe summationin f(x)is taken P x∈γ over finitely many points of γ which belong to the support of f. In [19], it was P shownthatΓwiththevaguetopologymaybemetrizableanditbecomesaPolish space (i.e., complete separable metric space). Corresponding to this topology, the Borel σ-algebra B(Γ) is the smallest σ-algebra for which all mappings Γ ∋ γ 7→|γ |∈N :=N∪{0} are measurable for any Λ∈B (Rd). Λ 0 b The space of n-point configurationsin an arbitraryY ∈B(Rd) is defined by Γ(n) := η ⊂Y |η|=n , n∈N. Y n (cid:12) o (cid:12) (cid:12) 3 WesetalsoΓ(0) :={∅}. Asaset,Γ(n)maybeidentifiedwiththesymmetrization Y Y of Yn = (x ,...,x )∈Yn x 6=x if k 6=l . 1 n k l Henceonecanintrodnucethecorrespondi(cid:12)ngBorelσ-algebroa,whichwedenote by B(Γ(n)). Thefspace of finite configurat(cid:12)(cid:12)ions in an arbitrary Y ∈ B(Rd) is Y defined by Γ := Γ(n). 0,Y Y nG∈N0 This space is equipped with the topology ofdisjoint unions. Therefore,one can introducethe correspondingBorelσ-algebraB(Γ ). Inthe caseofY =Rd we 0,Y will omit the index Y in the notation, namely, Γ0 :=Γ0,Rd, Γ(n) :=ΓR(nd). TherestrictionoftheLebesgueproductmeasure(dx)n to Γ(n),B(Γ(n)) we denote by m(n). We set m(0) := δ . The Lebesgue–Poisson measure λ on Γ {∅} (cid:0) (cid:1) 0 is defined by ∞ 1 λ:= m(n). (2.2) n! n=0 X For any Λ ∈ B (Rd) the restriction of λ to Γ := Γ will be also denoted b Λ 0,Λ by λ. The space Γ,B(Γ) is the projective limit of the family of spaces (Γ ,B(Γ )) . The Poisson measure π on Γ,B(Γ) is given as the Λ Λ Λ∈Bb(R(cid:0)d) (cid:1) p(cid:8)rojective lim(cid:9)it of the family of measures {πΛ}Λ∈Bb(R(cid:0)d), wher(cid:1)e πΛ := e−m(Λ)λ is the probabilitymeasureon Γ ,B(Γ ) . Here m(Λ)is the Lebesgue measure Λ Λ of Λ∈B (Rd). b For any measurable funct(cid:0)ion f : Rd(cid:1)→ R we define a Lebesgue–Poisson exponent e (f,η):= f(x), η ∈Γ ; e (f,∅):=1. (2.3) λ 0 λ x∈η Y Then, by (2.2), for f ∈L1(Rd,dx) we obtain e (f)∈L1(Γ ,dλ) and λ 0 e (f,η)dλ(η)=exp f(x)dx . (2.4) λ ZΓ0 (ZRd ) A set M ∈ B(Γ ) is called bounded if there exists Λ ∈ B (Rd) and N ∈ N 0 b such that M ⊂ N Γ(n). The set of bounded measurable functions with n=0 Λ bounded support we denote by B (Γ ), i.e., G ∈ B (Γ ) if G ↾ = 0 for F bs 0 bs 0 Γ0\M someboundedM ∈B(Γ ). AnyB(Γ )-measurablefunctionGonΓ ,infact,isa 0 0 0 sequence of functions G(n) where G(n) is a B(Γ(n))-measurable function n∈N0 on Γ(n). We consider also the set F (Γ) of cylinder functions on Γ. Each (cid:8) (cid:9) cyl F ∈ F (Γ) is characterized by the following relation: F(γ) = F ↾ (γ ) for cyl ΓΛ Λ some Λ∈B (Rd). b There is the following mapping from B (Γ ) into F (Γ), which plays the bs 0 cyl key role in our further considerations: KG(γ):= G(η), γ ∈Γ, (2.5) η⋐γ X 4 where G ∈ B (Γ ), see, e.g., [18, 26, 27]. The summation in (2.5) is taken bs 0 over all finite subconfigurations η ∈ Γ of the (infinite) configuration γ ∈ Γ; 0 we denote this by the symbol, η ⋐ γ. The mapping K is linear, positivity preserving, and invertible, with K−1F(η):= (−1)|η\ξ|F(ξ), η ∈Γ . (2.6) 0 ξ⊂η X We denote the restriction of K onto functions on Γ by K . 0 0 A measure µ ∈M1 (Γ) is called locally absolutely continuous with respect fm to (w.r.t. forshort)the Poissonmeasureπ iffor anyΛ∈B (Rd) the projection b of µ onto Γ is absolutely continuous w.r.t. the projection of π onto Γ . By Λ Λ [18], in this case, there exists a correlation functional k : Γ → R such that µ 0 + for any G∈B (Γ ) the following equality holds bs 0 (KG)(γ)dµ(γ)= G(η)k (η)dλ(η). (2.7) µ ZΓ ZΓ0 The restrictions k(n) of this functional on Γ(n), n ∈ N are called correlation µ 0 0 functions of the measure µ. Note that k(0) =1. µ We recall now without a proof the partial case of the well-known technical lemma (cf., [24]) which plays very important role in our calculations. Lemma 2.1. For any measurable function H :Γ ×Γ ×Γ →R 0 0 0 H(ξ,η\ξ,η)dλ(η)= H(ξ,η,η∪ξ)dλ(ξ)dλ(η) (2.8) ZΓ0ξ⊂η ZΓ0ZΓ0 X if only both sides of the equality make sense. 2.2 Non-equilibrium Glauber dynamics in continuum Let φ : Rd → R := [0;+∞) be an even non-negative function which satisfies + the following integrability condition C := 1−e−φ(x) dx<+∞. (2.9) φ ZRd (cid:0) (cid:1) For any γ ∈Γ, x∈Rd\γ we set Eφ(x,γ):= φ(x−y)∈[0;∞]. (2.10) y∈γ X Let us define the (pre-)generator of the Glauber dynamics: for any F ∈ F (Γ) we set cyl (LF)(γ):= F(γ\x)−F(γ) (2.11) x∈γ X(cid:2) (cid:3) +z F(γ∪x)−F(γ) exp −Eφ(x,γ) dx, γ ∈Γ. ZRd (cid:2) (cid:3) (cid:8) (cid:9) 5 Herez >0isthe activityparameter. NotethatforanyF ∈F (Γ)thereexists cyl a Λ ∈ B (Rd) such that F(γ\x) = F(γ) for all x ∈ γ and F(γ ∪x) = F(γ) b Λc for all x∈ Λc; note also that exp −Eφ(x,γ) ≤1, therefore, sum and integral in (2.11) are finite. (cid:8) (cid:9) For any fixed C > 1 we consider the following Banach space of B(Γ )- 0 measurable functions L := G:Γ →R kGk := |G(η)|C|η|dλ(η)<∞ . C 0 C (cid:26) (cid:12) ZΓ0 (cid:27) (cid:12) (cid:12) In [10, Proposition3.1], it w(cid:12)as shown that the mapping Lˆ :=K−1LK given on B (Γ ) by bs 0 (LˆG)(η)=−|η|G(η) (2.12) +z e−Eφ(x,ξ)G(ξ∪x)e (e−φ(x−·)−1,η\ξ)dx λ ξ⊂ηZRd X is a linear operator on L with the dense domain L ⊂L . If additionally, C 2C C z ≤min Ce−CCφ;2Ce−2CCφ , (2.13) then Lˆ,L is closable linear(cid:8)operator in L and i(cid:9)ts closure Lˆ,D(Lˆ) gen- 2C C erates a strongly continuous contraction semigroup Tˆ(t) on L (see [10, Theo- (cid:0) (cid:1) C (cid:0) (cid:1) rem 3.8] for details). Let us set dλ := C|·|dλ; then the dual space (L )′ = L1(Γ ,dλ ) ′ = C C 0 C L∞(Γ ,dλ ). The space (L )′ is isometrically isomorphic to the Banach space 0 C C (cid:0) (cid:1) K := k :Γ →R k·C−|·| ∈L∞(Γ ,λ) C 0 0 n (cid:12) o (cid:12) withthenormkkkKC :=kC−|·|k(·)kL∞(cid:12)(Γ0,λ) wheretheisomorphismisprovided by the isometry R C (L )′ ∋k 7−→R k:=k·C|·| ∈K . (2.14) C C C In fact, one may consider the duality between the Banach spaces L and C K given by the following expression C hhG, kii:= G·kdλ, G∈L , k ∈K (2.15) C C ZΓ0 with |hhG,kii| ≤ kGk · kkk . It is clear that k ∈ K implies |k(η)| ≤ C KC C kkk C|η| for λ-a.a. η ∈Γ . KC 0 Let Lˆ′,D(Lˆ′) be anoperatorin(L )′ whichis dualtothe closedoperator C Lˆ,D(Lˆ) . We consider also its image on K under the isometry R , namely, (cid:0) (cid:1) C C let Lˆ∗ = R Lˆ′R with the domain D(Lˆ∗) = R D(Lˆ′). It was noted in [7] (cid:0) (cid:1) C C−1 C 6 that Lˆ∗ is the dual operator to Lˆ w.r.t. the duality (2.15) and that for any k ∈D(Lˆ∗) (Lˆ∗k)(η)=−|η|k(η) (2.16) +z e−Eφ(x,η\x) e (e−φ(x−·)−1,ξ)k((η\x)∪ξ)dλ(ξ). λ x∈η ZΓ0 X Under condition (2.13), we consider the adjoint semigroup Tˆ′(t) in (L )′ C and its image Tˆ∗(t) in K . By the general results from [28, Sections 1.2, 1.3], C therestrictionTˆ⊙(t)ofthe semigroupTˆ∗(t)ontoits invariantBanachsubspace D(Lˆ∗) is a contraction strongly continuous semigroup. By [7, Proposition 3.1], forany α∈(0;1)we haveK ⊂D(Lˆ∗)and,moreover,by [7,Proposition3.3], αC there exists α =α (z,φ,C)∈(0;1) such that for any α∈(α ;1) the set K 0 0 0 αC will be also a Tˆ∗(t)-invariant linear subspace. As a result, for any D(Lˆ∗) the Cauchy problem in K C ∂ k =Lˆ∗k t t ∂t (2.17)  k =k  t t=0 0 is well-defined and solvable: kt =(cid:12)(cid:12) Tˆ∗(t)k0 = Tˆ⊙(t)k0 ∈ D(Lˆ∗); moreover, k ∈K implies k ∈K . 0 αC t αC 3 Vlasov-type scaling 3.1 Description of scaling We start from the explanation of the idea of the Vlasov-type scaling. We want to construct some scaling of the generator L, say, L , ε > 0, such that the fol- ε lowingschemeholds. SupposethatwehaveasemigroupTˆ (t)withgeneratorLˆ ε ε in some L . Consider the dual semigroupTˆ∗(t). Let us choosean initial func- Cε ε tion of the corresponding Cauchy problem with a big singularity by ε, namely, k(ε)(η) ∼ ε−|η|r (η), ε → 0, η ∈ Γ with some function r , independent of ε. 0 0 0 0 Our first demand to the scaling L 7→ L is that the semigroup Tˆ∗(t) preserves ε ε the order of the singularity: (Tˆ∗(t)k(ε))(η)∼ε−|η|r (η), ε→0, η ∈Γ . (3.1) ε 0 t 0 And the second one is that the dynamics r 7→ r should preserve Lebesgue– 0 t Poissonexponents, namely, if r (η)=e (ρ ,η) then r (η)=e (ρ ,η) and there 0 λ 0 t λ t exists explicit (nonlinear, in general) differential equation for ρ : t ∂ ρ (x)=υ(ρ )(x) (3.2) t t ∂t which we will call the Vlasov-type equation. 7 Now let us explain an informal way for the realization of this scheme. Let us consider for any ε>0 the following mapping (cf. (2.14)) on functions on Γ 0 (R r)(η):=ε|η|r(η). (3.3) ε This mapping is “self-dual” w.r.t. the duality (2.15), moreover, R−1 = R . ε ε−1 Then we have k(ε) ∼ R r , and we need r ∼ R Tˆ∗(t)k(ε) ∼ R Tˆ∗(t)R r . 0 ε−1 0 t ε ε 0 ε ε ε−1 0 Therefore,wehavetoshowthatforanyt≥0theoperatorfamilyR Tˆ∗(t)R , ε ε ε−1 ε>0 has limiting (in a proper sense) operator U(t) and U(t)e (ρ )=e (ρ ). (3.4) λ 0 λ t But, informally, Tˆ∗(t) = exp{tLˆ∗} and R Tˆ∗(t)R = exp{tR Lˆ∗R }. Let ε ε ε ε ε−1 ε ε ε−1 us consider the “renormalized” operator Lˆ∗ :=R Lˆ∗R . (3.5) ε,ren ε ε ε−1 Infact,weneedthatthereexistsanoperatorLˆ∗ suchthatexp{tR Lˆ∗R }→ V ε ε ε−1 exp{tLˆ∗}=:U(t) for which (3.4) holds. Therefore, a heuristic way to produce V such a scaling L7→L is to demand that ε ∂ lim e (ρ ,η)−Lˆ∗ e (ρ ,η) =0, η ∈Γ ε→0 ∂t λ t ε,ren λ t 0 (cid:18) (cid:19) if only ρ is satisfied (3.2). The point-wise limit of Lˆ∗ will be natural candi- t ε,ren date for Lˆ∗. V Notethat(3.5)impliesLˆ =R Lˆ R . Hence,wewillusethefollowing ε,ren ε−1 ε ε scheme to give rigorous meaning to all considerations above. We consider, for a proper scaling L , the “renormalized” operator Lˆ and prove that it is a ε ε,ren generatorofastronglycontinuouscontractionsemigroupTˆ (t)inL . Next, ε,ren C we show that the formal limit Lˆ of Lˆ is also a generator of a strongly V ε,ren continuouscontractionsemigroupTˆ (t)inL also. Then,weconsiderthedual V C semigroupsTˆ∗ (t)andTˆ∗(t)inthe properBanachsubspaceofthe spaceK . ε,ren V C Finally, we prove that Tˆ∗ (t)→Tˆ∗(t) strongly on this subspace and explain ε,ren V in which sense Tˆ∗(t) satisfies the properties above. Below we try to realize this V scheme. 3.2 Construction and convergence of the evolutions in LC Let us consider for any F ∈F (Γ), ε>0 cyl (L F)(γ):= F(γ\x)−F(γ) (3.6) ε x∈γ X(cid:2) (cid:3) +ε−1z F(γ∪x)−F(γ) exp −εEφ(x,γ) dx, γ ∈Γ. ZRd (cid:2) (cid:3) (cid:8) (cid:9) 8 We define also for any G∈B (Γ ), ε>0 bs 0 Lˆ G:=K−1L KG; Lˆ G:=R Lˆ R G. ε ε ε,ren ε−1 ε ε Let φ be integrable function on the whole Rd, namely, β := φ(x)dx<+∞. (3.7) ZRd We fix this notation for our considerations below. Then, by the elementary inequality 1−e−t ≤t, t≥0 (3.8) (which we will use often), φ will satisfy (2.9) and C ≤β. φ Proposition 3.1. For any G∈B (Γ ) bs 0 (Lˆ G)(η)=(L G)(η)+(L G)(η), (3.9) ε,ren 1 2,ε where (L G)(η)=−|η|G(η), 1 (L G)(η)=z e e−εφ(x−·),ξ 2,ε λ ξX⊂ηZRd (cid:16) (cid:17) e−εφ(x−·)−1 ×e ,η\ξ G(ξ∪x)dx. λ ε (cid:18) (cid:19) Moreover, the expression (3.9) defines a linear operator in L with dense do- C main L . 2C Proof. By (2.12), for any G∈B (Γ ) we have bs 0 (Lˆ G)(η)=−|η|G(η) (3.10) ε +ε−1z e−εEφ(x,ξ)G(ξ∪x)e (e−εφ(x−·)−1,η\ξ)dx. λ ξ⊂ηZRd X Then (Lˆ G)(η)=(R Lˆ R G)(η) ε,ren ε−1 ε ε =−ε−|η||η|ε|η|G(η) +ε−|η|ε−1z e−εEφ(x,ξ)ε|ξ∪x|G(ξ∪x)e (e−εφ(x−·)−1,η\ξ)dx λ ξ⊂ηZRd X =(L G)(η)+(L G)(η). 1 2,ε 9 Next, for any G∈L we obtain 2C kL Gk = |η||G(η)|C|η|dλ(η) 1 C ZΓ0 ≤ 2|η||G(η)|C|η|dλ(η)=kGk . (3.11) 2C ZΓ0 From (3.8) and the estimate e−φ ≤1 we get kL Gk 2,ε C 1−e−εφ(x−·) ≤z |G(ξ∪x)|e ,η\ξ dxC|η|dλ(η) λ ZΓ0ξ⊂ηZRd (cid:18) ε (cid:19) X ≤z |G(ξ∪x)|e (φ(x−·),η\ξ)dxC|η|dλ(η), λ ZΓ0ξ⊂ηZRd X then, by Lemma 2.1, one may continue, ≤z |G(ξ∪x)|e (φ(x−·),η)dxC|η|dλ(η)C|ξ|dλ(ξ) λ ZΓ0ZΓ0ZRd and (2.4) yields =zexp{Cβ} |G(ξ∪x)|dxC|ξ|dλ(ξ), ZΓ0ZRd then, using Lemma 2.1 again, =zexp{Cβ}C−1 |G(ξ)|·|ξ|C|ξ|dλ(ξ) ZΓ0 ≤zexp{Cβ}C−1kGk . (3.12) 2C The estimates (3.11) and (3.12) provide the statement. Proposition 3.2. Let for any G∈B (Γ ) bs 0 (Lˆ G)(η):= lim(Lˆ G)(η)=(L G)(η)+(LVG)(η), η ∈Γ , (3.13) V ε,ren 1 2 0 ε→0 where (LVG)(η)=z G(ξ∪x)e (−φ(x−·),η\ξ)dx. 2 λ ξ⊂ηZRd X Then, the expression (3.13) defines a linear operator in L with dense domain C L . 2C Proof. Since, by the definition, LVG ≤z |G(ξ∪x)|e (φ(x−·),η\ξ)dxC|η|dλ(η) 2 C λ ZΓ0ξ⊂ηZRd (cid:13) (cid:13) X (cid:13) (cid:13) the statement follows from (3.11) and (3.12). 10