Stochastic averaging principle for spatial Markov evolutions in the continuum
aa r X i v : . [ m a t h - ph ] D ec Stochastic averaging principle for spatial Markov evolutions inthe continuum
Martin Friesen ∗ Yuri Kondratiev † June 13, 2018
Abstract:
We study a spatial birth-and-death process on the phase space of locallyfinite configurations Γ ` ˆ Γ ´ over R d . Dynamics is described by an non-equilibriumevolution of states obtained from the Fokker-Planck equation and associated withthe Markov operator L ` p γ ´ q ` ε L ´ , ε ą
0. Here L ´ describes the environmentprocess on Γ ´ and L ` p γ ´ q describes the system process on Γ ` , where γ ´ indicatesthat the corresponding birth-and-death rates depend on another locally finite con-figuration γ ´ P Γ ´ . We prove that, for a certain class of birth-and-death rates, thecorresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evo-lution of states µ εt on Γ ` ˆ Γ ´ . Moreover, we give a sufficient condition such thatthe environment is ergodic with exponential rate. Let µ inv be the invariant measurefor the environment process on Γ ´ . In the main part of this work we establish thestochastic averaging principle, i.e. we prove that the marginal of µ εt onto Γ ` con-verges weakly to an evolution of states on Γ ` associated with the averaged Markovbirth-and-death operator L “ ş Γ ´ L ` p γ ´ q dµ inv p γ ´ q . AMS Subject Classification:
Keywords: spatial birth-and-death processes; Fokker-Planck equation; ergodicity; averaging;Random evolution
Many physical, ecological and biological phenomena can be modelled by spatial birth-and-deathprocesses. It is assumed that particles are located in a continuous space, say R d , are identical byproperties, indistinguishable and, randomly appear or disappear in the location space. Particularexamples can be found in [BCF ` ` R d the situation is much ∗ Department of Mathematics, Wuppertal University, Germany, [email protected] † Department of Mathematics, Bielefeld University, Germany, [email protected] R d . We supposethat the environment has, compared to the system, significantly large birth-and-death rates,i.e. we consider the scaling regime where its rates are scaled by ε ´ with ε !
1. Based onsimilar assumptions to [FK16b] we show that the corresponding Fokker-Planck equation for thecoupled model (system with environment) is well-posed. Moreover, we give a sufficient conditionfor the environment process to be ergodic (see Theorem 2.5). Roughly speaking, the stochasticaveraging principle asserts that the dynamics of the coupled model may, in the scaling regime ε Ñ
0, be accurately described by an one-component dynamics with rates obtained by averagingthe birth-and-death rates of the system w.r.t. the invariant measure of the environment. Suchscheme is well-known in the physical literature and falls into the class of Markovian limits (see[Spo88]).Such kind of problems are well-developed in the framework of stochastic differential equations(see e.g. [EK86],[Kur92],[Pin91],[SHS02]). The environments are typically relatively simpleprocesses and the system consists of finitely many particles. Recently, we have established in[FK16a] the stochastic averaging principle for a system consisting of finitely many particlesevolving in an infinite particle environment in equilibrium. The aim of this work is to extendthis result to the case where both, the system and environment, are infinite particle systemsand, moreover, the environment is not assumed to be in equilibrium.Since existence of a Markov process or an analysis of the backward Kolmogorov equationin the cases we consider is absent, we cannot directly apply the classical theory. Having inmind that solutions to the Fokker-Planck equations are constructed in terms of correlationfunctions, our main idea is to reformulate the problem in terms of correlation functions andthen seek to apply abstract semigroup methods such as [Kur73]. One important difficulty inthis approach is that, in contrast to finite particle systems, we cannot work with integrablecorrelation functions, i.e. correlation functions corresponding to infinite particle systems aretypically only Ruelle bounded. The collection of all correlation functions then forms a cone in aweighted L -space where by the Lotz Theorem [Lot85] any semigroup on such a space cannot bystrongly continuous. For this purpose we first study the pre-dual problem on a proper L -spacedescribing the evolution of so-called quasi-observables. Here we may apply [Kur73] and thendeduce our desired result on correlation functions by duality.This work is organized as follows. Our main results are formulated and discussed in thenext section. For this purpose we introduce some notation used throughout this work. Thenwe describe in some detail the environment process, the Fokker-Planck equation in Theorem 2.32nd give a sufficient condition for the ergodicity of the environment in Theorem 2.5. Afterwardswe briefly discuss the system and the limiting process. The stochastic averaging principle isstated in Theorem 2.8. Particular examples which show how this result can be applied are givenin the third section. The fourth section is devoted to the proofs of Theorem 2.3 and Theorem2.5. Finally, based on the results of section four, a proof of the main result is given in sectionfive. The following is mainly based on [KK02]. Each particle is completely described by its position x P R d and the corresponding (one-type) configuration space isΓ “ t γ Ă R d | | γ X Λ | ă 8 for all compacts Λ Ă R d u , where | Λ | denotes the number of elements in the set Λ. It is well-known that Γ is a Polishspace w.r.t. the smallest topology such that γ ÞÝÑ ř x P γ f p x q is continuous for any continuousfunction f having compact support (see [KK06]). Let B p Γ q be the Borel- σ -algebra on Γ. It isthe smallest σ -algebra such that Γ Q γ ÞÝÑ | γ X Λ | is measurable for any compact Λ Ă R d .The space of finite configurations is defined by Γ : “ t η Ă R d | | η | ă 8u . It is equipped withthe smallest σ -algebra such that Γ Q η ÞÝÑ | η X Λ | is measurable for any compact Λ Ă R d . Wedefine a measure λ on Γ by the relation ż Γ G p η q dλ p η q “ G pHq ` ÿ n “ n ! ż R dn G pt x , . . . , x n uq dx . . . dx n , where G is any non-negative measurable function. Let B bs p Γ q be the space of all boundedmeasurable functions with bounded support, i.e. G P B bs p Γ q iff G is bounded and there exists acompact Λ Ă R d and N P N with G p η q “
0, whenever η X Λ c ‰ H or | η | ą N . The K -transformis, for G P B bs p Γ q , defined by p KG qp γ q : “ ÿ η Ť γ G p η q , γ P Γ , (2.1)where Ť means that the sum is taken over all finite subsets of γ . Let µ be a probability measureon Γ ´ with finite local moments, i.e. ş Γ ´ | γ ´ X Λ | n dµ p γ ´ q ă 8 for all compacts Λ and n ≥ k µ : Γ ÝÑ R ` is defined by the relation ż Γ p KG qp γ q dµ p γ q “ ż Γ G p η q k µ p η q dλ p η q , G P B bs p Γ q . (2.2)The Poisson measure π z with intensity measure zdx , z ą
0, is main guiding example. It isuniquely determined by the relation π z pt γ P Γ | | γ X Λ | “ n uq “ m p Λ q n z n n ! where m p Λ q denotesthe Lebesgue measure of Λ. Then π z has correlation function k π z p η q “ z | η | .3elow we briefly describe how this notations are extended to the two-component case.Namely, let Γ : “ Γ ` ˆ Γ ´ be equipped with the product topology where Γ ˘ are two iden-tical copies of Γ. We let γ “ p γ ` , γ ´ q P Γ , where γ ` describes the particles of the system and γ ´ the particles of the environment, respectively. Similarly let Γ : “ Γ ` ˆ Γ ´ , η : “ p η ` , η ´ q and | η | : “ | η ` | ` | η ´ | . Define G P B bs p Γ q iff G is bounded and measurable and there existsa compact Λ Ă R d and N P N such that G p η q “
0, whenever | η | ą N or η ˘ X Λ c ‰ H . The K -transform is, for G P B bs p Γ q , defined by p K G qp γ q : “ ř η Ť γ G p η q , where ξ Ă η and η Ť γ are defined component-wise. Let µ be a probability measure on Γ (=: state) with finite localmoments, i.e. ş Γ | γ ´ X Λ | n | γ ` X Λ | n dµ p γ q ă 8 for all compacts Λ and n ≥
1. As before wedefine the correlation function k µ by ż Γ K G p γ q dµ p γ q “ ż Γ G p η q k µ p η q dλ p η q , G P B bs p Γ q . At this point it is worth to mention that not every non-negative function k on Γ is the correlationfunction of some state µ . It is necessary and sufficient that k pHq “ k is positivedefinite in the sense of Lenard (see [Len73], [Len75]). Particles, in the framework of spatial birth-and-death processes, may randomly disappear andnew particles may appear in the configuration γ ´ P Γ ´ . Death of a particle x P γ ´ is described bythe death rate d ´ p x, γ ´ q ≥
0. Similarly, b ´ p x, γ ´ q ≥ x P R d z γ ´ . In this work we assume that the environment is described by aMarkov operator (formally) given by p L ´ F qp γ ´ q “ ÿ x P γ ´ d ´ p x, γ ´ z x qp F p γ ´ z x q ´ F p γ ´ qq ` ż R d b ´ p x, γ ´ qp F p γ ´ Y x q ´ F p γ ´ qq dx (2.3)where F P F P p Γ ´ q : “ K p B bs p Γ ´ qq . For simplicity of notation we have let γ ´ z x, γ ´ Y x standfor γ ´ zt x u , γ ´ Y t x u . Note that in general we cannot expect that d ´ , b ´ are well-defined for all γ ´ P Γ ´ and x P R d . Below we discuss our assumptions on the birth-and-death rates.(E1) There exist measurable functions D ´ , B ´ : R d ˆ Γ ´ ÝÑ R such that d ´ p x, γ ´ q “ ÿ η ´ Ť γ ´ D ´ p x, η ´ q , b ´ p x, γ ´ q “ ÿ η ´ Ť γ ´ B ´ p x, η ´ q , x P R d , γ ´ P Γ ´ , (2.4)Moreover there exist constants A ą N P N and ν ≥ b ´ p x, η ´ q ` d ´ p x, η ´ q ≤ A p ` | η ´ |q N e ν | η ´ | , η ´ P Γ ´ , x P R d . C ´ ą a ´ P p , q such that c ´ p η ´ q ≤ a ´ M ´ p η ´ q , η ´ P Γ ´ holds, where M ´ p η ´ q “ ř x P η ´ d ´ p x, η ´ z x q and c ´ p η ´ q : “ ÿ x P η ´ ż Γ ´ ˇˇˇˇˇˇ ÿ ζ ´ Ă η ´ z x D ´ p x, ξ ´ Y ζ ´ q ˇˇˇˇˇˇ C | ξ ´ |´ dλ p ξ ´ q` C ´ ÿ x P η ´ ż Γ ´ ˇˇˇˇˇˇ ÿ ζ ´ Ă η ´ z x B ´ p x, ξ ´ Y ζ ´ q ˇˇˇˇˇˇ C | ξ ´ |´ dλ p ξ ´ q . A priori it is not clear that the sums in (2.4) are absolutely convergent, it is part of the assertionof Lemma 2.2. For the last condition let p R δ q δ ą Ă L p R d q X C p R d q with 0 ă R δ ≤ R δ p x q Õ x as δ Ñ δ ą V δ : Γ ´ ÝÑ R ` and constants c δ , ε δ ą p L ´ δ V δ qp η ´ q ≤ c δ p ` V δ p η ´ qq ´ ε δ D ´ δ p η ´ q , η ´ P Γ ´ where L ´ δ is obtained from L ´ with b ´ p x, ¨q replaced by R δ p x q b ´ p x, ¨q and D ´ δ p η ´ q “ M ´ p η ´ q ` ż R d R δ p x q b ´ p x, η ´ q dx. Let us give some additional comments on this assumptions.
Remark 2.1. (i) Here d ´ p x, Hq “ D ´ p x, Hq describes the constant mortality (without inter-actions) whereas D ´ p x, t y uq takes pair interactions (competition of x with y ) into account.General n -point interactions are described by D ´ p x, η ´ q where | η ´ | “ n .(ii) For many particular models, see the next section, the function c ´ can be computed explic-itly. Condition (E2) is satisfied, provided d ´ p x, Hq is large enough, i.e. it describes somesort of ”high-mortality regime”.(iii) Condition (E3) asserts the existence of some sort of Lyapunov function for the generator L ´ δ . The latter one describes a birth-and-death Markov process on Γ ´ and under the givencondition this process is conservative. A simpler sufficient condition for (E3) is given inthe next section; e.g. if b ´ p x, η ´ q ≤ A p ` | η ´ |q for some constant A ą , then (E3) holds. We study dynamics of the environment in terms of one-dimensional distributions, i.e. solu-tions to the Fokker-Planck equation ddt ż Γ ´ F p γ ´ q dµ ´ t p γ ´ q “ ż Γ ´ L ´ F p γ ´ q dµ ´ t p γ ´ q , µ ´ t | t “ “ µ ´ , t ≥ . (2.5)5he latter one is analyzed in the class P C ´ , where µ P P C ´ if and only if the correlation function k µ exists and satisfies for some constant A p µ q ą k µ p η ´ q ≤ A p µ q C | η ´ |´ , η ´ P Γ ´ . (2.6)It is worth to mention that under condition (2.6) the correlation function k µ uniquely determinesthe state µ (see [Len73]). The next lemma shows that L ´ is well-defined. Lemma 2.2.
Suppose that conditions (E1) and (E2) are satisfied. Then(a) For any µ P P C ´ and x P R d we have ż Γ ´ ` d ´ p x, γ ´ q ` b p x, γ ´ q ˘ dµ p γ ´ q ≤ max t , C ´ u A p µ q a ´ d ´ p x, Hq . In particular, the sums in (2.4) are absolutely convergent for µ -a.a. γ ´ P Γ ´ and all x P R d .(b) We have L ´ F P L p Γ ´ , dµ q for all µ P P C ´ and F P F P p Γ ´ q . A proof is given in section four. Following the general scheme described in [KKM08], westudy solutions to (2.5) in terms of correlation functions p k t q t ≥ . The latter ones should (at leastformally) satisfy a Markov analogue of the BBGKY-hierarchy known from physics (see equation(4.1)). Motivated by (2.6) we denote by K C ´ the Banach space of essentially bounded functions k on Γ ´ with norm } k } K C ´ “ ess sup η P Γ ´ | k p η ´ q| C ´| η ´ |´ . theorem 2.3. Suppose that conditions (E1) – (E3) are satisfied. Then for each µ ´ P P C ´ thereexists a family of states p µ ´ t q t ≥ Ă P C ´ with the following properties(a1) t ÞÝÑ ş Γ ´ p L ´ F qp γ ´ q dµ ´ t p γ ´ q is continuous for all F P F P p Γ ´ q .(a2) } k µ t } K C ´ ≤ } k µ } K C ´ for all t ≥ .(a3) t ÞÝÑ ş Γ ´ F p γ ´ q dµ ´ t p γ ´ q is continuously differentiable and (2.5) holds for all F P F P p Γ ´ q .Moreover, this solution is unique among all p ν ´ t q t ≥ Ă P C ´ which satisfy(b1) t ÞÝÑ ş Γ ´ p L ´ F qp γ ´ q dν ´ t p γ ´ q is locally integrable for all F P F P p Γ ´ q .(b2) sup t Pr ,T s } k ν t } K C ´ ă 8 holds for all T ą .(b3) t ÞÝÑ ş Γ ´ F p γ ´ q dν ´ t p γ ´ q is absolutely conditions and (2.5) holds for all F P F P p Γ ´ q anda.a. t ≥ . Classical solutions to the BBGKY-hierarchy (see (4.1)) with general rates have been firstobtained in [FKK12],[FKK15] in the class of functions obeying (2.6). The construction givenin this work relies on the same idea, but we use suitable perturbation theory for substochastic6emigroups on weighted L -spaces instead. At this point condition (E2) is used to show thatcertain operators involved in the construction are relatively bounded.In general a solution to the BBGKY-hierarchy does not need to determine uniquely anevolution of states. It is necessary and sufficient to show that such a solution is positive definitein the sense of Lenard (see (4.4) for the definition). Such a property was shown under morestringent assumptions in [FK16b]. Namely an additional technical assumption (stronger then(E2)) was imposed on c ´ ; the initial condition µ ´ was assumed to belong to some strictly smallersubspace P Ă P C ´ ; and finally condition (E3) was replaced by some more technical condition.Since the main ideas of the proof are the same as in [FK16b], we do not give a full proof. Insteadwe outline the most important steps in the first part of section four. Remark 2.4.
It is not difficult to adapt this result for a spatial birth-and-death process on Γ .This will be used, without proof, later on. Recently, we have studied in [Fri17] exponential ergodicity for a two-component Glauber-type process where particles are allowed to change their type at certain multiplicative rates. Inthis work we give a general result applicable for a birth-and-death process with Markov operator(2.3). theorem 2.5.
Suppose that conditions (E1) – (E3) are satisfied. Moreover, assume that inf $&% ÿ x P η ´ d ´ p x, η ´ z x q ˇˇˇˇ H ‰ η ´ P Γ ´ ,.- ą . (2.7) Then there exists µ inv P P C ´ with correlation function k inv such that:(a) µ inv is the unique stationary solution to (2.5) .(b) There exist constants A, λ ą such that for any µ ´ P P C ´ it holds that } k µ ´ t ´ k µ inv } K C ´ ≤ Ae ´ λt } k µ ´ ´ k µ inv } K C ´ , t ≥ where p µ ´ t q t ≥ Ă P C ´ is the unique solution to (2.5) . The proof uses some arguments similar to [Fri17], but since we work with more generalconditions additional technical steps have to be done. A proof of this statement is given insection four. Existence of the invariant measure is obtained from some type of generalizedKirkwood-Salsburg equation (see also [FKK12]). Ergodicity is then deduced by classical spectraltheory, i.e. we show that the generator for the dynmaics has a spectral gap.
Remark 2.6. (i) If b ´ p x, Hq “ , then µ inv “ δ H , i.e. the population gets extinct withexponential speed.(ii) If b ´ p x, Hq ą , then µ inv ‰ δ H . Particular examples such as the Sourgailis model with invariant measure π z and the Glauberdynamics with a Gibbs measure as the invariant measure have been studied in [Fin11a], [KKM10].The aggregation model considered in [FKKZ14] is a particular example where d ´ p x, Hq isbounded away from zero, but condition (2.7) does not hold.7 .3 The system Dynamics for the system is described by the birth-and-death rates d ` p x, γ q and d ` p x, γ q wherethe additional dependence on the parameter γ ´ takes interactions of the system with its envi-ronment into account. The Markov operator is formally given by p L ` F qp γ q “ ÿ x P η ` d ` p x, γ ` z x, γ ´ qp F p γ ` z x, γ ´ q ´ F p γ ` , γ ´ qq` ż R d b ´ p x, γ ` , γ ´ qp F p γ ` Y x, γ ´ q ´ F p γ ` , γ ´ qq dx where F P F P p Γ q “ K p B bs p Γ qq . Since no confusion may arise we also denote by λ the measure λ b λ . Similarly to (E1) – (E3) we impose the following conditions:(S1) There exist measurable functions D ` , B ` : R d ˆ Γ ÝÑ R such that d ` p x, γ q “ ÿ η Ť γ D ` p x, η q , b ` p x, γ q “ ÿ η Ť γ B ` p x, η q . Moreover there exist constants A ą N P N and ν ≥ b ` p x, η q ` d ` p x, η q ≤ A p ` | η |q N e ν | η | , η P Γ , x P R d . (S2) There exists C ` ą and a ` P p , q such that c ` p η q ≤ a ` M ` p η q , η P Γ holds where M ` p η q “ ř x P η ` d ` p x, η ` z x, η ´ q and c ` p η q : “ ÿ x P η ` ż Γ ˇˇˇˇˇˇˇˇ ÿ ζ `Ă η `z xζ ´Ă η ´ D ` p x, ζ ` Y ξ ` , ζ ´ Y ξ ´ q ˇˇˇˇˇˇˇˇ C | ξ ` |` C | ξ ´ |´ dλ p ξ q` C ` ÿ x P η ` ż Γ ˇˇˇˇˇˇˇˇ ÿ ζ `Ă η `z xζ ´Ă η ´ B ` p x, ζ ` Y ξ ` , ζ ´ Y ξ ´ q ˇˇˇˇˇˇˇˇ C | ξ ` |` C | ξ ´ |´ dλ p ξ q . (S3) Take p R δ q δ ą as in (E3). For any δ ą V δ : Γ ÝÑ R ` and constants c δ , ε δ ą p L ` δ V δ qp η q ≤ c δ p ` V δ p η qq ´ ε δ D ` δ p η q , η P Γ where L ` δ is given by L ` with b ` p x, ¨q replaced by R δ p x q b ` p x, ¨q and D ` δ p η q “ M ` p η q ` ż R d R δ p x q b ` p x, η q dx. P C be the space of all µ such that µ P P C iff its correlation function k µ exists and satisfiesthe Ruelle bound k µ p η q ≤ A p µ q C | η ` |` C | η ´ |´ , η P Γ . Arguing similarly to the proof of Lemma 2.2 we see thatsup x P R d ż Γ ` d ` p x, γ q ` b ` p x, γ q ˘ dµ p γ q ă 8 ,L ` F P L p Γ , dµ q for µ P P C and F P F P p Γ q and the following analogue of Theorem 2.3 holds. theorem 2.7. Suppose that (S1) – (S3) are satisfied. Then for any µ P P C there exists aunique solution p µ t q t ≥ Ă P C to ddt ż Γ F p γ q dµ t p γ q “ ż Γ L ` F p γ q dµ t p γ q , µ t | t “ “ µ , t ≥ , F P F P p Γ q . This describes the evolution of the system in the presence of a stationary environment fixedwith the choice of µ . Here and below we assume that conditions (E1) – (E3), (S1) – (S3) and (2.7) are satisfied.Consider the averaged Markov operator given by p LF qp γ ` q “ ÿ x P γ ` d p x, γ ` z x qp F p γ ` z x q ´ F p γ ` qq ` ż R d b p x, γ ` qp F p γ ` Y x q ´ F p γ ` qq dx where F P F P p Γ ` q . The birth-and-death rates are obtained by integration w.r.t. to the invariantmeasure of the environment, i.e. d p x, γ ` q : “ ż Γ d ` p x, γ q dµ inv p γ ´ q “ ÿ η ` Ť γ ` D p x, η ` q , (2.8) b p x, γ ` q : “ ż Γ b ` p x, γ q dµ inv p γ ´ q “ ÿ η ` Ť γ ` B p x, η ` q , (2.9)where D p x, η ` q “ ş Γ ´ D ` p x, η ` , η ´ q k inv p η ´ q dλ p η ´ q and B p x, η ` q “ ş Γ ´ B ` p x, η ` , η ´ q k inv p η ´ q dλ p η ´ q .In order to proceed it is necessary to assume that similar conditions to (E1) – (E3) with d ´ , b ´ replaced by d, b are satisfied, i.e. we assume:(AV1) There exist A ą N P N and ν ≥ d p x, η ` q ≤ A p ` | η ` |q N e ν | η ` | holds for all x P R d and η ` P Γ ` . 9AV2) There exists a constant a P p , q such that c p η ` q ≤ aM p η ` q , η ` P Γ ` holds where M p η ` q : “ ř x P η ` d p x, η ` z x q and c p¨q is defined as c ´ p¨q with D ´ , B ´ replacedby D, B and C ´ replaced by C ` .(AV3) Take p R δ q δ ą as in (E3). For any δ ą V δ : Γ ` ÝÑ R ` and constants c δ , ε δ ą p L δ V δ qp η ` q ≤ c δ p ` V δ p η ` qq ´ ε δ M δ p η ` q , η ` P Γ ` where L δ is given by L with b p x, ¨q replaced by R δ p x q b p x, ¨q and D δ p η ` q “ M p η ` q ` ż R d R δ p x q b p x, η ` q dx. One would expect that (S2), (S3) together with (2.8) and (2.9) already imply (AV2) and (AV3).Unfortunately we could not show that this is, indeed, the case. Particular examples consideredin the next section, however, show that conditions (AV1) – (AV3) are merely a restriction.
We consider the Fokker-Planck for the system and environment given by the Markov operator L ` ` ε L ´ . Here L ´ is extended to functions F : Γ ÝÑ R by its action only on the variable γ ´ . For a given state µ P P C the marginal on the first component µ ` is defined by ż Γ ` F p γ ` q dµ ` p γ ` q “ ż Γ F p γ ` q dµ p γ ` , γ ´ q . The following is our main result. theorem 2.8.
Suppose that conditions (E1) – (E3), (S1) – (S3), (AV1) – (AV3) and (2.7) aresatisfied. Then the following assertions hold:(a) For each µ P P C and each ε ą there exists a unique solution p µ εt q t ≥ Ă P C to ddt ż Γ F p γ q dµ εt p γ q “ ż Γ ˆ L ` ` ε L ´ ˙ F p γ q dµ εt p γ q , µ εt | t “ “ µ , t ≥ . (b) For each µ P P C ` there exists a unique solution p µ t q t ≥ Ă P C ` to ddt ż Γ ` F p γ ` q dµ t p γ ` q “ ż Γ ` p LF qp γ ` q dµ t p γ ` q , µ t | t “ “ µ , t ≥ . (2.10)10 c) For any F P F P p Γ ` q we have ż Γ F p γ ` q dµ εt p γ ` , γ ´ q ÝÑ ż Γ ` F p γ ` q dµ t p γ ` q , ε Ñ uniformly on compacts in t ≥ where p µ t q t ≥ is the unique solution to (2.10) with initialcondition µ “ µ ` . Note that assertion (a) extends Theorem 2.7 since here the environment does not need to bestationary. This result also extends [FK16a] where the system was a birth-and-death process onΓ ` and the environment was an equilibrium process on Γ ´ . Here the system and environmentare both infinite particle systems and secondly the environment is only assumed to be ergodic;it does not need to be in equilibrium. In a forthcoming work we will use the results obtained inthis work to extend the stochastic averaging principle for a model where conditions (E2), (S2)and (AV2) are relaxed. Many models of interacting particle systems are based on translation invariant rates (see e.g.[FKK15]). Such rates may result from an idealisation and simplification of the underlying phys-ical model under which particular properties can be studied and observed. However, biologicalsystems related with the description of tumour growth should, due to their complex spatialstructure, be modelled by space-inhomogeneous rates. The particular choice of such rates isoften based on ad-hoc assumptions and a deep understanding of the underlying nature of thedynamics involved. Using the stochastic averaging principle, we show that such rates can berigorously derived from the interaction with a Markovian environment. The birth-and-deathrates in the examples given below are build by relative energies E ϕ p x, γ ˘ q : “ ÿ y P γ ˘ ϕ p x ´ y q , x P R d , γ ˘ P Γ , where ϕ is a symmetric, non-negative and, integrable function. We will frequently use the combinatorial relation ż Γ ˘ ÿ ξ ˘ Ă η ˘ G p ξ ˘ , η ˘ z ξ ˘ q dλ p ξ ˘ q “ ż Γ ˘ ż Γ ˘ G p ξ ˘ , η ˘ q dλ p ξ ˘ q dλ p η ˘ q (3.1)provided one side of the equality is finite for | G | , cf. [Fri17]. Moreover by [Fin11b] we have λ ` t η ˘ P Γ ˘ | ξ ˘ X η ˘ ‰ Hu ˘ “ , ξ ˘ P Γ ˘ (3.2)11nd λ b λ ` t η P Γ | η ` X η ´ ‰ Hu ˘ “
0. For a given measurable function g we have ÿ ξ ˘ Ă η ˘ ź x P ξ ˘ g p x q “ ź x P η ˘ p ` g p x qq (3.3)and if | g | is integrable, then ż Γ ˘ ź x P η ˘ g p x q dλ p η ˘ q “ exp ¨˝ ż R d g p x q dx ˛‚ . (3.4)Let us finally give a sufficient condition for (E3). proposition 3.1. Suppose that for each δ ą there exists c δ ą and ε δ P p , q such that ż R d R δ p x q b ´ p x, η ´ q dx ≤ c δ p ` | η ´ |q ` ε δ ÿ x P η ´ d ´ p x, η ´ z x q , η ´ P Γ ´ . Then condition (E3) is satisfied. In particular, if b ´ p x, η ´ q ≤ A p `| η ´ |q holds for some constant A ą , then (E3) is satisfied.Proof. Write c δ “ c δ ` ε δ ą ε δ “ ´ ε δ ` ε δ P p , q where c δ “ c δ ` ε δ and ε δ “ ´ ε δ ` ε δ P p , q . Then ż R d R δ p x q b ´ p x, η ´ q dx ≤ c δ p ` | η ´ |q ` p ´ ε δ q ÿ x P η ´ d ´ p x, η ´ z x q ´ ε δ ż R d R δ p x q b ´ p x, η ´ q dx and hence we obtain for V δ p η ´ q “ | η ´ |p L ´ δ V δ qp η ´ q “ ´ ÿ x P η ´ d ´ p x, η ´ z x q ` ż R d R δ p x q b ´ p x, η ´ q dx ≤ c δ p ` | η ´ |q ´ ε δ ÿ x P η ´ d ´ p x, η ´ z x q ´ ε δ ż R d R δ p x q b ´ p x, η ´ q dx. A similar statement can be shown for (S3) and (AV3).
The environment process is assumed to be a Glauber dynamics in the continuum with thebirth-and-death rates d ´ p x, γ q “ ,b ´ p x, γ q “ z ´ exp p´ E ψ p x, γ ´ qq . (3.5)12uch dynamics has been studied in [FKKZ12]. The system process is another Glauber dynamicsgiven by d ` p x, γ q “ ,b ` p x, γ q “ z ` exp ` ´ E φ ´ p x, γ ´ q ˘ exp ` ´ E φ ` p x, γ ` q ˘ . A similar model has been studied in [FKKO15] and [Fri17]. For an integrable function f : R d ÝÑ R ` let β p f q : “ ż R d | e ´ f p x q ´ | dx P r , . Suppose that the following conditions are satisfied:(a) ψ, φ ˘ ≥ β p ψ q , β p φ ˘ q ă 8 and z ˘ ą C ˘ ą z ´ exp p C ´ β p ψ qq ă C ´ , (3.6) z ` exp ` C ´ β p φ ´ q ˘ exp ` C ` β p φ ` q ˘ ă C ` . Let us prove that conditions (E1) – (E3), (S1) – (S3) and (AV1) – (AV3) hold. Condition (E1)is obvious with the choice D ´ p x, η ´ q “ | η ´ | , B ´ p x, η ´ q “ z ´ ź y P η ´ ´ e ´ ψ p x ´ y q ´ ¯ . Condition (E3) immediately follows from b ´ p x, η ´ q ≤ z ´ . Concerning condition (E2) we firstobserve that for ξ ´ X ζ ´ “ H D ´ p x, ξ ´ Y ζ ´ q “ | ξ ´ | | ζ ´ | , B ´ p x, ξ ´ Y ζ ´ q “ z ´ ź y P ξ ´ ´ e ´ ψ p x ´ y q ´ ¯ ź w P ζ ´ ´ e ´ ψ p x ´ y q ´ ¯ . In view of (3.2), (3.3), (3.4) we obtain c ´ p η ´ q “ | η ´ | ` z ´ C ´ e C ´ β p ψ q ÿ x P η ´ e ´ E ψ p x,η ´ z x q ≤ | η ´ | ˆ ` z ´ C ´ e C ´ β p ψ q ˙ . Hence (E2) follows from assumption (b). Similarly we show that assumptions (S1) – (S3) hold.Namely, condition (S3) follows from b ` p x, η q ≤ z ` and (S1) holds with D ` p x, η q “ | η | , B ` p x, η q “ z ` ź y P η ` ´ e ´ φ ` p x ´ y q ´ ¯ ź w P η ´ ´ e ´ φ ´ p x ´ w q ´ ¯ . Hence we obtain c ` p η q “ | η ` | ` z ` C ` e C ` β p φ ` q e C ´ β p φ ´ q ÿ x P η ` e ´ E φ ` p x,η ` z x q e ´ E φ ´ p x,η ´ q ≤ | η ` | ˆ ` z ` C ` e C ` β p φ ` q e C ´ β p φ ´ q ˙ .
13n view of (b) condition (S2) holds. The unique invariant measure for the environment is givenby the Gibbs measure µ inv with activity z ´ and potential ψ . Moreover we have d p x, γ ` q “ ż Γ ´ d ` p x, γ ` , γ ´ q dµ inv p γ ´ q “ ,b p x, γ ` q “ ż Γ ´ b ` p x, γ ` , γ ´ q dµ inv p γ ´ q “ z ` λ p x q exp ` ´ E φ ` p x, γ ` q ˘ . where λ p x q “ ş Γ ´ e ´ E φ ´ p x,γ ´ q dµ inv p γ ´ q . Arguing as above we see that (AV1) and (AV3) holdwith D p x, η ` q “ | η ` | , B p x, η ` q “ z ` λ p x q ź y P η ` ´ e ´ φ ` p x ´ y q ´ ¯ . Hence c p η ` q ≤ ř x P η ` ´ ` z ` C ` λ p x q e C ` β p φ ` q ¯ ≤ | η ` | ´ ` z ` C ` e C ` β p φ ` q e C ´ β p φ ´ q ¯ implies (AV2). The environment is, as before, assumed to be a Glauber dynamics with birth-and-death rates(3.5). The system is assumed to be a BDLP-process and it is assumed that the environmentinfluences the system due to additional competition via the potential b ´ and particles from theenvironment may create new individuals within the system. More precisely we consider the rates d ` p x, γ q “ m ` ` ÿ y P γ ` a ´ p x ´ y q ` ÿ y P γ ´ b ´ p x ´ y q ,b ` p x, γ q “ ÿ y P γ ` a ` p x ´ y q ` ÿ y P γ ´ b ` p x ´ y q . Without influence of the environment ( b ˘ “
0) the system is simply an one-component BDLP-process studied in [BCF ` ψ ≥ β p ψ q ă 8 and there exists C ´ ą m ` , z ´ ą a ˘ , b ˘ ≥ C ` ą θ P p , C ` q and b ≥ ÿ x P η ` ÿ y P η ` z x a ` p x ´ y q ≤ θ ÿ x P η ` ÿ y P η ` z x a ´ p x ´ y q ` b | η ` | (3.7)holds and for some ϑ P p , C ` q we have ϑb ´ ≥ b ` (3.8) m ` ą C ´ } b ´ } L ` C ` } a ´ } L ` } a ` } L ` C ´ C ` } b ` } L ` bC ` . (3.9)14 emark 3.2. Condition (3.7) states that θa ´ ´ a ` is a stable potential in the sense of Ruelle.Some sufficient conditions are given e.g. in [KK16]. Let us show that (E1) – (E3), (S1) – (S3) and (AV1) – (AV3) are satisfied. First of all, previousexample shows that (E1) – (E3) are satisfied. Since b ` p x, η q ≤ | η ` |} a ` } L ` | η ´ |} b ` } L itfollows that (S3) holds. Condition (S1) is satisfied for the choice D ` p x, η q “ m ` | η | ` | η ´ | Γ ` p η ` q ÿ y P η ` a ´ p x ´ y q ` | η ` | Γ ´ p η ´ q ÿ y P η ´ b ´ p x ´ y q ,B ` p x, η q “ | η ´ | Γ ` p η ` q ÿ y P η ` a ` p x ´ y q ` | η ` | Γ ´ p η ´ q ÿ y P η ´ b ` p x ´ y q , where Γ ˘ : “ t η ˘ P Γ ˘ | | η ˘ | “ u . This gives for ζ ˘ X ξ ˘ “ H D ` p x, ζ ` Y ξ ` , ζ ´ Y ξ ´ q “ m ` | ξ | | ζ | ` | ζ | | ξ ` | Γ ´ p ξ ´ q ÿ y P ξ ´ b ´ p x ´ y q ` | ξ | | ζ ` | Γ ´ p ζ ´ q ÿ y P ζ ´ b ´ p x ´ y q` | ξ ´ | | ζ | Γ ` p ξ ` q ÿ y P ξ ` a ´ p x ´ y q ` | ξ | | ζ ´ | Γ ` p ζ ` q ÿ y P ζ ` a ´ p x ´ y q ,B ` p x, ζ ` Y ξ ` , ζ ´ Y ξ ´ q “ | ζ | | ξ ´ | Γ ` p ξ ` q ÿ y P ξ ` a ` p x ´ y q ` | ξ | | ζ ´ | Γ ` p ζ ` q ÿ y P ζ ` a ` p x ´ y q` | ζ | | ξ ` | Γ ´ p ξ ´ q ÿ y P ξ ´ b ` p x ´ y q ` | ξ | | ζ ` | Γ ´ p ζ ´ q ÿ y P ζ ´ b ` p x ´ y q . Using the definition of λ we get c ` p η q “ m ` | η ` | ` C ´ } b ´ } L | η ` | ` C ` } a ´ } L | η ` | ` ÿ x P η ` ÿ y P η ` z x a ´ p x ´ y q ` ÿ x P η ` ÿ y P η ´ b ´ p x ´ y q` | η ` |} a ` } L ` C ´ C ` } b ` } L | η ` | ` C ` ÿ x P η ` ÿ y P η ` z x a ` p x ´ y q ` C ` ÿ x P η ` ÿ y P η ´ b ` p x ´ y q . and hence by (3.7) and (3.8) c ` p η q ≤ ˆ m ` ` C ´ } b ´ } L ` C ` } a ´ } L ` } a ` } L ` C ´ C ` } b ` } L ` bC ` ˙ | η ` |` ˆ θC ` ` ˙ ÿ x P η ` ÿ y P η ` z x a ´ p x ´ y q ` ˆ ϑC ` ` ˙ ÿ x P η ` ÿ y P η ´ b ´ p x ´ y q ≤ p ` a ` q M ` p η q , where M ` p η q “ m ` | η ` | ` ř x P η ` ř y P η ` z x a ` p x ´ y q ` ř x P η ` ř y P η ´ b ´ p x ´ y q and a ` “ max C ´ } b ´ } L ` C ` } a ´ } L ` } a ` } L ` C ´ C ` } b ` } L ` bC ` m ` , θC ` , ϑC ` + . a ` P p , q which proves (S2).The unique invariant measure for the environment µ inv is the Gibbs measure with activity z ´ and potential ψ . The averaged rates are given by d p x, γ ` q “ m ` ` m p x q ` ÿ y P γ ` z x a ´ p x ´ y q ,b p x, γ ` q “ λ p x q ` ÿ y P γ ` a ` p x ´ y q , where λ p x q : “ ş Γ ´ ř y P γ ´ b ` p x ´ y q dµ inv p γ ´ q and m p x q : “ ş Γ ´ ř y P γ ´ b ´ p x ´ y q dµ inv p γ ´ q . Itremains to show that (AV1) – (AV3) are satisfied. First observe that (AV1) holds with D p x, η ` q “ | η ` | p m ` ` m p x qq ` Γ ` p η ` q ÿ y P η ` a ´ p x ´ y q ,B p x, η ` q “ λ p x q | η ` | ` Γ ` p η ` q ÿ y P η ` a ` p x ´ y q . Concerning condition (AV3) we first observe that λ p x q “ ż R d b ` p x ´ y q k inv pt y uq dy ≤ C ´ } k inv } K C ´ } b ` } L from which we obtain b p x, η ` q ≤ | η ` |} a ` } L ` C ´ } k inv } K C ´ } b ` } L . For the last condition weobtain similarly to (S2) c p η ` q “ ÿ x P η ` ˆ m ` ` C ` } a ´ } L ` } a ` } L ` bC ` ˙ ` ÿ x P η ` ˆ m p x q ` λ p x q C ` ˙ ` ˆ ` θC ` ˙ ÿ x P η ` ÿ y P η ` z x a ´ p x ´ y q ≤ p ` a q ÿ x P η ` ` m ` ` m p x q ˘ , where we have used (3.8) to obtain λ p x q ≤ ϑm p x q and by (3.9) a : “ max C ` } a ´ } L ` } a ` } L ` bC ` m ` , ϑC ` , θC ` + ≤ a ` P p , q . Under the given conditions we can apply Theorem 2.5 for L instead of L ´ . Let µ inv be thecorresponding unique invariant measure. Without interactions with the environment, i.e. m “ λ “
0, we have µ inv “ δ H . In the presence of interactions, however, the invariant measure isnon-degenerated, i.e. µ inv ‰ δ H . 16 .4 Density dependent branching in Glauber environment Suppose that the environment is, as before, a Glauber dynamics with parameters z ´ , ψ (see(3.5)). For the system we assume that d ` p x, γ q “ m ` exp ` E κ p x, γ ` q ˘ ,b ` p x, γ q “ ÿ y P γ ` exp ` ´ E φ p y, γ ´ q ˘ a ` p x ´ y q . This model describes a branching process with strong (exponential) killing rate where the branch-ing rate, in addition, can be slowed down by interaction with the environment. This model isa prototype of a branching process inside a ”delirious” environment. We make the followingassumptions on the parameters of the system:(a) z ´ ą ψ ≥ β p ψ q ă 8 and there exists C ´ ą m ` ą a ` , φ, κ ≥ β p φ q ă 8 and a ` is integrable. Moreover κ, a ` are bounded.(c) There exist constants ϑ ą b ≥ η ` P Γ ÿ x P η ` ÿ y P η ` z x a ` p x ´ y q ≤ ϑ ÿ x P η ` ÿ y P η ` z x κ p x ´ y q ` b | η ` | . Finally we have β p´ κ q ă 8 and there exists C ` ą e C ` β p´ κ q ` e C ´ β p φ q m ` C ` max t C ` } a ` } L ` b, ϑ u ă . Then conditions (E1) – (E3), (S1) – (S3) and (AV1) – (AV3) are satisfied. Conditions (E1) –(E3) have been shown in the first example. As before one can show that (S3) holds and (S1) issatisfied where D ` p x, η q “ | η ´ | m ` ź y P η ` ´ e κ p x ´ y q ´ ¯ ,B ` p x, η q “ Γ ` p η ` q ÿ y P η ` ź z P η ´ ´ e ´ φ p x ´ z q ´ ¯ a p x ´ y q . Then a computation shows that c ` p η q ≤ m ` e C ` β p´ κ q ÿ x P η ` e E κ p x,η ` z x q ` e C ´ β p φ q C ` ÿ x P η ` ÿ y P η ` z x a ` p x ´ y q ` C ` C ´ } a ` } L e C ´ β p φ q | η ` | ≤ m ` e C ` β p´ κ q ÿ x P η ` e E κ p x,η ` z x q ` ϑ e C ´ β p φ q C ` ÿ x P η ` ÿ y P η ` z x κ p x ´ y q` ˜ e C ´ β p φ q C ` b ` C ` C ´ } a ` } L e C ´ β p φ q ¸ | η ` | ≤ ˜ m ` e C ` β p´ κ q ` max t b ` C ` } a ` } L , ϑ u e C ´ β p φ q C ` ¸ ÿ x P η ` e E κ p x,η ` z x q , ÿ x P η ` e E κ p x,η ` z x q ≥ | η ` | ` ÿ x P η ` ÿ y P η ` z x κ p x ´ y q . This shows (S2). Let us show (AV1) – (AV3). The averaged birth-and-death rates are given by d p x, γ ` q “ m ` exp ` E κ p x, γ ` z x q ˘ ,b p x, γ ` q “ ÿ y P γ ` λ p y q a ` p x ´ y q with λ p y q : “ ş Γ ´ exp ` ´ E φ ´ p y, γ ´ q ˘ dµ inv p γ ´ q ≤
1. As before (AV1) and (AV3) hold with D p x, η ` q “ m ` ź y P η ` ´ e κ p x ´ y q ´ ¯ ,B p x, η ` q “ Γ ` p η ` q ÿ y P η ` λ p y q a ` p x ´ y q . Then for ξ ` X ζ ` “ H we get B p x, ζ ` Y ξ ` q “ | ξ ` | Γ ` p ζ ` q ÿ y P ζ ` λ p y q a ` p x ´ y q ` | ζ ` | Γ ` p ξ ` q ÿ y P ξ ` λ p y q a ` p x ´ y q and hence ˇˇˇˇˇˇ ÿ ζ ` Ă η ` z x B p x, ζ ` Y ξ ` q ˇˇˇˇˇˇ ≤ | ξ ` | ÿ y P η ` z x λ p y q a ` p x ´ y q ` Γ ` p ξ ` q ÿ y P ξ ` λ p y q a ` p x ´ y q . A similar computation for D gives, recall λ ≤ c p η ` q ≤ m ` e C ` β p´ κ q ÿ x P η ` e E κ p x,η ` z x q ` ÿ x P η ` ż R d λ p y q a ` p x ´ y q dy ` C ` ÿ x P η ` ÿ y P η ` z x λ p y q a ` p x ´ y q ≤ m ` e C ` β p´ κ q ÿ x P η ` e E κ p x,η ` z x q ` ˆ } a ` } L ` bC ` ˙ | η ` | ` ϑC ` ÿ x P η ` ÿ y P η ` z x κ p x ´ y q ≤ ˆ m ` e C ` β p´ κ q ` max " } a ` } L ` bC ` , ϑC ` *˙ ÿ x P η ` e E κ p x,η ` z x q . It follows from m ` e C ` β p´ κ q ` max " } a ` } L ` bC ` , ϑC ` * ≤ m ` e C ` β p´ κ q ` e C ´ β p φ q C ` max C ` } a ` } L ` b, ϑ ( that (AV2) is satisfied. 18 .5 Two interacting BDLP-models Suppose that the environment is given by an BDLP model with immigration parameter z ą d ´ p x, γ ´ q “ m ´ ` ÿ y P γ ´ a ´ p x ´ y q ,b ´ p x, γ ´ q “ ÿ y P γ ´ a ` p x ´ y q ` z. For the system we suppose that it is also an BDLP model with additional killing and branchingcaused by the environment at additive rates, i.e. d ` p x, γ q “ m ` ` ÿ y P γ ` b ´ p x ´ y q ` ÿ y P γ ´ ϕ ´ p x ´ y q ,b ` p x, γ q “ ÿ y P γ ` b ` p x ´ y q ` ÿ y P γ ´ ϕ ` p x ´ y q . We make the following assumptions on the parameters of the model(a) z, m ` , m ´ ą a ˘ , b ˘ , ϕ ˘ are non-negative, symmetric, integrable and bounded.(b) There exist constants b , b ≥ ϑ , ϑ , ϑ ą ÿ x P η ` ÿ y P η ` z x b ` p x ´ y q ≤ ϑ ÿ x P η ` ÿ y P η ` z x b ´ p x ´ y q ` b | η ` | ÿ x P η ´ ÿ y P η ´ z x a ` p x ´ y q ≤ ϑ ÿ x P η ´ ÿ y P η ´ z x a ´ p x ´ y q ` b | η ´ | , and ϕ ` ≤ ϑ ϕ ´ hold. Finally there exist C ` ą ϑ , ϑ and C ´ ą ϑ such that m ` ą C ` } b ´ } L ` C ´ } ϕ ´ } L ` b C ` ` } b ` } L ` } ϕ ` } L ,m ´ ą C ´ } a ´ } L ` b ` zC ´ ` } a ` } L . Then conditions (E1) – (E3), (S1) – (S3) and (AV1) – (AV3) are satisfied. First it is clear that(E1), (S1) and (E3), (S3) are satisfied with D ´ p x, η ´ q “ m ´ | η ´ | ` Γ ´ p η ´ q ÿ y P η ´ a ´ p x ´ y q ,B ´ p x, η ´ q “ z | η ´ | ` Γ ´ p η ´ q ÿ y P η ´ a ` p x ´ y q ,D ` p x, η q “ m ` | η | ` | η ´ | Γ ` p η ` q ÿ y P η ` b ´ p x ´ y q ` | η ` | Γ ´ p η ´ q ÿ y P η ´ ϕ ´ p x ´ y q ,B ` p x, η q “ | η ´ | Γ ` p η ` q ÿ y P η ` b ` p x ´ y q ` | η ` | Γ ´ p η ´ q ÿ y P η ´ ϕ ` p x ´ y q . c ´ p η ´ q ≤ ˆ m ´ ` C ´ } a ´ } L ` zC ´ ` } a ` } L ˙ | η ´ | ` ÿ x P η ´ ÿ y P η ´ z x a ´ p x ´ y q ` C ´ ÿ x P η ´ ÿ y P η ´ z x a ` p x ´ y q ≤ ˆ m ´ ` C ´ } a ´ } L ` zC ´ ` } a ` } L ` b C ´ ˙ | η ´ | ` ˆ ` ϑ C ´ ˙ ÿ x P η ´ ÿ y P η ´ z x a ´ p x ´ y q and likewise c ` p η q ≤ ˆ m ` ` C ` } b ´ } L ` C ´ } ϕ ´ } L ` } b ` } L ` C ´ C ` } ϕ ` } L ˙ | η ` |` ÿ x P η ` ÿ y P η ` z x b ´ p x ´ y q ` C ` ÿ x P η ` ÿ y P η ` z x b ` p x ´ y q` ÿ x P η ` ÿ y P η ´ ϕ ´ p x ´ y q ` C ` ÿ x P η ` ÿ y P η ´ ϕ ` p x ´ y q ≤ ˆ m ` ` C ` } b ´ } L ` C ´ } ϕ ´ } L ` } b ` } L ` C ´ C ` } ϕ ` } L ` b C ` ˙ | η ` |` ˆ ` ϕ C ` ˙ ÿ x P η ` ÿ y P η ` z x b ´ p x ´ y q ` ˆ ` ϕ C ` ˙ ÿ x P η ` ÿ y P η ´ ϕ ´ p x ´ y q . In view of the assumptions made on the parameters it is easily seen that (E2) and (S2) hold.Let us show that (AV1) – (AV3) hold. The averaged birth-and-death rates are given by d p x, γ ` q “ m ` ` ϕ ´ p x q ` ÿ y P γ ` z x b ´ p x ´ y q ,b p x, γ ` q “ ÿ y P γ ` b ` p x ´ y q ` ϕ ` p x q , where ϕ ˘ p x q “ ş Γ ´ ř y P γ ´ ϕ ˘ p x ´ y q dµ inv p γ ´ q . Clearly (AV1) and (AV3) hold with D p x, η ` q “ ` m ` ` ϕ ´ p x q ˘ | η ` | ` Γ ` p η ` q ÿ y P η ` b ´ p x ´ y q ,B p x, η ` q “ ϕ ` p x q | η ` | ` Γ ` p η ` q ÿ y P η ` b ` p x ´ y q . Then, as before, we get by assumption (c) c p η ` q ≤ ˆ m ` ` ϕ ´ p x q ` C ` } b ´ } L ` ϕ ` p x q C ` ` } b ` } L ˙ | η ` |` ÿ x P η ` ÿ y P η ` z x b ´ p x ´ y q ` C ` ÿ x P η ` ÿ y P η ` z x b ` p x ´ y q ≤ ˆ m ` ` ϕ ´ p x q ` C ` } b ´ } L ` ϕ ` p x q C ` ` } b ` } L ` b C ` ˙ | η ` | ` ˆ ` ϑ C ` ˙ ÿ x P η ` ÿ y P η ` z x b ´ p x ´ y q which shows condition (AV2). 20 Proof of Theorem 2.3 and Theorem 2.5
In this section we closely follow the arguments in [FK16b] (see also [Fri17]). Since the necessarycomputations are very similar to the latter works, we give only the main steps of proof. Let L C ´ be the Banach space of integrable functions with norm } G } L C ´ “ ż Γ ´ | G p η ´ q| C | η ´ |´ dλ p η ´ q . Define an operator p L ´ on the domain D p p L ´ q : “ t G P L C ´ | M ´ ¨ G P L C ´ u by p p L ´ G qp η ´ q “ ´ ÿ ξ ´ Ă η ´ G p ξ ´ q ÿ x P ξ ´ ÿ ζ ´ Ă ξ ´ z x D ´ p x, η ´ z ξ ´ Y ζ ´ q` ÿ ξ ´ Ă η ´ ż R d G p ξ ´ Y x q ÿ ζ ´ Ă ξ ´ B ´ p x, η ´ z ξ ´ Y ζ ´ q dx. We will see that this operator is related to L ´ via the K -transform. proposition 4.1. Suppose that (E1), (E2) are satisfied. Then p p L ´ , D p p L ´ qq is the generatorof an analytic semigroup of contractions p T ´ p t qq t ≥ on L C ´ . Moreover B bs p Γ ´ q is a core for p p L ´ , D p p L ´ qq .Proof. Consider the decomposition p L ´ “ A ` B where AG p η ´ q “ ´ M ´ p η q G p η ´ q and p BG qp η ´ q “ ´ ÿ ξ ´ Ĺ η ´ G p ξ ´ q ÿ x P ξ ´ ÿ ζ ´ Ă ξ ´ z x D ´ p x, η ´ z ξ ´ Y ζ ´ q` ÿ ξ ´ Ă η ´ ż R d G p ξ ´ Y x q ÿ ζ ´ Ă ξ ´ B ´ p x, η ´ z ξ ´ Y ζ ´ q dx. Observe that p A, D p p L ´ qq is the generator of a positive, analytic semigroup of contractions on L C ´ . Define another positive operator on D p p L ´ q by p B ˚ G qp η ´ q “ ´ ÿ ξ ´ Ĺ η ´ G p ξ ´ q ÿ x P ξ ´ ˇˇˇˇˇˇ ÿ ζ ´ Ă ξ ´ z x D ´ p x, η ´ z ξ ´ Y ζ ´ q ˇˇˇˇˇˇ ` ÿ ξ ´ Ă η ´ ż R d G p ξ ´ Y x q ˇˇˇˇˇˇ ÿ ζ ´ Ă ξ ´ B ´ p x, η ´ z ξ ´ Y ζ ´ q ˇˇˇˇˇˇ dx. By (E2) we can find r P p , q such that a ´ ă ` r ă
2. Then, by (3.1), a short computationshows that for 0 ≤ G P D p p L ´ q we have ż Γ ´ B ˚ G p η ´ q C | η ´ |´ dλ p η ´ q ≤ ż Γ ´ ` c ´ p η ´ q ´ M ´ p η ´ q ˘ | G p η ´ q| C | η ´ |´ dλ p η ´ q ≤ p a ´ ´ q} AG } L C ´ ş Γ ´ p A ` r B ˚ q G p η ´ q C | η ´ |´ dλ p η ´ q ≤
0. Hence by [TV06, Theorem 2.2] p A ` B ˚ , D p p L ´ qq is the generator of a positive semigroup V p t q of contractions on L C ´ . Applying[AR91, Theorem 1.1] together with | BG | ≤ B ˚ | G | it follows that p A ` B, D p p L ´ qq is the genera-tor of an analytic semigroup T ´ p t q on L C ´ . By [AR91, Theorem 1.2] we get | T ´ p t q G | ≤ V p t q| G | and since V p t q is a semigroup of contractions, the same holds true for T p t q . For the last assertionlet G P D p p L ´ q and set G n p η ´ q “ | η ´ | ≤ n η ´ Ă B n G p η ´ q where B n denotes the ball of diameter n around zero. Then it is easily seen that G n ÝÑ G and p L ´ G n ÝÑ p L ´ G in L C ´ .Let us now prove Lemma 2.2. Proof. (Lemma 2.2) (a) Fix µ P P C ´ . First note that the K-transform has an unique extensionto a bounded linear operator K : L p Γ ´ , k µ dλ q ÝÑ L p Γ ´ , dµ q such that (2.1) is absolutelyconvergent for G P L p Γ ´ , k µ dλ q and µ -a.a. γ ´ (see [KK02]). The assertion now follows fromthe following estimates ż Γ ´ ` d ´ p x, γ ´ q ` b ´ p x, γ ´ q ˘ dµ p γ ´ q “ ż Γ ´ ` D ´ p x, η ´ q ` B ´ p x, η ´ q ˘ k µ p η ´ q dλ p η ´ q ≤ ż Γ ´ ` | D ´ p x, η ´ q| ` | B ´ p x, η ´ q| ˘ k µ p η ´ q dλ p η ´ q ≤ } k µ } K C ´ max t , C ´ u c ´ pt x uq ≤ } k µ } K C ´ max t , C ´ u a ´ d ´ p x, Hq . (b) Since for µ P P C ´ and G P B bs p Γ ´ q we have p LG P L C ´ Ă L p Γ ´ , k µ dλ q it follows that K p LG P L p Γ ´ , dµ q . Let p K ´ F qp η ´ q “ ř ξ ´ Ă η ´ p´ q | η ´ z ξ ´ | F p ξ ´ q , η ´ P Γ ´ be the inversetransformation to (2.1). Using the properties of K ´ together with (E1) we obtain p K ´ d ´ p x, ¨ Y ξ ´ z x qqp η ´ z ξ ´ q “ ÿ ζ ´ Ă η ´ z ξ ´ p´ q |p η ´ z ξ ´ qz ζ ´ | ÿ α Ă ζ ´ Y ξ ´ z x D ´ p x, α q“ ÿ ζ ´ Ă η ´ z ξ ´ p´ q |p η ´ z ξ ´ qz ζ ´ | ÿ α Ă ζ ´ ÿ α Ă ξ ´ z x D ´ p x, α Y α q“ ÿ α Ă ξ ´ z x D ´ p x, η ´ z ξ ´ Y α q and similarly p K ´ b ´ p x, ¨ Y ξ ´ qqp η ´ z ξ ´ q “ ř α Ă ξ ´ B ´ p x, η ´ z ξ ´ Y α q . Hence we have shownthat p p L ´ G qp η ´ q “ ´ ÿ ξ ´ Ă η ´ G p ξ ´ q ÿ x P ξ ´ p K ´ d ´ p x, ¨ Y ξ ´ z x qqp η ´ z ξ ´ q` ÿ ξ ´ Ă η ´ ż R d G p ξ ´ Y x qp K ´ b p x, ¨ Y ξ ´ qqp η ´ z ξ ´ q dx. Now we may deduce from [FKK12, Proposition 3.1] that K p L ´ “ L ´ KG for G P B bs p Γ ´ q .22et p L C ´ q ˚ be the dual Banach space to L C ´ . Using the duality x G, k y : “ ş Γ ´ G p η ´ q k p η ´ q dλ p η ´ q it can be identified with K C ´ . For k P K C ´ let p L ∆ , ´ k qp η ´ q “ ´ ÿ x P η ´ ż Γ ´ k p η ´ Y ξ ´ q ÿ ζ ´ Ă η ´ z x D ´ p x, ζ ´ Y ξ ´ q dλ p ξ ´ q` ÿ x P η ´ ż Γ ´ k p η ´ Y ξ ´ z x q ÿ ζ ´ Ă η ´ z x B ´ p x, ζ ´ Y ξ ´ q dλ p ξ ´ q . Note that L ∆ , ´ k is λ -a.e. well-defined, satisfies for any k P K C ´ | L ∆ , ´ k p η ´ q| ≤ } k } K C ´ C | η ´ |´ c ´ p η ´ q , η ´ P Γ ´ , but, in general, L ∆ , ´ k R K C ´ . Lemma 4.2.
Suppose that conditions (E1) and (E2) are satisfied. Let p p L ˚ , ´ , D p p L ˚ , ´ qq be theadjoint operator to p p L ´ , D p p L ´ qq on K C ´ . Then L ∆ , ´ k “ p L ˚ , ´ k for any k P D p p L ˚ , ´ q and D p p L ˚ , ´ q “ t k P K C ´ | L ∆ , ´ k P K C ´ u . Proof.
Arguing similarly to [FK16b, Lemma 3.5] one can show that x p L ´ G, k y “ x
G, L ∆ , ´ k y , G P D p p L ´ q , k P K C ´ from which one can readily deduce the assertion.The Cauchy problem ddt k ´ t “ L ∆ , ´ k ´ t , k ´ t | t “ “ k ´ (4.1)is a Markov analogue of the BBGKY-hierarchy and describes the evolution of correlation func-tions corresponding to the Fokker-Planck equation. Denote by p T ´ p t q ˚ the adjoint semigroup to T ´ p t q . The next proposition gives existence and uniqueness of weak solutions to this hierarchy. proposition 4.3. Suppose that (E1) and (E2) are satisfied.(a) For any k ´ P K C ´ the function k ´ t “ T ´ p t q ˚ k ´ satisfies x G, k ´ t y “ x G, k ´ y ` t ż x p L ´ G, k ´ s y ds, t ≥ for any G P B bs p Γ ´ q .(b) Let p r t q t ≥ Ă K C ´ with r “ k ´ satisfy (4.2) and suppose that sup t Pr ,T s } r t } K C ´ ă 8 , @ T ą . (4.3) Then r t “ T ´ p t q ˚ k . roof. Note that assertion (a) readily follows from the properties of the adjoint semigroup T ´ p t q ˚ . Let us prove (b). It follows from [WZ06, Theorem 2.1] that there exists at most onesolution to (4.2) such that t ÞÝÑ r t is continuous w.r.t. the topology C . Here C is the topologyof uniform convergence on compact sets of L C ´ given by a basis of neighbourhoods with r P K C ´ : |x G, r y ´ x
G, k y| ă ε, @ G P K ( where ε ą k P K C ´ and K is a compact subset of L C ´ . Using p L ´ G P L C ´ , (4.3) and (4.2)we see that t ÞÝÑ x
G, r t y is continuous for any G P B bs p Γ ´ q . Since B bs p Γ ´ q is dense in L C ´ , by(4.3) we can show that t ÞÝÑ r t is continuous w.r.t. σ p K C ´ , L C ´ q . By [WZ06, Lemma 1.10] itfollows that t ÞÝÑ r t is also continuous w.r.t. C which proves the assertion.In the next step we prove the equivalence between solutions to (2.5) and (4.2). Lemma 4.4.
Suppose that conditions (E1) – (E2) are satisfied. Let p µ t q t ≥ Ă P C ´ , denote by p k t q t ≥ the corresponding correlation functions and assume that sup t Pr ,T s } k t } K C ´ ă 8 , @ T ą . Then p µ t q t ≥ satisfies (2.5) if and only if p k t q t ≥ satisfies (4.2) .Proof. Recall that K : L p Γ , k µ t dλ q ÝÑ L p Γ ´ , dµ t q is a bounded linear operator. Hence for F “ KG with G P B bs p Γ ´ q we get L ´ F “ K p L ´ G P L p Γ ´ , dµ t q . Moreover, by (2.2) we get ż Γ ´ G p η ´ q k t p η ´ q dλ p η ´ q “ ż Γ ´ F p γ ´ q dµ t p γ ´ q , ż Γ ´ p p L ´ G qp η ´ q k t p η ´ q dλ p η ´ q “ ż Γ ´ p L ´ F qp γ ´ q dµ t p γ ´ q which then implies the assertion.Let µ ´ P P C ´ with correlation function k ´ and let k ´ t be the unique solution to (4.2). Itremains to show that there exist µ ´ t P P C ´ such that k µ ´ t “ k ´ t . For this purpose we show that k ´ t is positive definite in the sense of Lenard, i.e. ż Γ ´ G p η ´ q k ´ t p η ´ q dλ p η ´ q ≥ , @ G P B bs p Γ ´ q with KG ≥ . (4.4)First we prove the following lemma. Lemma 4.5.
Let p u δt q t ≥ be such that | u δt p η ´ q| ≤ AC | η ´ |´ ź x P η ´ R δ p x q , η P Γ ´ , t ≥ nd suppose that for any bounded measurable function F : Γ ´ ÝÑ R x F, H u δt y “ x F, H u δ y ` t ż x L ´ δ F, H u δs y ds, t ≥ holds, where L ´ δ F is given by (2.3) with b ´ replaced by R δ p x q b ´ and p H u δs qp η ´ q “ ż Γ ´ p´ q | ξ ´ | u δs p η ´ Y ξ ´ q dλ p ξ ´ q , η ´ P Γ ´ . Then t ÞÝÑ u t P L p Γ ´ , dλ q is continuous w.r.t. the norm.Proof. Take 0 ≤ s ă t and let F be bounded and measurable with } F } L ≤
1. Then ˇˇˇˇˇˇˇ ż Γ ´ F p η ´ q H u δt p η ´ q dλ p η ´ q ´ ż Γ ´ F p η ´ q H u δs p η ´ q dλ p η ´ q ˇˇˇˇˇˇˇ ≤ t ż s ż Γ ´ |p L ´ δ F qp η ´ q|| H u δr p η ´ q| dλ p η ´ q dr ≤ p t ´ s q c sup r Pr s,t s ż Γ ´ e c | η ´ | | H u δr p η ´ q| dλ p η ´ q for some constants c , c ą ż Γ ´ e c | η ´ | | H u δr p η ´ q| dλ p η ´ q ≤ ż Γ ´ ż Γ ´ e c | η ´ | | u δr p η ´ Y ξ ´ q| dλ p η ´ q dλ p ξ ´ q“ ż Γ ´ ÿ η ´ Ă ξ ´ e c | η ´ | | u δr p ξ ´ q| dλ p ξ ´ q ≤ C ż Γ ´ p ` e c q | ξ ´ | C | ξ ´ |´ ź x P ξ ´ R δ p x q dλ p ξ ´ q ≤ C ÿ k “ C k ´ p ` e c q k } R δ } kL k ! ă 8 . Taking the supremum over all such F gives } H u δt ´ H u δs } L ≤ A p t ´ s q for some constant A ą .2 Proof of Theorem 2.5 Using (2.2) together with Lemma 4.2 we see that any invariant measure µ inv P P C ´ satisfies0 “ ż Γ ´ p KG qp γ ´ q dµ inv p γ ´ q “ ż Γ ´ p p L ´ G qp η ´ q k inv p η ´ q dλ p η ´ q “ ż Γ ´ G p η ´ qp L ∆ , ´ k inv qp η ´ q dλ p η ´ q for any G P B bs p Γ ´ q and hence L ∆ , ´ k inv “
0. The next lemma states that this equation has,indeed, exactly one solution.
Lemma 4.6.
The equation L ∆ , ´ k inv “ , k inv pHq “ has a unique solution k inv P K C ´ . Moreover, for any ρ P R the function k ρ p η ´ q “ ρ | η ´ | ` ρk inv p η ´ q is the unique solution to L ∆ , ´ k ρ “ , k ρ pHq “ ρ. (4.5)The following proof is based on the ideas from [FKK12]. Proof.
Consider the decomposition K C ´ “ K C ´ ‘ K ≥ C ´ with K C ´ “ t k P K C ´ | k p η ´ q “ k pHq | η ´ | u , K ≥ C ´ “ t k P K C ´ | k pHq “ u . (4.6)For k P K C ´ write k “ | η ´ | ρ ` r k with r k P K ≥ C ´ and ρ “ k pHq . Hence obtain p L ∆ , ´ k qp η ´ q “ ´ M ´ p η ´ q r k p η ´ q ´ ÿ x P η ´ ż Γ ´ zH r k p η ´ Y ξ ´ q ÿ ζ ´ Ă η ´ z x D ´ p x, ζ ´ Y ξ ´ q dλ p ξ ´ q` ÿ x P η ´ ż Γ ´ r k p η ´ Y ξ ´ z x q ÿ ζ ´ Ă η ´ z x B ´ p x, ζ ´ Y ξ ´ q dλ p ξ ´ q ` ρ Γ ´ p η ´ q ÿ x P η ´ B ´ p x, Hq . We let S r k pHq “ η ´ ‰ Hp S r k qp η ´ q “ ´ M ´ p η ´ q ÿ x P η ´ ż Γ ´ zH r k p η ´ Y ξ ´ q ÿ ζ ´ Ă η ´ z x D ´ p x, ζ ´ Y ξ ´ q dλ p ξ ´ q` M ´ p η ´ q ÿ x P η ´ ż Γ ´ r k p η ´ Y ξ ´ z x q ÿ ζ ´ Ă η ´ z x B ´ p x, ζ ´ Y ξ ´ q dλ p ξ ´ q where the latter expression is well-defined due to M ´ p η ´ q ą η ´ ‰ H . Using p L ∆ , ´ k qpHq “
0, it is easily seen that k satisfies (4.5) if and only if r k p η ´ q ´ p S r k qp η ´ q “ ρ Γ ´ p η ´ q ÿ x P η ´ B ´ p x, Hq D ´ p x, Hq . (4.7)It is not difficult to see that S is a bounded linear operator such that } S } L p K C ´ q ă emark 4.7. Equation (4.7) is an analogue of the Kirkwood-Salsburg equation.
Next we prove that p L ´ has a spectral gap. For this purpose we introduce the same decom-position as (4.6) for L C ´ , i.e. L C ´ “ L C ´ ‘ L ≥ C ´ with projection operators P G p η ´ q “ | η ´ | G pHq , P ≥ G p η ´ q “ p ´ | η ´ | q G p η ´ q . proposition 4.8. Suppose that conditions (E1), (E2) and (2.7) are satisfied and let ω : “ sup " ω P ” , π ı ˇˇˇˇ a ´ ă ` cos p ω q * . (4.8) Then the following statements hold(a) The point is an eigenvalue for p p L ´ , D p p L ´ qq with eigenspace L C ´ .(b) Let λ : “ p ´ a ´ q M ˚ ą where M ˚ : “ inf H‰ η ´ P Γ ´ M ´ p η q , then I : “ t λ P C | Re p λ q ą ´ λ uzt u ,I : “ " λ P C ˇˇˇˇ | arg p λ q| ă π ` ω * zt u both belong to the resolvent set ρ p p L ´ q of p L ´ on L C ´ .Proof. Observe that p L ´ P “ p L ´ P ≥ “ p L ´ . Thus we obtain the decomposition p L ´ “ L P ≥ ` L P ≥ , where D p L q “ D p p L ´ q X L ≥ C ´ and L : D p L q ÝÑ L C ´ , L G “ P p L ´ GL : D p L q ÝÑ L ≥ C ´ , L G “ P ≥ p L ´ G. Using the definition of p L ´ we see that p L G qp η ´ q “ | η ´ | ş R d G p x q B ´ p x, Hq dx and L “ A ` B with p A G qp η ´ q “ ´ M ´ p η ´ q G p η ´ qp B G qp η ´ q “ ´ ÿ ξ ´ Ĺ η ´ G p ξ ´ q ÿ x P ξ ´ ÿ ζ ´ Ă ξ ´ z x D ´ p x, η ´ z ξ ´ Y ζ ´ q` ÿ ξ ´ Ă η ´ ż R d G p ξ ´ Y x q ÿ ζ ´ Ă ξ ´ B ´ p x, η ´ z ξ ´ Y ζ ´ q dx. Let us first show that p L , D p L qq is invertible on L ≥ C ´ . Denote by } ¨ } L C ´ the norm on L ≥ C ´ .Since M ´ p η ´ q ≥ M ˚ for all | η ´ | ≥
1, we obtain for any λ “ u ` iw , u ≥ w P R ˇˇˇˇ Gλ ` M ´ p η ´ q ˇˇˇˇ ≤ | G | a p u ` M ˚ q ` w ≤ | G | min ˜ | λ | , a M ˚ ` w ¸ . λ P ρ p A q and } R p λ ; A q G } L C ´ ≤ min ˜ | λ | , a M ˚ ` w ¸ } G } L C ´ . (4.9)A simple computation shows that for any G P L ≥ C ´ } B G } L C ´ ≤ ż Γ ´ | B G p η ´ q| C | η ´ |´ d λ p η ´ q ≤ p a ´ ´ q} A G } L C ´ . Hence we see that p ´ B R p λ ; A qq is invertible on L ≥ C ´ and using p λ ´ L q “ p ´ B R p λ ; A qqp λ ´ A q . (4.10)we obtain λ P ρ p L q with R p λ ; L q “ R p λ ; A qp ´ B R p λ ; A qq ´ . (4.11)In particular, we obtain for λ “ u ` iw , u ≥ w P R by (4.9) and (4.11) } R p λ ; L q G } L C ´ ≤ min ˆ | λ | , ? M ˚ ` w ˙ ´ a ´ } G } L C ´ and for λ “ iw , w P R } R p iw, L q G } L C ´ ≤ a M ˚ ` w ´ ´ a ´ } G } L C ´ . For λ “ u ` iw , 0 ą u ą ´ λ and w P R write p u ` iw ´ L q “ p ` uR p iw ; L qqp iw ´ L q . Then, by | u | ă λ and | u | ? M ˚ ` w ´ a ´ ≤ | u | λ ă λ P ρ p L q and } R p λ ; L q G } L C ´ ≤ a M ˚ ` w ´ ´ a ´ ˆ ´ | u | λ ˙ ´ } G } L C ´ . Therefore, I belongs to the resolvent set of L . For I let λ “ u ` iw P I and u ă
0. Then,there exists ω P p , ω q such that | arg p λ q| ă π ` ω and hence | w | “ | cot p arg p λ q ´ π q|| u | ≥ cot p ω q| u | . This implies for η ´ ‰ H| λ ` M ´ p η ´ q| “ p u ` M ´ p η ´ qq ` w ≥ p u ` M ´ p η ´ qq ` cot p ω q u . u “ ´ M ´ p η ´ q ` cot p ω q which yields | λ ` M ´ p η ´ q| ≥ M ´ p η ´ q ˜ˆ cot p ω q ` cot p ω q ˙ ` cot p ω q p ` cot p ω q q ¸ “ M ´ p η ´ q cot p ω q ` cot p ω q “ M ´ p η ´ q cos p ω q . Then } B R p λ ; A q G } L C ´ ≤ p a ´ ´ q} A R p λ ; A q G } L C ´ ≤ a ´ ´ p ω q } G } L C ´ . Finally by (4.10) together with a ´ ´ ă cos p ω q (see (4.8)) we obtain I Ă ρ p L q . Moreover,for each λ “ u ` iw such that π ă | arg p λ q| ă π ` ω and, for some ω P p , ω q , } R p λ ; L q G } L C ´ ≤ a p u ` M ˚ q ` w ´ ´ a ´ ´ p ω q } G } L C ´ ≤ p ´ a ´ ´ p ω q q ´ | w | } G } L C ´ ≤ ? p ´ a ´ ´ p ω q q ´ | λ | } G } L C ´ , where we have used | w | ≥ | λ |? .Let us prove (a). Take ψ P D p p L ´ q and consider the decomposition ψ “ P ψ ` P ≥ ψ “ ψ ` ψ with ψ P L C ´ and ψ P D p L q . Then0 “ p L ´ ψ “ L ψ ` L ψ P L C ´ ‘ L ≥ C ´ and hence L ψ “
0. Since 0 P ρ p L q we obtain ψ “ λ P I Y I and H “ H ` H P L C ´ ‘ L ≥ C ´ . Then, we have to find G P D p p L ´ q such that p λ ´ p L q G “ H . Using again the decomposition p L ´ “ L P ≥ ` L P ≥ ,above equation is equivalent to the system of equations λG ´ L G “ H p λ ´ L q G “ H . Since λ P I Y I Ă ρ p L q the second equation has a unique solution on L ≥ C ´ given by G “ R p λ ; L q H . Therefore, G is given by G “ λ p H ` L R p λ ; L q H q . Define a projection operator p P : L C ´ ÝÑ L C ´ by p P G p η ´ q “ ż Γ ´ G p ξ ´ q k inv p ξ ´ q dλ p ξ ´ q | η ´ | Then x p P G, k y “ x G, p P ˚ k y where p P ˚ k p η ´ q “ k inv p η ´ q k pHq . The next proposition completes theproof of Theorem 2.5. 29 roposition 4.9. There exists a unique invariant measure µ inv P P C ´ with correlation function k inv . Moreover, the following holds.(a) T ´ p t q is uniformly ergodic with exponential rate and projection operator p P .(b) T ´ p t q ˚ is uniformly ergodic with exponential rate and projection operator p P ˚ .Proof. Using p L ´ P “ T ´ p t q P “ P and hence T ´ p t q “ P ` P T ´ p t q P ≥ ` P ≥ T ´ p t q P ≥ , t ≥ . (4.12)The L ≥ C ´ part of T ´ p t q is given by P ≥ T ´ p t q P ≥ and has the generator L . The proof ofprevious proposition shows that for any ε ą ω “ ω p ε q P p , π q such thatΣ p ε q : “ " λ P C ˇˇˇˇ | arg p λ ` λ ´ ε q| ≤ π ` ω * Ă I Y I Y t u and there exists M p ε q ą } R p λ ; L q G } L ≥ C ≤ M p ε q| λ | } G } L C ´ for all λ P Σ p ε qzt u . Moreover, p L , D p L qq is a sectorial operator of angle ω on L ≥ C ´ . Denoteby r T ´ p t q the bounded analytic semigroup on L ≥ C ´ given by r T ´ p t q “ πi ż σ e ζt R p ζ ; L q dζ, t ą , (4.13)where the integral converges in the uniform operator topology, see [Paz83]. Here σ denotes anypiecewise smooth curve in " λ P C ˇˇˇˇ | arg p λ q| ă π ` ω * zt u running from e ´ iθ to e iθ for θ P p π , π ` ω q . Then r T ´ p t q “ P ≥ T ´ p t q P ≥ .The spectral properties stated above and (4.13) imply that for any ε ą C p ε q ą t ≥ G P L ≥ C ´ } r T ´ p t q G } L C ´ ≤ C p ε q e ´p λ ´ ε q t } G } L C ´ . Repeat, e.g., the arguments in [KKM10]. By duality and (4.12), we see that the adjoint semi-group p T ´ p t q ˚ q t ≥ admits the decomposition T ´ p t q ˚ “ P ` P ≥ T ´ p t q ˚ P ` r T ´ p t q ˚ , t ≥ , where r T ´ p t q ˚ P L p K ≥ C ´ q is the adjoint semigroup to p r T ´ p t qq t ≥ . Hence } T ´ p t q ˚ k } K C ´ ≤ C p ε q e ´p λ ´ ε q t } k } K C ´ , k P K ≥ C ´ . k P K C ´ , then k ´ p P ˚ k P K ≥ C ´ . Using T ´ p t q ˚ p P ˚ “ p P ˚ T ´ p t q ˚ “ p P ˚ we obtain } T ´ p t q ˚ k ´ p P ˚ k } K C ´ “ } T ´ p t q ˚ p k ´ p P ˚ k q} K C ´ ≤ C p ε q e ´p λ ´ ε q t } k ´ p P ˚ k } K C ´ . This shows that T ´ p t q ˚ is uniformly ergodic with exponential rate. Duality implies that T ´ p t q is uniformly ergodic with exponential rate. Let µ P P C ´ , µ t P P C ´ the associated evolution ofstates and k µ t P K C ´ its correlation function for t ≥
0. By x G, k µ t y “ x G, T ´ p t q ˚ k µ y “ x T ´ p t q G, k µ y ≥ G P B bs p Γ ´ q with KG ≥ T ´ p t q we see that k inv is positivedefinite. Thus, there exists a unique measure µ inv P P C ´ having k inv as its correlation function. In this section we suppose that (E1) – (E3), (S1) – (S3), (AV1) – (AV3) and (2.7) are satisfied.Introduce the Banach space L C of equivalence classes of integrable functions on Γ equippedwith the norm } G } L C “ ż Γ | G p η q| C | η ` |` C | η ´ |´ dλ p η q . Then L C – L C ` p b π L C ´ where p b π denotes the projective tensor product of Banach spaces. Givenbounded linear operators A on L C ` and A on L C ´ , the product A b A on L C is defined asthe unique linear extension of the operator p A b A q G p η q “ A G p η ` q A G p η ´ q , G P X . This definition satisfies }p A b A q G } L C “ } A G } L C ` } A G } L C ´ . Since L C – L C ` p b π L C ´ , onecan show that such extension exists (see [Rya02]). For A being the identity operator we usethe notation A b and for A being the identity we use the notation b A respectively. Step 1. Construction of isolated, ergodic environment
In this step we study, in contrast to Theorem 2.3 and Theorem 2.5, the environment process onΓ . We study the extension of p L ´ (introduced in previous section) onto L C . Namely, let p p L ´ b q G p η q : “ ´ ÿ ξ ´ Ă η ´ G p η ` , ξ ´ q ÿ x P ξ ´ ÿ ζ ´ Ă ξ ´ z x D ´ p x, η ´ z ξ ´ Y ζ ´ q (5.1) ` ÿ ξ ´ Ă η ´ ż R d G p η ` , ξ ´ Y x q ÿ ζ ´ Ă ξ ´ B ´ p x, η ´ z ξ ´ Y ζ ´ q dx. be defined on the domain D p b p L ´ q “ t G P L C | M ´ ¨ G P L C u . Since K is defined component-wise, it is easily seen that K p b p L ´ q “ L ´ K holds on B bs p Γ q , where L ´ is extended onto F P p Γ q in the obvious way. Let X : “ t G b G | G P L C ` , G P L C ´ u Ă L C where p G b G qp η q : “ G p η ` q G p η ´ q . Then, lin p X q Ă L C is dense, where lin denotes the linear span of agiven subset of L C . 31 emma 5.1. The following assertions hold(a) b T ´ p t q is an analytic semigroup of contractions on L C with generator b p L ´ and core B bs p Γ q .(b) Let D : “ t G b G P X | G P D p p L ´ qu . Then lin p D q is a core for the generator p b p L ´ , D p b p L ´ qq .Proof. Following the same arguments as given in Proposition 4.1 we easily deduce that b p L ´ is the generator of an analytic semigroup U ´ p t q of contractions on L C and that B bs p Γ q is acore. It follows from the definition of b p L ´ that p b p L ´ qp G b G q “ G b p p L ´ G q , G b G P D . Hence G t : “ G b T ´ p t q G is a solution to the Cauchy problem ddt G t “ p b p L ´ q G t , G t | t “ “ G b G P D on L C . Since for G b G P D Ă D p b p L ´ q this Cauchy problem has the unique solution on L C given by U ´ p t qp G b G q , it follows that G b T ´ p t q G “ U ´ p t qp G b G q , which gives U ´ p t q “ b T ´ p t q . From this we easily deduce that D , and hence lin p D q , is invariant for b T ´ p t q . Thus it is a core for the generator b p L ´ .The following is the main estimate for the first step. proposition 5.2. There exist constants
C, λ ą such that for any G P L C }p b T ´ p t qq G ´ p P G } L C ≤ Ce ´ λt } G } L C , t ≥ , (5.2) where k inv is the correlation function for µ inv and p b p P q G p η q “ ż Γ ´ G p η ` , ξ ´ q k inv p ξ ´ q dλ p ξ ´ q | η ´ | . (5.3) Proof.
First observe that due to L C – L C ` p b π L C ´ and by [Rya02] we have L C – G “ ÿ n “ G n b G n ˇˇˇˇ p G n q n P N Ă L C ` , p G n q n P N Ă L C ´ , ÿ n “ } G n } L C ` } G n } L C ´ ă 8 + . Let G “ ř n “ G n b G n P L C and set G N : “ ř Nn “ G n b G n P lin p X q . By (5.3) we get p b p P q G N “ ř Nn “ G n ¨ p P G n and similarly p b T ´ p t qq G N “ ř Nn “ G n ¨ T ´ p t q G n . Thus weobtain }p b T ´ p t qq G N ´ p b p P q G N } L C ≤ N ÿ n “ } G n } L C ` } T ´ p t q G n ´ p P G n } L C ´ ≤ Ce ´ λt N ÿ n “ } G n } L C ` } G n } L C ´ , N Ñ 8 yields by G N ÝÑ G in L C }p b T ´ p t qq G ´ p P G } L C ≤ Ce ´ λt ÿ n “ } G n } L C ` } G n } L C ´ . (5.4)Since (see [Rya02]) } G } L C “ inf ÿ n “ } G n } L C ` } G n } L C ´ ˇˇˇˇ G “ ÿ n “ G n b G n P L C + we find a sequence G k “ ř n “ G n,k b G n,k with ř n “ } G n,k } L C ` } G n,k } L C ´ ÝÑ } G } L C , as k Ñ 8 . Applying (5.4) to G k gives }p b T ´ p t qq G k ´ p b p P q G k } L C ≤ Ce ´ λ t ÿ n “ } G n,k } L C ` } G n,k } L C ´ . Taking the limit k Ñ 8 yields (5.2).
Step 2. Construction of scaled dynamics
Consider the scaled operator L ε “ L ` ` ε L ´ and let p L ε “ p L ` ` ε p L ´ where p L ´ is given by (5.1)and p p L ` G qp η q “ ´ ÿ ξ Ă η G p ξ q ÿ x P ξ ` ÿ ζ `Ă ξ `z xζ ´Ă ξ ´ D ` p x, η z ξ Y ζ q ` ÿ ξ Ă η ż R d G p ξ ` Y x, ξ ´ q ÿ ζ Ă ξ B ` p x, η z ξ Y ζ q dx. is defined on D p p L ` q “ t G P L C | M ` ¨ G P L C u with M ` p η q “ ř x P η ` d ` p x, η ` z x, η ´ q . Lemma 5.3.
The following assertions hold(a) The operator p p L ` , D p p L ` qq is the generator of an analytic semigroup p T ` p t qq t ≥ of con-tractions on L C . Moreover, the assertions from Theorem 2.7 are satisfied.(b) D p p L ` q X D p b p L ´ q is a core for the operator p b p L ´ , D p b p L ´ qq .Proof. (a) This follows by the same arguments as given in section four (see also [FK16b]).(b) Let G P D p b p L ´ q and set G λ “ λλ ` M ` G , λ ą
0. Then G λ P D p p L ` q X D p b p L ´ q andobviously G λ ÝÑ G in L C as λ Ñ 8 . Moreover, we have }p b p L ´ q G λ ´ p b p L ´ q G } L C ≤ a ´ } M ´ ¨ p G λ ´ G q} L C “ a ´ ż Γ M ´ p η ´ q M ` p η q λ ` M ` p η q | G p η q| C | η ` |` C | η ´ |´ dλ p η q which tends by dominated convergence to zero as λ Ñ 8 .Finally we may show the following. 33 roposition 5.4.
For every ε ą the operator p p L ` ` ε p L ´ , D p p L ` q X D p p L ´ qq is the generatorof an analytic semigroup of contractions on L C . Moreover, the assertions from Theorem 2.8.(a)are satisfied.Proof. For ε ą η P Γ we get c ` p η q ` ε c ´ p η ´ q ≤ a ` M ` p η q ` a ´ ε M ´ p η ´ q ≤ max t a ´ , a ` u ˆ M ` p η q ` ε M ´ p η ´ q ˙ . The assertion can be now deduced by the same arguments as given in section four (see also[FK16b]).
Step 3. Construction of averaged dynamics
Define a linear operator by p LG p η q “ ´ ÿ ξ ` Ă η ` G p ξ ` q ÿ x P ξ ` ÿ ζ ` Ă ξ ` z x D p x, η ` z ξ ` Y ζ ` q` ÿ ξ ` Ă η ` ż R d G p ξ ` Y x q ÿ ζ ` Ă ξ ` B p x, η ` z ξ ` Y ζ ` q dx. on the domain D p p L q “ t G P L C ` | G ¨ M P L C ` u with M p η ` q “ ř x P η ` d p x, η ` z x q . Followingline to line the arguments in section four we readily deduce Theorem 2.8.(b) and, in particular,the next proposition. proposition 5.5. The operator p p L, D p p L qq is the generator of an analytic semigroup T p t q on L C ` . Moreover, T p t q is a contraction operator and B bs p Γ ` q is a core for the generator. Step 4. Stochastic averaging principle
Using the definition of correlation functions and previous steps, we immediately see that themain result, Theorem 2.8.(c), follows from the next proposition. proposition 5.6.
Let T ε p t q be the semigroup generated by p p L ` ` ε p L ´ , D p p L ` q X D p p L ´ qq and let T p t q be the semigroup generated by p p L, D p p L qq . Then for any T ą and G P L C ` lim ε Ñ sup t Pr ,T s } T ε p t qp G b ´ q ´ p T p t q G q b ´ } L C “ , (5.5) where p G b ´ qp η q “ G p η ` q | η ´ | and for any k P K C lim ε Ñ sup t Pr ,T s ˇˇˇˇˇˇˇ ż Γ ` G p η ` qp T ε p t q ˚ k qp η ` , Hq dλ p η ` q ´ ż Γ ` G p η ` q ` T p t q ˚ k p¨ , Hq ˘ p η ` q dλ p η ` q ˇˇˇˇˇˇˇ “ . (5.6) Proof.
Suppose that the following properties are satisfied34i) p p L ` , D p p L ` qq is the generator of a strongly continuous contraction semigroup.(ii) p b p L ´ , D p b p L ´ qq is the generator of a strongly continuous contraction semigroup b T ´ p t q on L C . Moreover b T ´ p t q is ergodic with projection operator b p P and D p p L ` q X D p b p L ´ q is a core for the generator.(iii) p p L, D p p L qq is the generator of a strongly continuous contraction semigroups T p t q on L C ` .(iv) The averaged operator p b p P q p L ` equipped with the domain Ran p b p P q X D p p L ` q isclosable and its closure p PL ` , D p PL ` qq satisfies p PL ` G qp η q “ | η ´ | p p LG qp η ` q , G b ´ P D p PL ` q where D p PL ` q “ t G b ´ | G P D p p L qu .Then we may apply [Kur73, Theorem 2.1] and from that readily deduce (5.5). By duality thisyields x G b ´ , T ε p t q ˚ k y “ x T ε p t qp G b ´ q , k y ÝÑ xp T p t q G q b ´ , k y uniformly on compacts in t ≥
0. Convergence (5.6) then follows from ż Γ ` T p t q G p η ` q k p η ` , Hq dλ p η ` q “ ż Γ ` G p η ` q ` T p t q ˚ k p¨ , Hq ˘ p η ` q dλ p η ` q . Properties (i) – (iii) have been checked in step 1 – step 3. It remains to prove (iv). First observethat Ran p b p P q “ L C ` b L C ´ and hence by definition of D p p L ` q Ran p b p P q X D p p L ` q “ t G “ | η ´ | G P L C ` b L C ´ | M ` p¨ , Hq ¨ G P L C ` u . For such G p η q “ G p η ` q | η ´ | P Ran p b p P q X D p p L ` q we obtain p p L ` G qp η q “ ´ ÿ ξ ` Ă η ` G p ξ q ÿ x P ξ ` ÿ ζ ` Ă ξ ` z x D ` p x, η ` z ξ ` Y ζ ` , η ´ q` ÿ ξ ` Ă η ` ż R d G p ξ ` Y x q ÿ ζ ` Ă ξ ` B ` p x, η ` z ξ ` Y ζ ` , η ´ q dx. and hence applying b p P yields by the definitions (2.8) and (2.9) pp b p P q p L ` G qp η q “ ´ | η ´ | ÿ ξ ` Ă η ` G p ξ ` q ÿ x P ξ ` ÿ ζ ` Ă ξ ` z x D p x, η ` z ξ ` Y ζ ` q` | η ´ | ÿ ξ ` Ă η ` ż R d G p ξ ` Y x q ÿ ζ ` Ă ξ ` B p x, η ` z ξ ` Y ζ ` q dx “ p p LG qp η ` q | η ´ | . t G b ´ P L C ` b L C ´ | G P D p p L qu is the generatorof the analytic semigroup U p t q of contractions on L C ` b L C ´ given by p U p t qp G b ´ qqp η q : “ T p t q G p η ` q | η ´ | . The generator has clearly D : “ t G b ´ P L C ` b L C ´ | G P B bs p Γ ` qu as a core. Since D Ă Ran p b p P q X D p p L ` q the assertion follows. Acknowledgements
Financial support through CRC701, project A5, at Bielefeld University is gratefully acknowl-edged. The authors would like to thank the anonymous referee for his remarks which lead to asignificant improvement of this work. 36 eferences [AR91] W. Arendt and A. Rhandi. Perturbation of positive semigroups.
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