Evolution of states and mesoscopic scaling for two-component ... - arXiv

arXiv:1608.06560v1 [math.PR] 22 Aug 2016

Evolution of states and mesoscopic scaling for two-component birth-and-death dynamics in continuum Martin Friesen˚ Oleksandr Kutoviy: August 24, 2016

Abstract: Two coupled spatial birth-and-death Markov evolutions on Rd are obtained as the unique weak solutions to the associated Fokker-Planck equations. Such solutions are constructed by its associated sequence of correlation functions satisfying the so-called Ruelle-bound. Using the general scheme of Vlasov scaling we are able to derive the system of non-linear, non-local mesoscopic equations describing the effective density of the particle system. The results are applied to several models of ecology and biology. AMS Subject Classification: 35Q84, 35Q83, 47D06, 60K35, 82C22 Keywords: interacting particle systems, Fokker-Planck equations, Vlasov scaling, mesoscopic scaling

1

Introduction

In the last years we observe an increasing interest to interacting particle systems in the continuum in regard to particular models of ecology, biology and social sciences, cf. [BP99, DL05, SEW05, BCF` 14, FFH` 15]. An important part of the general theory of interacting particle systems in the continuum is related to the construction of the associated dynamics and the study of their scalings. In particular, from the point of view of applications it is worthy to investigate the corresponding mesoscopic equations. Within this framework, each particle is described by its position x P Rd and the collection of all particles by the configuration γ. Assuming that each two particles cannot occupy the same position ˚ :

Department of Mathematics, Bielefeld University, Germany, [email protected] Department of Mathematics, Bielefeld University, Germany, [email protected]

1

and all particles are indistinguishable, we will choose the configuration space Γ to be the collection of all admissible particle configurations γ, i.e. Γ “ tγ Ă Rd | |γ X K| ă 8 for all compacts K Ă Rd u. Here and subsequently |A| denotes the number of elements in the set A Ă Rd . It is assumed that the microscopic dynamics consists of two elementary events. Namely, the death of a particle pγ ÞÝÑ γztxuq and the birth of a particle pγ ÞÝÑ γ Y txuq. A detailed analysis of such birth-and-death dynamics can be found in [FKK12], see also the references therein. However, many particular models from ecology and biology require that we have at least two different types of particles. The aim of this work is to extend the known results for two-component interacting particle systems. Since two particles of different types cannot occupy the same position, a proper state space for the dynamics is naturally given by the configuration space Γ2 :“ tpγ ` , γ ´ q P Γ ˆ Γ | γ ` X γ ´ “ Hu, cf. [FKO13]. The stochastic time-evolution is incorporated in the particular form of the associated Markov (pre-)generator L. The fact that particles can only die or create new particles leads to the special form of the corresponding generator L “ L` ` L´ , where both operators L˘ are given on polynomially bounded cylinder functions F on Γ2 by ÿ d´ px, γ ` , γ ´ ztxuqpF pγ ` , γ ´ ztxuq ´ F pγqq (1.1) pL´ F qpγq “ xPγ ´

`

ż

b´ px, γqpF pγ ` , γ ´ Y txuq ´ F pγqqdx

Rd

and ÿ

pL` F qpγq “

d` px, γ ` ztxu, γ ´ qpF pγ ` ztxu, γ ´ q ´ F pγqq

(1.2)

xPγ `

`

ż

b` px, γqpF pγ ` Y txu, γ ´ q ´ F pγqqdx.

Rd

For simplicity of notation, we write γ “ pγ ` , γ ´ q, γ ˘ zx and γ ˘ Yx instead of γ ˘ Ytxu and γ ˘ ztxu, respectively. The birth-and-death Markov process associated to the operator L, provided it exists, consists of two elementary events. The death intensities d˘ px, γq ≥ 0 determine the probability that the particle x P γ ˘ disappears from the configuration γ ˘ . The birth intensities b˘ px, γq ≥ 0 determine the probability for a new particle to appear at x P Rd . Multi-species birth-and-death dynamics in the continuum are especially important in ecological, biological and physical applications. For example, the famous 2

Widom-Rowlinson approach to the phase transition in the continuum is based on a twocomponent classical gas model. This approach was extended to more general Potts-type systems (see [GH96]). Recently, the dynamical versions of these models were studied in [FKKO15]. The corresponding Markov statistical dynamics was constructed with the help of Glauber dynamics on Γ. The dynamical phase transition was shown there in the mesoscopic limit of this dynamics. In contrast to that work we are able to construct the corresponding dynamics for all t ≥ 0. Different kinds of other models, additional comments and explanations for the modelling of tumour growth can be found in [FFH` 15] and the references therein. The Markov process associated with the (pre-)generator L may be obtained as the unique solution to certain stochastic differential equations, cf. [GK06]. Unfortunately the conditions given in the latter paper are too restrictive for some applications. A different, functional analytic, approach to the construction of the processes is related to the construction of solutions to the (backward) Kolmogorov equation on functions F : Γ2 ÝÑ R BFt “ LFt , Ft |t“0 “ F0 , t ≥ 0. Bt

(1.3)

However, we cannot expect to solve (1.3) in any space of continuous functions on Γ2 and hence obtain a process for any initial configuration γ P Γ2 , cf. [KS06]. This difficulty comes from the necessity to control the number of particles in any bounded domain in Rd . One therefore tries to construct an associated evolution of probability measures on Γ2 , that is the one-dimensional distributions of the Markov process. A probability measure µ on Γ2 describes in this setting the distribution of particles located in Rd . Functions F : Γ2 ÝÑ R are called observables and ż xF, µy :“ F pγqdµpγq Γ2

are considered as measurable quantities of the particle system. Above duality yields a weak formulation for the dual equation to (1.3), i.e. d xF, µty “ xLF, µt y, µt |t“0 “ µ0 , t ≥ 0. dt

(1.4)

It is known as the (forward) Kolmogorov equation. In the physical literature (1.4) is also called Fokker-Planck equation. Because of the Markovian property of the operator L we expect that solutions to (1.4) can be constructed in the class of probability measures on Γ2 and hence determine an evolution of states. Solutions to the Fokker-Planck equation determine the distributions of the associated Markov process, provided, of course, it exists. Hence pµt qt≥0 is referred to as the ’statistical description’ of the birth-and-death process. For many interesting, from the point of view of applications, interacting particle systems it seems impossible to solve (1.4) for any initial state µ0 . It is necessary to restrict 3

ourselves to a certain set of admissible initial states. Here we see a crucial difference comparing with Markov stochastic processes framework where the evolution can be constructed for any initial data. One possible class of admissible initial states is given by the collection of all sub-Poissionian states, cf. [KKM08]. Such states are described in pn,mq terms of their associated correlation functions pkµ q8 n,m“0 . For the convenience of the reader we repeat the definition of correlation functions and their properties in the next section, see also [KK02] and the references therein. The collection of correlation functions pn,mq pkµ q8 n,m“0 is called sub-Poissonian if it satisfies the Ruelle bound kµpn,mq px1 , . . . , xn ; y1 , . . . , ym q ≤ Aeαn eβm , n, m P N0 for some constants A ą 0 and α, β P R. A sub-Poissonian state µ is therefore a probability measure on Γ2 for which the associated correlation functions exist and are sub-Poissonian. The Cauchy problem (1.4) is then formally equivalent to a system of Banach space-valued differential equations pn,mq

Bkt Bt

pn,mq

“ pL∆ kt qpn,mq , kt

pn,mq

|t“0 “ k0

, n, m P N0 ,

(1.5)

where L∆ can be seen as an infinite operator-valued matrix. We provide, under some reasonable conditions, existence and uniqueness of weak solutions to (1.5) and consequently derive existence and uniqueness of weak solutions to (1.4). Note that a solution pn,mq 8 pkt qn,m“0 to (1.5), in general, does not need to be the correlation function of some state µt on Γ2 . For such property additional analysis is required and was only achieved for a few models, see e.g. [KKP08, KKM08, KK16]. This work is organized as follows. Notations and preliminary results are introduced in the second section. The third section is devoted to the construction of solutions to the pre-dual equation to (1.5). To this end we first prove that the pre-dual equation is well-posed and show that solutions can be obtained by the action of an analytic semigroup of contractions, see Theorem 3.1. The adjoint semigroup is related to existence and uniqueness of solutions to (1.5). It is shown that such solutions are the correlation functions corresponding to an evolution of states, see Theorem 3.7 and Proposition 3.12. The fourth section extends the Vlasov scaling, cf. [FKK12], to the case of two-component Markov evolutions, see Theorem 4.1 – 4.3. Examples are considered in the last section.

2

Preliminaries

Space of finite configuration. One-component case Denote by Γ0 the configuration space of all finite subsets of Rd , i.e. Γ0 “ tη Ă Rd | |η| ă 8u, 4

where |η| denotes the number of particles in the set η. This space has a natural decom8 Ů pnq pnq position into n-particle spaces, Γ0 “ Γ0 , where Γ0 “ tη Ă Rd | |η| “ nu, n ≥ 1 n“0

Č d qn the space of all sequences and in the case n “ 0 we set “ tHu. Denote by pR pnq Č d qn via the px1 , . . . , xn q P pRd qn with xi ‰ xj for i ‰ j. Γ0 can be identified with pR symmetrization map p0q Γ0

Č d qn ÝÑ Γpnq , px , . . . , x q ÞÝÑ tx , . . . , x u, symn : pR 1 n 1 n 0 pnq

pnq

which defines a topology on Γ0 . Namely, a set A Ă Γ0 is open if and only if sym´1 n pAq Ă Č d n pR q is open. On Γ0 we define the topology of disjoint unions, i.e. a set A Ă Γ0 is open pnq pnq iff A X Γ0 is open in Γ0 for all n P N. Then Γ0 is a locally compact Polish space. Let BpΓ0 q stand for the Borel-σ-algebra on Γ0 . Denote by dx the Lebesgue measure on Rd pnq and by dbn x the product measure on pRd qn . Let dpnq x be the image measure Γ0 w.r.t. symn . The Lebesgue-Poisson measure is defined by λ “ δH `

8 ÿ 1 pnq d x. n! n“1

Given a measurable function G : Γ0 ˆ Γ0 ÝÑ R, then ż ż ż ÿ Gpξ, ηzξqdλpηq “ Gpξ, ηqdλpξqdλpηq Γ0

ξĂη

(2.1)

Γ0 Γ0

holds, provided one side of the equality is finite for |G|. For a given measurable function f : Rd ÝÑ R the Lebesgue-Poisson exponential is defined by ź eλ pf ; ηq :“ f pxq xPη

and satisfies the combinatorial formula ÿ eλ pf ; ξq “ eλ p1 ` f ; ηq. ξĂη

For computations we will use the identity ż

Γ0

whenever f P L1 pRd q.

¨

eλ pf ; ηqdλpηq “ exp ˝

ż

Rd

5

˛

f pxqdx‚,

Space of finite configurations. Two-component case In this part we want to provide a brief overview for the two-component configuration space Γ20 , see [Fin13, FKO13] and the references therein. We suppose that two different particles cannot occupy the same location x P Rd and therefore define the two-component state space by Γ20 “ tpη ` , η ´ q P Γ0 ˆ Γ0 | η ` X η ´ “ Hu. Here and in the following we simply write η instead of pη ` , η ´ q P Γ20 if no confusion may arise. Set operations ξ Ă η, ξ Y η and ηzξ are defined component-wise, i.e. by ξ ˘ Ă η ˘ , etc. For η P Γ20 we let |η| :“ |η ` | ` |η ´ |. The space Γ20 has the natural decomposition Γ20 “

8 ğ

pn,mq

Γ0

,

n,m“0 pn,mq

where Γ0 “ tpη ` , η ´ q Ă Rd ˆ Rd | η ` X η ´ “ H, |η ` | “ n, |η ´ | “ mu. The topology pn,mq pnq on Γ0 and Γ20 is defined in the same way as for Γ0 and Γ0 . It is not difficult to see that this topology is the same as the subspace topology of the product topology on Γ0 ˆ Γ0 . In particular Γ20 is a Polish space. The Lebesgue-Poisson measure λ2 on Γ20 is defined as the restriction of λ b λ to Γ20 . Since no confusion may arise we use the same notation λ for the Lebesgue-Poisson measure λ2 on Γ20 and λ on Γ0 . We see that λ b λptpη ` , η ´ q P Γ0 ˆ Γ0 | η ` X η ´ ‰ Huq “ 0 holds. Hence integrals w.r.t. integrable functions G : Γ20 ÝÑ R can be also written as ż ż ż Gpηqdλpηq “ Gpη ` , η ´ qdλpη ` qdλpη ´ q. Γ20

Γ0 Γ0

Similarly to (2.1) the two-component Lebesgue-Poisson measure satisfies for any measurable function G : Γ20 ˆ Γ20 ÝÑ R ż ÿ ż ż Gpξ, ηzξqdλpηq “ Gpξ, ηqdλpξqdλpηq (2.2) Γ20

ξĂη

Γ20 Γ20

provided one side of the equality is finite for |G|. A set M Ă Γ20 is called bounded if there exist a compact Λ Ă Rd and N P N0 such that M Ă tpη ` , η ´ q P Γ20 | η ˘ Ă Λ, |η| ≤ Nu. A function G is said to have bounded support if it is supported on a bounded set. Denote by Bbs pΓ20 q the space of all bounded, measurable functions having bounded support. We say that H : Γ20 ÝÑ R is locally integrable ş if it is integrable for any bounded set. This is the same as regarding that the integral Gpηq|Hpηq|dλpηq is finite for all non-negative Γ20

functions G P

Bbs pΓ20 q.

6

Space of locally finite configurations For two-component systems the state space is defined as the direct product of two copies of Γ Γ2 :“ tpγ ` , γ ´ q P Γ ˆ Γ | γ ` X γ ´ “ Hu, cf. [FKO13]. For simplicity of notation we write γ :“ pγ ` , γ ´ q. Likewise we use for η P Γ20 the notation η Ă γ and γzη by which we mean that η ` Ă γ ` , η ´ Ă γ ´ and γ ` zη ` , γ ´ zη ´ . The restriction of the product topology of Γ ˆ Γ onto Γ2 yields a topology for which Γ2 is a Polish space. The Poisson measure πα,β is defined for α, β P R as the unique probability measure πα,β on Γ2 having the Laplace transform ¨ ˛ ¨ ˛ ż ř f pxq ř gpxq ż ż exPγ` exPγ´ dπα,β pγq “ exp ˝eα pef pxq ´ 1qdx‚exp ˝eβ pegpxq ´ 1qdx‚, Γ2

Rd

Rd

where f, g : Rd ÝÑ R are continuous with compact support. For G P Bbs pΓ20 q define the K-transform by ÿ Gpηq, (2.3) pKGqpγq “ ηŤγ

where η Ť γ means that the sum only runs over all finite subsets η of γ. Then KG is a polynomially bounded cylinder function, i.e. there exists a compact Λ Ă Rd and constants C ą 0, N P N such that pKGqpγ ` , γ ´ q “ pKGqpγ ` X Λ, γ ´ X Λq and |pKGqpγq| ≤ Cp1 ` |γ ` X Λ| ` |γ ´ X Λ|qN , γ P Γ2 holds. The K-transform K : Bbs pΓ20 q ÝÑ F PpΓ2 q :“ KpBbs pΓ20 qq is a positivity preserving isomorphism with inverse given by ÿ pK´1 F qpηq :“ p´1q|ηzξ| F pξq, η P Γ20 . ξĂη

Denote by K0 the restriction of K determined by evaluating KG only on Γ20 . Its inverse 2 is then denoted by K´1 0 . Given a probability measure µ on Γ , the correlation function kµ : Γ20 ÝÑ R` is defined by the relation ż ż KGpγqdµpγq “ Gpηqkµ pηqdλpηq, G P Bbs pΓ20 q, (2.4) Γ2

Γ20

provided it exists. In such a case the K-transform can be uniquely extended to a bounded linear operator K : L1 pΓ20 , kµ dλq ÝÑ L1 pΓ2 , dµq such that }KG}L1 pΓ2 ,dµq ≤ }G}L1 pΓ20 ,kµ dλq ` ´ and (2.3) holds for µ-a.a. γ P Γ2 . Let Lkµ :“ L1 pΓ20 , kµ dλq, for kµ pηq :“ eα|η | eβ|η | we 7

also write Lkµ ” Lα,β . A function G P Lα,β is called positive definite if KG ≥ 0. Denote by L` α,β the cone of all positive ş definite functions. The duality xG, ky :“ Gpηqkpηqdλpηq is used for the identification L˚α,β – Kα,β . Γ20

Here Kα,β stands for the Banach space of all equivalence classes of functions k : Γ20 ÝÑ R equipped with norm ` ´ }k}Kα,β “ ess sup |kpηq|e´α|η | e´β|η | . ηPΓ20

` A function k P Kα,β is said to be positive definite if xG, ky ≥ 0 holds for all G P Bbs pΓ20 q :“ ` ` 2 Bbs pΓ0 q X Lα,β . Let Kα,β be the space of all positive definite functions in Kα,β . The next statement shows a one-to-one correspondence between certain classes of probability ` measures on Γ2 and elements in Kα,β .

Theorem 2.1. The following assertions are satisfied. 1. Let µ be a probability measure on Γ2 having the correlation function kµ . Then kµ pHq “ 1 and kµ is positive definite. ` 2. Conversely, let k P Kα,β and assume that kpHq “ 1 holds. Then there exists a unique probability measure µ on Γ2 with k as its correlation function.

For given α, β P R let Pα,β be the space of all probability measures µ such that for ` each µ there exists an associated correlation function kµ P Kα,β . Let µ P Pα,β , then by Fubini’s theorem ż ż ż KGpγqdµpγq “ KGpγ ` , γ ´ qdµpγ ` , γ ´q Γ2

Γ Γ

holds for all G P Lα,β .

3

Construction of dynamics

Let L “ L´ ` L` be given by (1.1) and (1.2). We suppose that the intensities satisfy the condition given below. (A) There exists a measurable set Γ8 Ă Γ2 such that for all x P Rd Rd ˆ Γ28 Q px, γq ÞÝÑ d` px, γ ` zx, γ ´ q, d´ px, γ ` , γ ´ zxq, b` px, γq, b´ px, γq

(3.1)

are measurable, finite and for any compact Λ Ă Rd and bounded set M Ă Γ20 ż ż pd` px, ηq ` d´ px, ηq ` b` px, ηq ` b´ px, ηqqdλpηqdx ă 8 (3.2) Λ M

holds. Moreover, any measure µ P Pα,β is supported on Γ28 . 8

3.1

Quasi-observables

Introduce the cumulative death intensity by ÿ ÿ d` px, η ` zx, η ´ q d´ px, η ` , η ´ zxq ` Mpηq :“ xPη`

xPη´

and set cpL, α, β; ηq “ cpηq “ cpα, β; ηq by ÿ ż ` ´ ´ ` ´ eα|ξ | eβ|ξ | |K´1 cpL, α, β; ηq :“ 0 d px, ¨ Y η , ¨ Y η zxq|pξqdλpξq xPη´

`

Γ20

ÿ ż

xPη` ´β

`e

` ` ´ eα|ξ | eβ|ξ | |K´1 0 d px, ¨ Y η zx, ¨ Y η q|pξqdλpξq `

Γ20

ÿ ż

xPη´ ´α

`e

´

`

´

Γ20

ÿ ż

xPη`

´ ` ´ eα|ξ | eβ|ξ | |K´1 0 b px, ¨ Y η , ¨ Y η zxq|pξqdλpξq

` ` ´ eα|ξ | eβ|ξ | |K´1 0 b px, ¨ Y η zx, ¨ Y η q|pξqdλpξq. `

´

Γ20

´1 2 ˚ p Define on Bbs pΓ# 0 q the operator L :“ K0 LK0 . Denote by 1 the function given by 1, |η| “ 0 1˚ pηq :“ 0|η| “ . Using the methods proposed in [FKK12, FKO13] we 0, otherwise p It has the form L p “ A ` B. The latter operators are well-defined on Lα,β can compute L. 2 for any G P Bbs pΓ0 q and are given by pAGqpηq “ ´MpηqGpηq, where ÿ ÿ d` px, η ` zx, η ´ q ≥ 0 d´ px, η ` , η ´ zxq ` Mpηq “ xPη`

xPη´

and by pBGqpηq “ ´

ÿ

Gpξq

´

Gpξq

ξĹη

`

ÿż

ξĂη

`

ÿ

` ` ´ pK´1 0 d px, ¨ Y ξ zx, ¨ Y ξ qqpηzξq

xPξ ` ´ ` ´ Gpξ ` , ξ ´ Y xqpK´1 0 b px, ¨ Y ξ , ¨ Y ξ qqpηzξqdx

Rd

ÿż

ξĂη

´ ` ´ pK´1 0 d px, ¨ Y ξ , ¨ Y ξ zxqqpηzξq

xPξ ´

ξĹη

ÿ

ÿ

` ` ´ Gpξ ` Y x, ξ ´ qpK´1 0 b px, ¨ Y ξ , ¨ Y ξ qqpηzξqdx.

Rd

9

Below we construct a semigroup associated to the Cauchy problem for quasi-observables BGt p t , Gt |t“0 “ G0 P Lα,β . “ LG Bt

(3.3)

Note that solutions to (1.3) are formally related to (3.3) by Ft “ KGt . Theorem 3.1. Suppose that (A) is satisfied and assume that there exists α, β P R and a constant a “ apα, βq P p0, 2q such that cpα, β; ηq ≤ apα, βqMpηq, η P Γ20

(3.4)

holds. Then the following assertions are true: p DpLqq p of pL, p Bbs pΓ2 qq is the generator of an analytic semigroup (a) The closure pL, 0 pTpptqqt≥0 of contractions on Lα,β such that Tpptq1˚ “ 1˚ holds.

(b) Suppose that there exist α˚ ă α˚ and β˚ ă β ˚ with α P pα˚ , α˚q and β P pβ˚ , β ˚ q such that for all α1 P pα˚ , α˚q and β 1 P pβ˚ , β ˚q condition (3.4) is satisfied. Denote by Tpα1 ,β 1 ptq the associated semigroup on Lα1 ,β 1 . Then for any α ą α1 and β ą β 1 the space Lα,β is invariant for Tpα1 ,β 1 ptq and Tpptq “ Tpα1 ,β 1 ptq|Lα,β holds.

p :“ tG P Lα,β | M ¨ G P Lα,β u. Then, since M ≥ 0, the operator Proof. (a) Set DpLq p is the generator of an analytic (of angle π ), positive C0 -semigroup pe´tM qt≥0 pA, DpLqq 2 on Lα,β , see [EN00]. Let B 1 be defined for any G P Bbs pΓ20 q by ÿ ÿ ´ ` ´ |K´1 pB 1 Gqpη ` , η ´ q :“ Gpξq 0 d px, ¨ Y ξ , ¨ Y ξ zxq|pηzξq xPξ ´

ξĹη

`

ÿ

Gpξq

ξĹη

`

ÿż

ξĂη

`

` ` ´ |K´1 0 d px, ¨ Y ξ zx, ¨ Y ξ q|pηzξq

xPξ ` ´ ` ´ Gpξ ` , ξ ´ Y xq|K´1 0 b px, ¨ Y ξ , ¨ Y ξ q|pηzξqdx

Rd

ÿż

ξĂη

ÿ

` ` ´ Gpξ ` Y x, ξ ´ q|K´1 0 b px, ¨ Y ξ , ¨ Y ξ q|pηzξqdx.

Rd

p is resolvent positive, Then, since |BG| ≤ B 1 |G|, it is enough to show that pA ` B 1 , DpLqq p is the generator of a cf. [AR91, Theorem 1.1]. To this end we show that pA ` B 1 , DpLqq p DpLqq p is the closure of pL, p Bbs pΓ0 qq. positive semigroup. We will prove afterwards that pL,

10

p see So fix r P p0, 1q, cf. (3.4), such that apα, βq ă 1 ` r ă 2. For each 0 ≤ G P DpLq, (2.2), we obtain (3.4) ż ż ` ´ 1 α|η` | β|η´ | B Gpηqe e dλpηq “ pcpα, β; ηq ´ MpηqqGpηqeα|η | eβ|η | dλpηq Γ20

Γ20

≤r

ż

`

´

MpηqGpηqeα|η | eβ|η | dλpηq

Γ20

and hence

ş`

Γ20

˘ ` ´ A ` 1r B 1 Gpηqeα|η | eβ|η | dλpηq ≤ 0 holds. Therefore by [TV06, Theorem

p is the generator of a sub-stochastic semigroup pUpsqqs≥0 2.2] the operator pA ` B 1 , DpLqq p “ pL, p DpLqq p is the generator and by [AR91, Theorem 1.1, Theorem 1.2] also pA`B, DpLqq of an analytic semigroup pTppsqqs≥0 such that |TppsqG| ≤ Upsq|G| holds for all G P Lα,β . Since Upsq is a contraction operator, so is Tppsq. p DpLqq p on Lα,β . Lemma 3.2. Bbs pΓ20 q is a core for the generator pL,

p An Ă Γ2 an increasing sequence of bounded sets with Proof. Let G P DpLq, 0

Ť

n≥1

An “ Γ20

and let Gn pηq :“ 1An pηq1|G|≤n pηqGpηq. Then |Gn | ≤ |G|, Gn ÝÑ G a.e. and by dominated p n ÝÑ LG p as n Ñ 8 almost everywhere. Moreover, by convergence also LG p n | ≤ M|Gn | ` B 1 |Gn | ≤ pM ` B 1 q|G| P L1 pΓ20 , dλq |LG

p n ÝÑ LG p in Lα,β . Therefore Bbs pΓ2 q Ă DpLq p and dominated convergence we obtain LG 0 is dense in the graph norm.

(b) Let α1 ă α and β 1 ă β such that (3.4) also holds for pα1 , β 1 q. Denote by p p Dα1 ,β 1 pLqq pTpα1 ,β 1 psqqs≥0 the corresponding semigroup on Lα1 ,β 1 constructed in (a). Let pL, be the generator of Tpα1 ,β 1 ptq. By previous construction it is simply given by the action of p on the domain the operator L p :“ tG P Lα1 ,β 1 | M ¨ G P Lα1 ,β 1 u. Dα1 ,β 1 pLq

We have to show that Lα,β is invariant for Tpα1 ,β 1 psq and

TppsqG “ Tpα1 ,β 1 psqG, G P Lα,β , s ≥ 0.

(3.5)

To this end we define a linear isomorphism

1

`

1

´

S : Lα,β ÝÑ Lα1 ,β 1 , pSGqpηq “ epα´α q|η | epβ´β q|η | Gpηq 11

1

`

1

with inverse S ´1 given by pS ´1 Gqpηq “ e´pα´α q|η | e´pβ´β q|η| Gpηq. Define on Lα1 ,β 1 a new p1 :“ S LS p ´1 equipped with the domain operator by L p “ tG P Lα1 ,β 1 | MS ´1 G P Lα,β u. p1 q “ tG P Lα1 ,β 1 | S ´1 G P DpLqu Dα1 ,β 1 pL

p Let us show that p1 q “ Dα1 ,β 1 pLq. Since }MS ´1 G}Lα,β “ }MG}Lα1 ,β 1 we obtain Dα1 ,β 1 pL p1 qq is the generator of a C0 -semigroup on Lα1 ,β 1 . The definition of S and S ´1 p1 , Dα1 ,β 1 pL pL p1 “ A ` B1 where A is the same as for L p and B1 is given by implies L pB1 Gqpηq “ ÿ ÿ 1 ` ` 1 ´ ´ ´ ` ´ pK´1 ´ Gpξqepα´α q|η zξ | epβ´β q|η zξ | 0 d px, ¨ Y ξ , ¨ Y ξ zxqqpηzξq xPξ ´

ξĹη

´

ÿ

1

Gpξqepα´α q|η

ξĹη 1

` e´pβ´β q

ÿż

ξĂη 1

` e´pα´α q

1

epβ´β q|η

ÿ

´ zξ ´ |

` ` ´ pK´1 0 d px, ¨ Y ξ zx, ¨ Y ξ qqpηzξq

xPξ ` 1

` zξ ` |

1

` zξ ` |

Gpξ ` , ξ ´ Y xqepα´α q|η

´ pK´1 0 b px, ¨ Y ξqqpηzξqdx

1

´ zξ ´ |

1

´ zξ ´ |

epβ´β q|η

Rd

ÿż

ξĂη

` zξ ` |

Gpξ ` Y x, ξ ´ qepα´α q|η

epβ´β q|η

` pK´1 0 b px, ¨ Y ξqqpηzξqdx.

Rd

Define analogously to B 1 the positive operator B11 such that |B1 G| ≤ B11 |G|, then for any p1 q we obtain non-negative function G P Dα1 ,β 1 pL ż ż 1 ` 1 ´ 1 α1 |η` | β 1 |η´ | B1 Gpηqe e dλpηq “ pcpα, β; ηq ´ MpηqqGpηqeα |η | eβ |η | dλpηq. Γ20

Γ20

p1 qq is the The same arguments as for the construction of Tpptq show that pA ` B11 , Dα1 ,β 1 pL p p generator of a sub-stochastic semigroup and hence pL1 , Dα1 ,β 1 pL1 qq is the generator of a C0 -semigroup. Now [Paz83, Chapter 4, Theorem 5.5, Theorem 5.8] implies that Lα,β is invariant for Tpα1 ,β 1 ptq and the restriction to Lα,β is a C0 -semigroup given by Trα,β ptq :“ p in Lα,β , that p Dα1 ,β 1 pLqq Tpα1 ,β 1 ptq|Lα,β . The generator of Trα,β ptq is given by the part of pL, is by p X Lα,β | LG p P Lα,β u p L :“ tG P Dα1 ,β 1 pLq Dα1 ,β 1 pLq| α,β p P Lα,β u. “ tG P Lα,β | M ¨ G P Lα1 ,β 1 , LG

p L q is an p L and hence pL, p Dα1 ,β 1 pLq| p Ă Dα1 ,β 1 pLq| Condition (3.4) therefore implies DpLq α,β α,β p p p p p r p the extension of pL, DpLqq. Denote by Rpλ; Lq the resolvent for pL, DpLqq and by Rpλ; Lq 12

p L q. For sufficiently large λ ą 0 it follows that Rpλ, LqG p p Dα1 ,β 1 pLq| P resolvent for pL, α,β p p DpLq Ă Dα1 ,β 1 pLq|Lα,β for any G P Lα,β and thus r LqG p ´ Rpλ; LqG p “ Rpλ; r Lqppλ p p ´ pλ ´ LqqRpλ; p p “ 0, Rpλ; ´ Lq LqG

p the action of the generators is given by the where we have used that for elements in DpLq p “ A ` B and hence coincide. formulas for L

For one-component models, i.e. b´ “ 0 “ d´ , a similar construction was already done in [FKK12]. The main assumption was that each term in cpα, β; ηq is bounded by 23 Mpηq and it was not clear whether Tpptq is a contraction operator for t ≥ 0. The next example shows that the constant 2 in (3.4) is optimal.

Theorem 3.3. The constant 2 in condition (3.4) is optimal in the sense that it cannot be increased. Proof. It suffices to find a model with apα, βq ą 2 and show that the Cauchy problem (3.3) does not admit a solution in Lα,β . Take d´ “ 1, b´ “ z ą 0 constant and b` “ d` “ 0, then condition (3.4) can be restated to z ă eβ and α P R is arbitrary. The evolution equation (3.3) is in this case exactly solvable and hence has for every initial condition G0 the solution pGt qt≥0 given by ż ` ˘ ´t|η´ | Gt pηq “ e Gpη ` , η ´ Y ξ ´ qeλ zp1 ´ e´t q; ξ ´ dλpξ ´ q, η P Γ20 , Γ0

see [Fin11] for the one-component case. If condition (3.4) is satisfied, then Gt P Lα,β . Suppose that apα, βq ą 2, i.e. z ą eβ and let t0 ą 0 such that p1 ´ e´t qz ą eβ for all t ≥ t0 and hence zp1 ´ e´t qpe´β ` e´t q ≥ zp1 ´ e´t qe´β ą 1. Take 0 ≤ G P Lα,β such that G R Lα,β 1 for any β 1 ą β. The unique solution Gt is then positive and satisfies ż ż ´ ` ´ }Gt }Lα,β “ e´t|η | Gpη ` , η ´ Y ξ ´ qeλ pzp1 ´ e´t q; ξ ´ qeα|η | eβ|η | dλpξ ´ qdλpηq Γ20 Γ0

“

ż

Γ20

` ˘|ξ´ | α|η` | β|ξ´ | Gpη ` , ξ ´ q pe´β ` e´t qzp1 ´ e´t q e e dλpη ` , ξ ´ q “ 8.

Remark 3.4. The Trotter-Kato approximation theorem can be used to show that Tpptq p and hence on the birth-and-death intensities. depends continuously on the operator L 13

3.2

Correlation functions

Suppose that condition (A) and (3.4) are fulfilled. Denote by Tpptq˚ the adjoint semigroup p˚ , DpL p˚ qq its generator. This is, by definition, the adjoint operator to on Kα,β and by pL p DpLqq, p i.e. xLG, p ky “ xG, L p˚ ky for G P DpLq p and k P DpL p˚ q. pL,

Remark 3.5. Let α1 ă α and β 1 ă β be such that condition (3.4) holds for α1 , β 1 and α, β. Let pTpα1 ,β 1 psqqs≥0 be the analytic semigroup constructed in Theorem 3.1. Then by (3.5) for any G P Lα,β Ă Lα1 ,β 1 and k P Kα1 ,β 1 Ă Kα,β we obtain xG, Tpα1 ,β 1 ptq˚ ky “ xTpα1 ,β 1 ptqG, ky “ xTpptqG, ky “ xG, Tpptq˚ ky

and hence Tpptq˚ k “ Tpα1 ,β 1 ptq˚ k holds.

We consider the linear operator L∆ ÿ ż ´ ` ´ ∆ kpη Y ξqpK´1 pL kqpηq “ ´ 0 d px, ¨ Y η , ¨ Y η zxqqpξqdλpξq xPη´

´

ÿ ż

xPη`

`

´ ` ´ kpη ` Y ξ ` , η ´ Y ξ ´ zxqpK´1 0 b px, ¨ Y η , ¨ Y η zxqqpξqdλpξq

Γ20

ÿ ż

xPη`

` ` ´ kpη Y ξqpK´1 0 d px, ¨ Y η zx, ¨ Y η qqpξqdλpξq

Γ20

ÿ ż

xPη´

`

Γ20

` ` ´ kpη ` Y ξ ` zx, η ´ Y ξ ´ qpK´1 0 b px, ¨ Y η zx, ¨ Y η qqpξqdλpξq.

Γ20

on the maximal domain DpL∆ q “ tk P Kα,β | L∆ k P Kα,β u. Introduce the following additional conditions. (B’) There exist constants A ą 0, N P N and τ ≥ 0 such that d` px, ηq ` d´ px, ηq ≤ Ap1 ` |η|qN eτ |η| , x P Rd , η P Γ20 holds. (C) There exist α1 , β 1 P R with α1 ` τ ă α, β 1 ` τ ă β and apα1 , β 1 q ą 0 such that cpα1 , β 1; ηq ≤ apα1 , β 1qMpηq, η P Γ20 holds. p˚ , DpL p˚ qq “ pL∆ , DpL∆ qq. Lemma 3.6. Suppose that (3.4) and (A) are fulfilled, then pL If in addition conditions (B’) and (C) hold, then L∆ considered as an operator Kα1 ,β 1 ÝÑ Kα,β is bounded. In particular Kα1 ,β 1 Ă DpL∆ q holds. 14

p and k P DpL p˚ q Proof. It is not difficult to see that for any G P DpLq ż ż ż ˚ p p GpηqpL kqpηqdλpηq “ pLGqpηqkpηqdλpηq “ GpηqpL∆ kqpηqdλpηq, Γ20

Γ20

Γ20

p˚ k P Kα,β and hence DpL p˚ q Ă DpL∆ q. Conversely let k P DpL∆ q, see (2.2). Thus L∆ k “ L p above equality implies k P DpL p˚ q. For the second assertion observe then for any G P DpLq that for k P Kα1 ,β 1 1

`

1

´

`

´

|L∆ kpηq| ≤ }k}Kα1 ,β 1 apα1 , β 1qA|η|N `1 e´pα´α ´τ q|η | e´pβ´β ´τ q|η | eα|η | eβ|η | ,

hence the assertion follows by p|η ` | ` |η ´ |qN `1 ≤ 2N p|η ` |N `1 ` |η ´ |N `1 q and 1

`

1

|η|N `1 e´pα´α ´τ q|η | e´pβ´β ´τ q|η

´|

≤

2N pN ` 1qN `1 e´pN `1q 2N pN ` 1qN `1 e´pN `1q ` . pα ´ α1 ´ τ qN `1 pβ ´ β 1 ´ τ qN `1

Since Lα,β is not reflexive, Tpptq˚ does not need to be strongly continuous. However, it is continuous w.r.t. the topology σppLα,β q˚ , Kα,β q “ σpKα,β , Lα,β q. Here σpKα,β , Lα,β q is the smallest topology such that for each k P Kα,β , Lα,β Q G ÞÝÑ xG, ky is continuous. d It is well-known that Tpptq˚ is strongly continuous on the proper subspace Kα,β “ DpL∆ q pd k “ L∆ k, and its restriction Tpptqd :“ Tpptq˚ |Kd is a C0 -semigroup with generator L α,β

pd q “ tk P DpL∆ q | L∆ k P Kd u. DpL α,β

Hence we obtain existence and uniqueness of strong solutions to (1.5) on the Banach d space Kα,β . Unfortunately this space depends on the generator and does not provide uniqueness for the weak solutions. Another possibility is to change the topology on Kα,β . Let CpKα,β , Lα,β q “: C be the topology of uniform convergence on compact subsets of Lα,β . A basis of neighbourhoods around 0 is given by sets of the form tk P Kα,β | sup |xG, ky| ă εu, GPK

with ε ą 0 and a compact K Ă Lα,β , see [WZ02, WZ06, Lem10] and the references therein. The semigroup pTpptq˚ qt≥0 becomes continuous w.r.t. C and its generator w.r.t. C is exactly the adjoint operator pL∆ , DpL∆ qq, cf. [WZ06, Theorem 1.4]. The next theorem provides existence, uniqueness and regularity of solutions to the Cauchy problem d p kt y, kt |t“0 “ k0 , G P DpLq. p xG, kt y “ xLG, dt 15

(3.6)

Theorem 3.7. Suppose that (3.4) and (A) are satisfied. Then for any k0 P Kα,β the equation (3.6) has a unique given by kt “ Tpptq˚ k0 . This means that kt is continuous w.r.t. the topology C and satisfies żt

p ks yds, G P DpLq. p xG, kty “ xG, k0 y ` xLG,

(3.7)

0

Assume that (B’) and (C) are fulfilled. Then the following assertions are true: 1. If k0 P Kα1 ,β 1 , then kt is continuous w.r.t. to the norm in Kα,β . 2. If k0 P Kα1 ,β 1 and α1 ` 2τ ă α, β 1 ` 2τ ă β, then kt is also continuously differentiable w.r.t. to the norm in Kα,β and the unique classical solution to (1.5). Proof. Existence and uniqueness for the Cauchy problem (3.7) follows from [WZ06, Thep kt y orem 2.1] and Theorem 3.1. Since Tpptq˚ is continuous w.r.t. σpKα,β , Lα,β q, t ÞÝÑ xLG, is continuous and hence (3.7) implies (3.6). d 1. If k0 P Kα1 ,β 1 , then by Lemma 3.6 L∆ k0 P Kα,β and hence k0 P DpL∆ q Ă Kα,β which implies the assertion. 2. Suppose that α1 ` 2τ ă α, β 1 ` 2τ ă β and let α2 P pα1 , αq, β 2 P pβ 1 , βq be such that α1 ` τ ă α2 , α2 ` τ ă α and β 1 ` τ ă β 2 , β 2 ` τ ă β. By Lemma 3.6 the operator L∆ is bounded as Kα1 ,β 1 Ñ Kα2 ,β 2 and Kα2 ,β 2 Ñ Kα,β . Therefore k0 P DpL∆ q and pd q implies that kt is continuously differentiable L∆ k0 P Kα2 ,β 2 Ă DpL∆ q. Thus k0 P DpL w.r.t. the norm in Kα,β and it is a classical solution to (1.5).

p one may replace DpLq p in (3.7) by Bbs pΓ2 q. Remark 3.8. Since Bbs pΓ20 q is a core for L, 0

3.3

Positive definiteness

Suppose that (A) and (3.4) are satisfied. We start with the definition of solutions to the Fokker-Planck equation (1.4). Definition 3.9. A family of Borel probability measures pµt qt≥0 Ă Pα,β is said to be a weak solution to (1.4) if for any F P F PpΓ2 q, t ÞÝÑ xLF, µt y is locally integrable and satisfies żt

xF, µty “ xF, µ0 y ` xLF, µs yds, t ≥ 0.

(3.8)

0

Uniqueness is stated in the next theorem. Theorem 3.10. (Uniqueness) Suppose that (A) and (3.4) are fulfilled. Then equation (1.4) has at most one solution pµt qt≥0 Ă Pα,β such that its correlation functions pkt qt≥0 satisfy sup }kt }Kα,β ă 8, @T ą 0. tPr0,T s

16

Proof. Let pµt qt≥0 Ă Pα,β be a solution to (1.4), and denote by pkt qt≥0 Ă Kα,β the asp such that sociated correlation functions. Let F P F PpΓ2 q and G P Bbs pΓ20 q Ă DpLq ` ´ p P Lα,β Ă Lkt . Since F “ KG. Then by kt pηq ≤ }kt }Kα,β eα|η | eβ|η | it follows that G, LG p belong to K : Lkt ÝÑ L1 pΓ2 , dµt q is continuous it follows that F “ KG, LKG “ KLG 1 2 L pΓ , dµt q for any t ≥ 0. Moreover, p µt y “ xLG, p kt y xLF, µt y “ xKLG,

p kt y is locally integrable and hence shows that t ÞÝÑ xLG, żt

p ks yds, t ≥ 0, G P Bbs pΓ2 q xG, kt y “ xG, k0y ` xLG, 0 0

holds. Thus kt is continuous w.r.t. σpKα,β , Lα,β q and since kt is norm-bounded on r0, T s [WZ06, Lemma 1.10] implies that kt is also continuous w.r.t. the topology C. It remains p To this end let G P DpLq, p then there to show that pkt qt≥0 solves (3.7) for any G P DpLq. 2 p p exists Gn P Bbs pΓ0 q such that Gn ÝÑ G and LGn ÝÑ LG in Lα,β . Passing in żt

p n , ks yds, n P N xGn , kt y “ xGn , k0y ` xLG 0

to the limit n Ñ 8 shows (3.7). As a consequence pkt qt≥0 is a weak solution to (3.7) and hence given by kt “ Tpptq˚ k0 .

Remark 3.11. Let k0 P Kα,β be positive definite and suppose that kt :“ Tpptq˚ k0 P Kα,β is positive definite. Then pkt qt≥0 is a weak solution to (3.7) and for each t ≥ 0 there exists p kt y “ a unique µt P Pα,β having correlation function kt . By xG, kt y “ xF, µty and xLG, xLF, µt y it follows that pµt qt≥0 is a weak solution to (1.4).

Above considerations show that for existence of weak solutions to (1.4), it suffices to show that Tpptq˚ preserves the cone of positive definite functions. To this and we approximate kt “ Tpptq˚ k0 by an auxiliary evolution Tpδ ptq˚ k0 and prove that Tpδ ptq˚ k0 is positive definite. Such approximation scheme was used for particular models in [FKKZ12, KK16]. Let pRδ qδą0 be a sequence of continuous integrable functions with 0 ă Rδ ≤ 1 and Rδ pxq 1 1 as δ Ñ 0 for all x P Rd . In the following we simply say that pRδ qδą0 is ` a localization sequence. Define new birth intensities by b` δ px, ηq :“ Rδ pxqb px, ηq and ´ d 2 2 b´ δ px, ηq :“ Rδ pxqb px, ηq for all x P R and η P Γ0 . Then for any η P Γ0 and δ ą 0 ż ` ` ˘ bδ px, ηq ` b´ (3.9) δ px, ηq dx ă 8 Rd

17

´ 2 holds. Denote by Lδ the operator L with b` , b´ replaced by b` δ , bδ . Then for each η P Γ0 2 there exists an (minimal) birth-and-death process pηt qt≥0 on Γ0 associated to Lδ and starting from η. The following are the main assumptions for the existence of weak solutions to the Fokker-Planck equation.

(B) There exist constants A ą 0, τ ≥ 0 and N P N such that for all x P Rd and η P Γ20 d` px, ηq ` d´ px, ηq ` b` px, ηq ` b´ px, ηq ≤ Ap1 ` |η|qN eτ |η|

(3.10)

holds. (D) There exists a localization sequence pRδ qδą0 such that the (minimal) birth-and-death process associated to Lδ is conservative, i.e. it has no explosion starting from any initial point η P Γ20 . The next statement provides existence and uniqueness of solutions to (1.4). Proposition 3.12. (Existence) Suppose that (A) – (D) and (3.4) are fulfilled. Then ` Tpptq˚ Kα`1 ,β 1 Ă Kα,β . In particular for any µ0 P Pα1 ,β 1 there exists exactly one weak solution pµt qt≥0 Ă Pα,β to (1.4). The correlation functions are given by kµt “ Tpptq˚ kµ0 . If ` ` conditions (B) and (C) hold for all τ ą 0, then Tpptq˚ Kα,β Ă Kα,β holds. Existence of an associated Markov function is stated in the next corollary, cf. [KKM08].

Corollary 3.13. Suppose that (A) – (D) hold for any τ ą 0 and assume that (3.4) holds. Then for any µ P Pα,β there exists a Markov function pXtµ qt≥0 on the configuration space Γ2 with the initial distribution µ associated with the generator L. The rest ş ´ is devoted to the proof of Proposition 3.12. Let Dδ pηq “ ş ` of this section Mpηq ` bδ px, ηqdx ` bδ px, ηqdx and define Rd

Rd

Qδ Rpηq “

ż

Rd

`

´

`

´

d px, ηqRpη , η Y xqdx ` ÿ

ż

d` px, ηqRpη ` Y x, η ´ qdx

Rd ` ´ ` ´ b´ δ px, η , η zxqRpη , η zxq `

ÿ

` ´ ` ´ b` δ px, η zx, η qRpη zx, η q.

xPη`

xPη´

The linear operator Iδ “ ´Dδ ` Qδ is considered on the domain DpIδ q “ tR P L1 pΓ20 , dλq | Dδ R P L1 pΓ20 , dλqu. ş ş Since Qδ preserves positivity and satisfies Qδ Rpηqdλpηq “ Dδ pηqRpηqdλpηq for any Γ20

Γ20

0 ≤ R P DpIδ q, it follows that pIδ , DpIδ qq has an extension which is the generator of a 18

minimal sub-stochastic semigroup pSδ ptqqt≥0 on L1 pΓ20 , dλq, cf. [TV06]. Condition (D) implies that pIδ , DpIδ qq is closable and the closure is the generator of the C0 -semigroup pSδ ptqqt≥0 , cf. [FK16]. Therefore, for any R0 P DpIδ q there exists exactly one solution to BRtδ “ Iδ Rtδ , Rtδ |t“0 “ R0 Bt

(3.11)

and this solution is given by Sδ ptqR0 . For technical reasons we will need the adjoint semigroup on L8 pΓ20 , dλq. Let pJδ , DpJδ qq be the adjoint operator to pIδ , DpIδ qq on L8 pΓ20 , dλq. The next lemma follows by standard arguments and (2.2). Lemma 3.14. For any F P DpJδ q it holds that Jδ F “ Lδ F . The next statement follows by Rδ pxq ≤ 1, a repetition of the arguments used for the construction of Tpptq and Trotter-Kato approximation.

Lemma 3.15. For any δ ą 0 Theorem 3.1 and 3.7 hold with L replaced by Lδ . Let Tpδ ptq and Tpδ ptq˚ be the semigroups on Lα,β and Kα,β , respectively. Then for any G P Lα,β Tpδ ptqG ÝÑ TpptqG, δ Ñ 0

is satisfied.

Let Bα,β be the Banach space of all equivalence classes of functions G with norm ż ` ´ ~G~Bα,β “ |Gpηq|eλ pRδ ; η ` qeλ pRδ ; η ´ qeα|η | eβ|η | dλpηq. Γ20

˚ Its dual Banach space Bα,β can be identified with the Banach space of all equivalence classes of functions k with norm

|kpηq|

˚ “ ess sup ~k~Bα,β

` ´ α|η` | eβ|η´ | ηPΓ20 eλ pRδ ; η qeλ pRδ ; η qe

.

The same arguments as for the proof of Theorem 3.1 and 3.7 show that we can replace ˚ Lα,β , Kα,β also by Bα,β and Bα,β . Denote by Uδ ptq and Uδ ptq˚ the corresponding semigroups ˚ pδ , D B pLqq p be the generator of Uδ ptq. The proofs of on Bα,β and Bα,β , respectively. Let pL Theorem 3.1 and 3.7 show that Thus the Cauchy problem

p “ tG P Bα,β | M ¨ G P Bα,β u. D B pLq

d pδ G, uδt y, uδt |t“0 “ u0 , @G P D B pLq p xG, uδt y “ xL dt

˚ ˚ has for every u0 P Bα,β a unique weak solution in Bα,β given by Uδ ptq˚ u0 .

19

(3.12)

˚ Lemma 3.16. Let k0 P Bα,β , then Tpδ ptq˚ k0 “ Uδ ptq˚ k0 holds.

˚ Proof. First observe that Bα,β Ă Kα,β continuously and hence k0 P Kα,β . In particular uδt :“ Uδ ptq˚ k0 and ktδ :“ Tpδ ptq˚ k0 are well-defined. Moreover, since also Lα,β Ă Bα,β p Ă D B pLq, p i.e. pL pδ , D B pLqq p is an extension of pL pδ , DpLqq. p continuously we obtain DpLq δ δ δ Therefore put qt≥0 is also a weak solution to (3.7) and thus by uniqueness ut “ kt , t ≥ 0.

˚ Lemma 3.17. Let k0 P Bα˚1 ,β 1 be positive definite. Denote by uδt P Bα,β the unique weak δ solution to (3.12), then ut is positive definite for any t ≥ 0. ş ˚ Proof. Define for any u P Bα,β a linear operator Hupηq :“ p´1q|ξ|upη Y ξqdλpξq. Then Γ20

Hu is well-defined and satisfies for any C` , C´ ą 0 ż |η` | |η´ | |Hupηq|C` C´ dλpηq Γ20

˚ ≤ ~u~Bα,β

ż

`

´

`

´

p1 ` C` q|η | p1 ` C´ q|η | eα|η | eβ|η | eλ pRδ ; η ` qeλ pRδ ; η ´ qdλpηq,

Γ20 ˚ i.e. H : Bα,β ÝÑ LlogpC` q,logpC´ q is continuous. Let G P Bα,β be arbitrary, then for any ˚ u P Bα,β we get by Fubini’s theorem and (2.2) that

xK0 G, Huy “ xG, uy

(3.13)

holds. We can apply Fubini’s theorem and (2.2) since ż ż ż |Gpξq||upη Y ξ Y ζq|dλpζqdλpξqdλpηq Γ20 Γ20 Γ20

2eα xRδ y 2eβ xRδ y

˚ e ≤ }u}Bα,β

e

ż

`

´

|Gpξq|eα|ξ | eβ|ξ | eλ pRδ ; ξ ` qeλ pRδ ; ξ ´ qdλpξq

Γ20

is satisfied, where xRδ y :“

ş

Rd

pδ G “ Lδ K0 G and K0 L

p we obtain by (3.13) Rδ pxqdx. For the same u and G P D B pLq

pδ G, uy “ xK0 L pδ G, Huy “ xLδ K0 G, Huy. xL

˚ Now let Uδ ptq˚ k0 “ uδt P Bα,β , then

żt

pδ G, uδ yds, G P D B pLq. p xG, uδt y “ xG, u0y ` xL s 0

20

(3.14)

p Hence by (3.13) and (3.14) it Observe that condition (B) implies Klogp2q,logp2q Ă D B pLq. δ δ 1 2 follows for Rt :“ Hut P L pΓ0 , dλq, t ≥ 0 that xK

żt

“ xK0 G, R0 y ` xLδ K0 G, Rsδ yds, G P Klogp2q,logp2q

δ 0 G, Rt y

0

|η| holds. For any F P DpJδ q Ă L pΓ0 , dλq we get |K´1 and hence 0 F pηq| ≤ }F }L8 2 DpJδ q Ă K0 Klogp2q,logp2q . Thus we can find G P Klogp2q,logp2q such that K0 G “ F P DpJδ q. Lemma 3.14 therefore implies 8

xF, Rtδ y

żt

“ xF, R0 y ` xJδ F, Rsδ yds, F P DpJδ q. 0

Bα˚1 ,β 1

Since k0 P we get by Theorem 3.7.1 that uδt is continuous in t ≥ 0 w.r.t. the norm in ˚ ˚ Bα,β . Because H : Bα,β ÝÑ L1 pΓ20 , dλq is continuous, Rtδ “ Huδt is continuous w.r.t. t ≥ 0 on L1 pΓ20 , dλq. Hence pRtδ qt≥0 is a weak solution to (3.11). The main result from [Bal77] ` therefore implies Rtδ “ Sδ ptqR0 ≥ 0. Finally, for any G P Bbs pΓ20 q we get xG, uδt y “ xK0 G, Rtδ y ≥ 0, t ≥ 0. We are now prepared to complete the proof of positive definiteness. Proof. (Proposition 3.12) Let µ0 P Pα1 ,β 1 with correlation function k0 P Kα1 ,β 1 . Define k0,δ pηq :“ k0 pηqeλ pRδ ; η ` qeλ pRδ ; η ´ q, δ ą 0, η P Γ20 , then k0,δ P Bα˚1 ,β 1 and it is positive definite, cf. [Fin11, Fin13]. By Lemma 3.16 we get ˚ Tpδ ptq˚ k0,δ “ Uδ ptq˚ k0,δ P Bα,β and by Lemma 3.17 the latter expression is positive definite. ` 2 Let G P Bbs pΓ0 q. Then it suffices to show that xG, Tpδ ptq˚ k0,δ y ÝÑ xG, Tpptq˚ k0 y, δ Ñ 0.

To this end observe that

xG, Tpδ ptq˚ k0,δ y “ xTpδ ptq˚ G ´ TpptqG, k0,δ y ` xTpptqG, k0,δ y.

The first term can be estimated by

}Tpδ ptqG ´ TpptqG}Lα,β }k0 }Kα,β

and hence tends by Lemma 3.15 to zero. The second term tends by dominated convergence to xTpptqG, k0 y “ xG, Tpptq˚ k0 y, which implies that Tpptq˚ k0 is positive definite. If conditions (B) and (C) hold for all τ ą 0, then k0,δ pηq :“ e´δ|η| k0 pηq belongs to Kα´δ,β´δ for any δ ą 0. Consequently, above considerations imply that Tpptq˚ k0,δ P Kα,β is positive definite. Taking the limit δ Ñ 0 yields the assertion. 21

Remark 3.18. Suppose instead of (B) the following to be satisfied: There exist A ą 0, N P N and νb ≥ 0, ν1 , ν2 ≥ 0 such that for all x P Rd and η P Γ20 : b` px, ηq ` b´ px, ηq ≤ Ap1 ` |η|qN eνb |η| d` px, ηq ≤ Ap1 ` |η|qN eν1 |η| d´ px, ηq ≤ Ap1 ` |η|qN eν2 |η| . Then for any positive definite k0 P Kα1 ,β 1 the evolution Tpptq˚ k0 is positive definite, provided (C) holds for α1 ` ν1 ă α, β 1 ` ν2 ă β.

4

Vlasov scaling

General description of Vlasov scaling Let us briefly explain the Vlasov scaling in the two-component case. A motivation and additional explanations can be found in [FKK10]. Let L be a Markov (pre-)generator on ∆ Γ2 , the aim is to find a scaling Ln such that the following scheme holds. Let Tn∆ ptq “ etLn be the (heuristic) representation of the scaled evolution of correlation functions, see (1.5). The particular choice of L Ñ Ln should preserve the order of singularity, that is the limit n´|η| Tn∆ ptqn|η| k ÝÑ TV∆ ptqk, n Ñ 0

(4.1)

should exist and the evolution TV∆ ptq should preserve Lebesgue-Poisson exponentials, i.e. ` ´ ` ´ ` ∆ ´ ` if r0 pηq “ eλ pρ´ 0 , η qeλ pρ0 ; η q, then TV ptqr0 pηq “ eλ pρt , η qeλ pρt ; η q. In such a case ` ρ´ t , ρt satisfy the system of non-linear integro-differential equations Bρ´ t ` “ v´ pρ´ t , ρt q Bt

Bρ` t ` “ v` pρ´ t , ρt q. Bt

(4.2)

The functionals v´ , v` can be computed explicitly for a large class of models and (4.2) is ` the system of kinetic equations for the densities ρ´ t , ρt for the particle system. Instead of ´|η| ∆ |η| investigating the limit (4.1), we define renormalized operators L∆ Ln n and n,ren :“ n ∆ study the behaviour of the semigroups Tn,ren ptq when n Ñ 8. In such a case one can compute a limiting operator ∆ L∆ n,ren ÝÑ LV

(4.3)

∆ and show that L∆ V is associated to a semigroup TV ptq. The limit (4.1) is then obtained by showing the convergence ∆ Tn,ren ptq ÝÑ TV∆ ptq

in a proper sense. 22

(4.4)

Scaling of two-component model ´ ` ´ Consider the scaled intensities d` n , dn , bn , bn ≥ 0 and suppose they satisfy condition (A) ` for any n P N. Let Ln “ L´ n ` Ln where ÿ ` ´ ` ´ ` ´ d´ L´ F pγq “ n px, γ , γ zxqpF pγ , γ zxq ´ F pγ , γ qq n xPγ ´

`n

ż

` ´ ` ´ ` ´ b´ n px, γ , γ qpF pγ , γ Y xq ´ F pγ , γ qqdx

Rd

and ÿ

L` n ptqF pγq “

` ´ ` ´ ` ´ d` n px, γ zx, γ qpF pγ zx, γ q ´ F pγ , γ qq

xPγ `

`n

ż

` ´ ` ´ ` ´ b` n px, γ , γ qpF pγ Y x, γ q ´ F pγ , γ qqdx.

Rd

Introduce cn pα, β; ηq :“ `

ÿ ż

xPη´

ÿ ż

xPη`

`e

`

´

Γ20 ` ` ´ |ξ| α|ξ | β|ξ | e dλpξq |K´1 0 dn px, ¨ Y η zx, ¨ Y η q|pξqn e `

Γ20

´β

´ ` ´ |ξ| α|ξ | β|ξ | e dλpξq |K´1 0 dn px, ¨ Y η , ¨ Y η zxq|pξqn e

ÿ ż

´

´ ` ´ |ξ| α|ξ | β|ξ | |K´1 e dλpξq 0 bn px, ¨ Y η , ¨ Y η zxq|pξqn e `

´

xPη´ Γ

` e´α

0 ÿ ż

` ` ´ |ξ| α|ξ | β|ξ | |K´1 e dλpξq 0 bn px, ¨ Y η zx, ¨ Y η q|pξqn e `

´

xPη` Γ

0

and Mn pηq :“

ř

xPη´

` ´ d´ n px, η , η zxq `

conditions to be satisfied:

ř

xPη`

` ´ d` n px, η zx, η q. We will suppose the following

(V1) There exists apα, βq P p0, 2q such that for all η P Γ20 and n P N cn pα, β; ηq ≤ apα, βqMn pηq is satisfied.

23

(V2) For all ξ P Γ20 and x P Rd the following limits exist in Lα,β and are independent of ξ ´ |¨| ´1 ´ V,´ lim n|¨| pK´1 0 dn px, ¨ Y ξqq “ lim n pK0 dn px, ¨qq “: Dx nÑ8

nÑ8

lim n pK |¨|

nÑ8

´1 ` 0 dn px, ¨

` V,` Y ξqq “ lim n|¨| pK´1 0 dn px, ¨qq “: Dx nÑ8

lim n pK

V,´ ´ Y ξqq “ lim n|¨| pK´1 0 bn px, ¨qq “: Bx

lim n pK

` V,` Y ξqq “ lim n|¨| pK´1 0 bn px, ¨qq “: Bx .

|¨|

´1 ´ 0 bn px, ¨

nÑ8

|¨|

nÑ8

(V3) Let MV pηq :“

ř

xPη`

´1 ` 0 bn px, ¨

Dx` pHq `

ř

nÑ8

xPη´

nÑ8

Dx´ pHq, then there exists σ ą 0 such that either

Mn pηq ≤ σMV pηq, η P Γ20 , n P N

or Mn pηq ≤ σMV pηq, η P Γ20 , n P N are satisfied. pn :“ K´1 Ln K0 and the renormalized operators L pn,ren :“ Rn L pn Rn´1 , where Define L 0 |η| 2 pn,ren G, ky “ Rα Gpηq “ α Gpηq. Then for any G P Bbs pΓ0 q and k P Kα,β the relation xL ∆ xG, Ln,ren ky holds.

Statements The next statement provides the existence and uniqueness of an evolution of quasiobservables and correlation functions for any fixed n P N. Theorem 4.1. Suppose that condition (V1) is satisfied. Then for any fixed n P N the following assertions are true: pn,ren , Bbs pΓ2 qq is given by pL pn,ren , DpL pn,renqq, where 1. The closure of pL 0 pn,renq “ tG P Lα,β | Mn ¨ G P Lα,β u. DpL

It is the generator of an analytic semigroup pTpn,ren psqqs≥0 of contractions on Lα,β .

∆ 2. Let Tpn,renptq˚ be the adjoint semigroup. The generator is given by pL∆ n,ren , DpLn,ren qq with the (maximal) domain ∆ DpL∆ n,ren q “ tk P Kα,β | Ln,ren k P Kα,β u.

For any n P N and k0 P Kα,β , there exists a unique weak solution to B pn,ren G, kt,n y, kt,n |t“0 “ k0 , G P DpL pn,renq xG, kt,n y “ xL Bt given by kt,n “ Tpn,ren ptq˚ k0 . 24

The case n “ 1 is covered by the results obtained in Theorem 3.7. Following the arguments there, it is not difficult to adopt the proofs to this case. In the next step we construct the limiting dynamics when n Ñ 8. Condition (V2) suggests to consider pn,ren G ÝÑ L pV G, as n Ñ 8. The operator L pV :“ AV ` BV is given by the limit L AV Gpηq “ ´MV pηqGpηq, where ÿ ÿ DxV,´ pHq DxV,` pHq ` MV pηq “ xPη`

BV Gpηq “ ´

ÿ

ÿ

Gpξq

`

ξĂη

`

ÿ

DxV,` pηzξq ´

Gpξ Y x, ξ

Gpξq

ξ` Ĺη ` ξ´ Ĺη ´

xPξ `

ξ` Ĺη ` ξ´ Ĺη ´

ÿż

xPη´

´

qBxV,` pηzξqdx

`

DxV,´ pηzξq

xPξ ´

ÿż

ξĂη

Rd

ÿ

Gpξ ` , ξ ´ Y xqBxV,´ pηzξqdx.

Rd

pV q :“ tG P Lα,β | MV ¨ G P Lα,β u, define Let DpL

cV pα, β; ηq :“ ÿ ż ÿ ż ` ´ V,` α|ξ ` | β|ξ ´ | ´α |BxV,` pξq|eα|ξ | eβ|ξ | dλpξq |Dx pξq|e e dλpξq ` e ` xPη`

`

ÿ ż

xPη´

xPη`

Γ20

` ´ |DxV,´ pξq|eα|ξ | eβ|ξ | dλpξq

´β

`e

ÿ ż

xPη´

Γ20

Γ20

`

´

|BxV,´ pξq|eα|ξ | eβ|ξ | dλpξq

Γ20

and finally pL∆ V kqpηq

:“ ´

ÿ ż

xPη`

´

`

ÿ ż

xPη`

Γ20

ÿ ż

xPη´

kpη Y

ξqDxV,`pξqdλpξq

kpη Y

ξqDxV,´pξqdλpξq

`

Γ20

ÿ ż

xPη´

Γ20

kpη ` zx Y ξ ` , η ` Y ξ ` qBxV,` pξqdλpξq kpη ` Y ξ ` , η ´ zx Y ξ ´ qBxV,´ pξqdλpξq.

Γ20

Theorem 4.2. Assume that conditions (V1), (V2) are satisfied. Then the following assertions are true: pV , DpL pV qq is the generator of an analytic semigroup pTpV ptqqt≥0 of 1. The operator pL contractions on Lα,β . 2. Let pTpV ptq˚ qt≥0 be the adjoint semigroup on Kα,β , then for any r0 P Kα,β there exists a unique solution rt “ TpV ptq˚ r0 to the Cauchy problem B pV G, rt y, rt |t“0 “ r0 , G P DpL pV q. xG, rty “ xL Bt 25

(4.5)

3. Let r0 pηq “

ś

xPη` β

ρ` 0 pxq

e . Assume that

ś

´ ` ´ ` α 8 d ρ´ 0 pxq and ρ0 , ρ0 P L pR q with }ρ0 }L8 ≤ e , }ρ0 }L8 ≤

xPη´ ` ´ pρt , ρt q is

a classical solution to ż Bρ´ t ` ´ ´ V,´ ´ pxq “ ´ eλ pρ` t ; ξ qeλ pρt ; ξ qDx pξqdλpξqρt pxq Bt Γ20

`

ż

` ´ ´ V,´ eλ pρ` t ; ξ qeλ pρt ; ξ qBx pξqdλpξq

Γ20

Bρ` t pxq “ ´ Bt

ż

` ´ ´ V,` ` eλ pρ` t ; ξ qeλ pρt ; ξ qDx pξqdλpξqρt pxq

Γ20

`

ż

` ´ ´ V,` eλ pρ` t ; ξ qeλ pρt ; ξ qBx pξqdλpξq

Γ20 ` ` ´ ´ ` ´ α β with initial conditions t“0 “ ρ0 , ρt |t“0 “ ρ0 and }ρt }L8 ≤ e and }ρt }L8 ≤ e . ś ` ρt |ś Then rt pηq :“ ρt pxq ρ´ t pxq is a weak solution to (4.5) in Kα,β . xPη`

xPη´

Proof. By conditions (V1) and (V2) it follows that cV pα, β; ηq ≤ apα, βqMV pηq holds. pV q by Define a positive operator BV1 on DpL ÿ ÿ ÿ ÿ |DxV,´ pηzξq| Gpξq |DxV,` pηzξq| ` Gpξq BV1 Gpηq “ `

ÿż

ξĂη

`

Gpξ Y x, ξ

xPξ ´

ξ` Ĺη ` ξ´ Ĺη ´

xPξ `

ξ` Ĺη ` ξ´ Ĺη ´

´

q|BxV,` pηzξq|dx

`

ÿż

ξĂη

Rd

Gpξ ` , ξ ´ Y xq|BxV,´ pηzξq|dx.

Rd

pV q Then it is not difficult to see that for any 0 ≤ G P DpL ż ż ` ´ 1 α|η` | β|η´ | BV Gpηqe e dλpηq ≤ papα, βq ´ 1q MV pηqGpηqeα|η | eβ|η | dλpηq Γ20

Γ20

is fulfilled. The same arguments as in the proof of Theorem 4.1 yield existence, analyticity and the contraction property of the semigroup TpV ptq. For the last assertion we only show that rt is continuous w.r.t. C. The other assertions are simple computations, see e.g. ` ´ [FKK10]. First observe that by |rt pηq| ≤ eα|η | eβ|η | the function rt is norm-bounded and hence it suffices to show that it is continuous w.r.t. σpKα,β , Lα,β q. But this function is continuous in t ≥ 0 for any η and hence the assertion follows by dominated convergence.

26

Theorem 4.3. Suppose that conditions (V1) – (V3) are fulfilled. Then Tpn,ren ptq ÝÑ TpV ptq holds strongly in Lα,β and uniformly on compacts in t ≥ 0.

Proof. We are going to apply [FKK12, Lemma 4.3] and Trotter-Kato approximation. Fix λ ą 0 and denote by Rpλ; An q and Rpλ, AV q the resolvent for Aε and AV , respectively. Then it follows that }Rpλ; An q}LpLα,β q , }Rpλ; AV q}LpLα,β q ≤ λ1 , }Bn Rpλ; An qG}Lα,β ≤ papα, βq´1q}G}Lα,β and likewise }BV Rpλ; AV qG}Lα,β ≤ papα, βq´1q}G}Lα,β . Since Mn ÝÑ MV as n Ñ 8, it is easy to show by dominated convergence that Rpλ; An q ÝÑ Rpλ; AV q holds strongly in Lα,β as n Ñ 8. Hence it remains to show the convergence Bn Rpλ; An qG ÝÑ BV Rpλ; AV qG, n Ñ 8.

(4.6)

This can be proved, similarly to [FKK12], by dominated convergence. Remark 4.4. The proof shows that condition (V3) can be replaced by ` N τ |η| d´ , x P Rd , η P Γ20 , n P N n px, ηq ` dn px, ηq ≤ Ap1 ` |η|q e

for some constants A ą 0, N P N and τ ≥ 0.

5

Examples

We introduce four models describing the stochastic behaviour of particle systems. It is assumed that there are two types of particles. Namely, the first type is described by the configuration γ ` , whereas the second type by γ ´ . The time evolution of the corresponding particle system is assumed to be a solution to the associated Fokker-Planck equation (1.4). Interactions of particles, of the same and also of different type, are modelled by the relative energy function ÿ ϕpx ´ yq, x P Rd , γ ˘ P Γ, Eϕ px, γ ˘ q :“ yPγ ˘

where ϕ is a symmetric, non-negative and integrable function. To assure condition (A), we always suppose that either ϕ is compactly supported or there exists an integrable function g ≥ 0 such that for each x P Rd there exists Cx ą 0 and ϕpx ´ yq ≤ Cx gpyq, y P Rd holds. Associated to above relative energy is the following functional ż Cpϕq :“ |e´ϕpxq ´ 1|dx. Rd

Since ϕ is non-negative, we obtain Cpϕq ≤

ş

Rd

27

ϕpxqdx “: xϕy.

(5.1)

5.1

Two-interacting BDLP-model

Suppose that the death intensities are given by ÿ d´ px, γ ` , γ ´ zxq “ m´ ` a´ px ´ yq yPγ ´ zx

d` px, γ ` zx, γ ´ q “ m` `

ÿ

b´ px ´ yq `

yPγ ` zx

ÿ

ϕ´ px ´ yq.

yPγ ´

This means that the particles in γ ˘ have a random lifetime determined by the parameters m` , m´ ą 0. The additional terms describe the competition of particles for resources. Namely, each particle x P γ ´ may die due to the competition for resources with another particle y P γ ´ zx from the same type. The rate of this event is determined by a´ px ´ yq. Likewise each particle x P γ ` may be die due to the interaction with another particle by y P γ ` zx with the rate b´ px ´ yq. Moreover, this particle x P γ ` may also die due to the competition for resources with another particle y P γ ´ of different type. The rate of this event is described by the interaction potential ϕ´ px ´ yq. The birth intensities are assumed to be given by ÿ a` px ´ yq ` z b´ px, γq “ yPγ ´

b` px, γq “

ÿ

b` px ´ yq `

ÿ

ϕ` px ´ yq.

yPγ ´

yPγ `

The terms corresponding to a` and b` describe the free proliferation of particles γ ˘ independently of each other. Additionally, each particle y P γ ´ may create a new particle at position x P Rd of opposite type. The spatial distribution of the new particle is determined by the function ϕ` . By definition, such events occur independently of each other. The term including z ≥ 0 describes the additional creation of particles by an outer source. Suppose that a˘ , b˘ , ϕ˘ are non-negative, symmetric and integrable. Theorem 5.1. Suppose that a˘ , b˘ , ϕ˘ are bounded and there exist constants b1 , b2 ≥ 0 and ϑ1 , ϑ2 , ϑ3 ą 0 such that ÿ ÿ ÿ ÿ b´ px ´ yq ` b1 |η ` | b` px ´ yq ≤ ϑ1 xPη` yPη` zx

xPη` yPη` zx

ÿ

ÿ

xPη´ yPη´ zx

a` px ´ yq ≤ ϑ2

ÿ

ÿ

a´ px ´ yq ` b2 |η ´ |,

xPη´ yPη´ zx

and ϕ` ≤ ϑ3 ϕ´ hold. Moreover, assume that the parameters satisfy the relations ϑ1 , ϑ3 ă eα , ϑ2 ă eβ , m` ą eα xb´ y ` eβ xϕ´ y ` e´α b1 ` xb` y ` xϕ` y m´ ą eβ xa´ y ` e´β pb2 ` zq ` xa` y. Then conditions (A) – (D) hold for τ “ 0 and (3.4) is fulfilled. 28

Suppose that the conditions given above are fulfilled. Then (V1) – (V3) are satisfied and after Vlasov scaling we arrive at the kinetic equations Bρ´ t ´ ´ ´ ` ´ pxq “ ´ m´ ρ´ t pxq ´ ρt pxqpa ˚ ρt qpxq ` pa ˚ ρt qpxq ` z Bt ` ˘ ` Bρ` t ` ´ ` pxq “ ´ m` ` pϕ´ ˚ ρ´ t qpxq ρt pxq ´ ρt pxqpb ˚ ρt qpxq Bt ` ´ ` pb` ˚ ρ` t qpxq ` pϕ ˚ ρt qpxq. ş Here pf ˚ gqpxq :“ f px ´ yqgpyqdy denotes the usual convolution of functions on Rd . Rd

5.2

Two interacting Glauber-models

Suppose that the death intensities are given by ˘ ` d´ px, γ ` , γ ´ zxq “ exp ´sEψ` px, γ ` q ˘ ` d` px, γ ` zx, γ ´ q “ exp ´sEψ´ px, γ ´ q ,

where s P r0, 21 s and ψ ` , ψ ´ are symmetric, non-negative and integrable. The birth intensities are assumed to be of the form ˘ ˘ ` ` b´ px, γq “ z ´ exp ´p1 ´ sqEψ` px, γ ` q exp ´Eφ´ px, γ ´ q ˘ ˘ ` ` b` px, γq “ z ` exp ´p1 ´ sqEψ´ px, γ ´ q exp ´Eφ` px, γ ` q ,

where z ´ , z ` ą 0 and φ´ , φ` are assumed to be non-negative, symmetric and integrable. This model describes the time-evolution of two interacting types of particles. In contrast to the previous model, the proliferation of this particles is replaced by an additional source. The next theorem provides an evolution of states. Theorem 5.2. Let φ` , φ´ , ψ ` , ψ ´ be symmetric, non-negative and integrable and assume that the paramters satisfy the relations ee

α Cpsψ ` q

` e´β z ´ ee

α Cpp1´sqψ ` q

ee

ee

β Cpsψ ´ q

` e´α z ` ee

β Cpp1´sqψ ´ q

ee

β Cpφ´ q

α Cpφ` q

ă2

(5.2)

ă 2.

(5.3)

Then conditions (A) – (D) are satisfied for τ “ 0 and (3.4) holds. In the case s “ 0 conditions (5.2) and (5.3) simplify to z ´ ee

α Cpψ ` q

eβ Cpψ´ q

z` e

ee

β Cpφ´ q

eα Cpφ` q

e

29

ă eβ

(5.4)

ă eα .

(5.5)

Of particular interest is the special case φ` “ 0 “ φ´ , also known as the Widom-Rowlinson model. The non-equilibrium dynamics for this model has recently been analysed in [FKKO15], but without conditions (5.4) and (5.5) only existence of a local evolution of correlation functions could have been shown. Conditions (5.4) and (5.5) are satisfied for e´α “ Cpψ ` q and e´β “ Cpψ ´ q if 1 1 z´ ă and z ` ă ´ eCpψ q eCpψ ` q are satisfied. Vlasov scaling For simplicity we consider the case s “ 0, hence the death intensities need not to be scaled, i.e. are given by d´ px, γ ` , γ ´ zxq “ 1 “ d` px, γ ` zx, γ ´ q. The scaled birth intensities are given by ˆ ˙ ˆ ˙ 1 1 ´ ´ ` ´ bn px, γq “ z exp ´ Eψ` px, γ q exp ´ Eφ´ px, γ q n n ˙ ˆ ˙ ˆ 1 1 ´ ` ` ` bn px, γq “ z exp ´ Eψ´ px, γ q exp ´ Eφ` px, γ q . n n

All previous results can be applied which yields the mesoscopic equations, cf. (4.2) ` Bρ´ ` t ´ ´pφ´ ˚ρ´ t qpxq e´pψ ˚ρt qpxq pxq “ ´ρ´ t pxq ` z e Bt ´ Bρ` ´ t ` ´pφ` ˚ρ` t qpxq e´pψ ˚ρt qpxq . pxq “ ´ρ` t pxq ` z e Bt

5.3

(5.6) (5.7)

BDLP-model in Glauber environment

Let us consider death intensities given by d´ px, γ ` , γ ´ zxq “ 1 d` px, γ ` zx, γ ´ q “ m` `

ÿ

a´ px ´ yq `

ÿ

φpx ´ yq,

yPγ ´

yPγ ` zx

where m` ą 0 and 0 ≤ a´ , φ P L1 pRd q are symmetric. The birth intensities are assumed to be of the form ` ˘ b´ px, γq “ z ´ exp ´Eψ px, γ ´ q ÿ ÿ b` px ´ yq, a` px ´ yq ` b` px, γq “ yPγ ´

yPγ `

where z ´ ą 0 and 0 ≤ ψ, a` , b` P L1 pRd q are symmetric. 30

Theorem 5.3. Suppose that a˘ , b` , φ are bounded and there exist θ P p0, eα q and b ≥ 0 such that ÿ ÿ ÿ ÿ a´ px ´ yq ` b|η ` | (5.8) a` px ´ yq ≤ θ xPη` yPη` zx

xPη` yPη` zx

is satisfied. Moreover, assume that for some ϑ P p0, eα q and ϑφ ≥ b`

` ˘ eβ ą z ´ exp eβ Cpψq `

´

α

β

(5.9) (5.10) `

`

´α

m ą e xa y ` e xφy ` xa y ` xb y ` e

(5.11)

b

hold. Then conditions (A) – (D) are satisfied with τ “ 0 and (3.4) holds. Vlasov scaling Suppose that a˘ , b` , φ, ψ are bounded and (5.8) – (5.11) with z ´ ee (5.10) hold. Scaling of the potentials by n1 yields for the death

β xψy

ă eβ instead of

d´ px, γ ` , γ ´ zxq “ 1 ` ´ ` d` n px, γ zx, γ q “ m `

1 ÿ 1 ÿ ´ φpx ´ yq. a px ´ yq ` n yPγ ` zx n yPγ ´

For the birth we obtain ˆ ˙ 1 ´ “ z exp ´ Eψ px, γ q n ÿ 1 ÿ ` 1 ` a px ´ yq ` b px ´ yq. b` px, γq “ n n yPγ ` n yPγ ´ b´ n px, γq

´

Taking n Ñ 8 yields DxV,´ pηq “ 0|η| and ÿ ÿ ´ ` DxV,` pηq “ 0|η| m` ` 0|ξ | 1Γp1q pξ ` q a´ px ´ yq ` 0|ξ | 1Γp1q pξ ´ q φpx ´ yq. yPξ `

yPξ ´

For the birth intensity of the environment we obtain

K

´1 ´ 0 bn px, ¨

and hence

`

´

1 ´ ´n Eψ px,η´ zxq

Y η , ¨ Y η zxqpξq “ z e

¯ ` ´ 1 ´ n ψpx´¨q ´ eλ e ´ 1; ξ 0|ξ |

` ˘ ` BxV,´ pηq “ z ´ eλ ´ψpx ´ ¨q; ξ ´ 0|ξ | . 31

Similarly for the birth intensity of the system ÿ ÿ ` ´ a` px ´ yq ` 0|ξ | 1Γp1q pξ ´ q b` px ´ yq. BxV,` pηq “ 0|ξ | 1Γp1q pξ ` q yPξ `

yPξ ´

The kinetic equation is therefore given by Bρ´ t ´ ´pψ˚ρ´ t qpxq pxq “ ´ ρ´ t pxq ` z e Bt ` ˘ ` Bρ` t ` ´ ` ` ` ` ´ pxq “ ´ m` ` pφ ˚ ρ´ t qpxq ρt pxq ´ ρt pxqpa ˚ ρt qpxq ` pa ˚ ρt qpxq ` pb ˚ ρt qpxq. Bt

5.4

Density dependent branching in Glauber environment

Suppose that the death intensities are given by d´ px, γ ` , γ ´ zxq “ 1

˘ ` d` px, γ ` zx, γ ´ q “ m` exp Eφ` px, γ ` zxq ,

where m` ą 0. The birth intensities are given by ˘ ` b´ px, γq “ z ´ exp ´Eφ´ px, γ ´ q ÿ ˘ ` exp ´Eψ´ py, γ ´ q a` px ´ yq b` px, γq “ yPγ `

with z ´ ą 0 and a` , φ´ , φ` , ψ ´ symmetric, non-negative and integrable. Theorem 5.4. Suppose that 0 ‰ φ` , a` are bounded, there exist constants κ ą 0 and b ≥ 0 such that for all η ` P Γ0 ÿ ÿ ÿ ÿ φ` px ´ yq ` b|η ` | (5.12) a` px ´ yq ≤ ϑ xPη` yPη` zx

xPη` yPη` zx

and the parameters satisfy the relations ` ˘ eβ ą z ´ exp eβ Cpφ´ q maxtxa` y ` be´α , ϑe´α u eβ Cpψ´ q α ` 2 ą ee Cp´φ q ` e . m` Then conditions (A) – (D) hold with τ “ }φ` }8 and (3.4) is satisfied.

32

Vlasov scaling 1 n

gives d´ px, γ ` , γ ´ zxq “ 1, ˆ ˙ 1 ` ` ´ ` ` d px, γ zx, γ q “ m exp Eφ` px, γ zxq n

Scaling all potentials by

and for the birth intensities ˙ ˆ 1 ´ b px, γq “ z exp ´ Eφ´ px, γ q n ˆ ˙ ÿ 1 1 b` px, γq “ exp ´ Eψ´ py, γ ´ q a` px ´ yq. n yPγ ` n ´

´

Suppose that 0 ‰ φ` , a` , φ´ , ψ ´ , a` are bounded, (5.12) holds and the parameters satisfy the stronger relations ` ˘ eβ ą z ´ exp eβ xφ´ y maxtxa` y ` be´α , ϑe´α u eβ xψ´ y α ` e . 2 ą ee xφ y ` m` Then conditions (V1) – (V3) are satisfied. This yields the kinetic equations Bρ´ t ´ ´pφ´ ˚ρ´ t qpxq pxq “ ´ ρ´ t pxq ` z e Bt ´ Bρ` ´ t pφ` ˚ρ` t qpxq ` pa` ˚ ρ` qpxqe´pψ ˚ρt qpxq . pxq “ ´ m` ρ` t pxqe t Bt

33

References [AR91]

W. Arendt and A. Rhandi. Perturbation of positive semigroups. Arch. Math. (Basel), 56(2):107–119, 1991.

[Bal77]

J. M. Ball. Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc., 63(2):370–373, 1977.

[BCF` 14] B. Bolker, S. Cornell, D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and O. Ovaskainen. A general mathematical framework for the analysis of spatio-temporal point processes. Theoretical Ecology, 7(1):101–113, 2014. [BP99]

B. Bolker and S. W. Pacala. Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal. The American Naturalist, 153(6):575– 602, 1999.

[DL05]

U. Dieckmann and R. Law. Relaxation projections and the method of moments. The Geometry of Ecological Interactions: Symplifying Spatial Complexity, pages 412–455, 2005.

[EN00]

K. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.

[FFH` 15] D. Finkelshtein, M. Friesen, H. Hatzikirou, Y. Kondratiev, T. Kr¨ uger, and O. Kutoviy. Stochastic models of tumour development and related mesoscopic equations. Inter. Stud. Comp. Sys., 7:5–85, 2015. [Fin11]

D. Finkelshtein. Functional evolutions for homogeneous stationary death-immigration spatial dynamics. Methods Funct. Anal. Topology, 17(4):300–318, 2011.

[Fin13]

D. Finkelshtein. On convolutions on configuration spaces. I. Spaces of finite configurations. Ukrainian Math. J., 64(11):1752–1775, 2013.

[FK16]

M. Friesen and Y. Kondratiev. Weak-coupling limits in ergodic environments. in preperation, 2016.

[FKK10]

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Vlasov scaling for stochastic dynamics of continuous systems. J. Stat. Phys., 141(1):158–178, 2010.

[FKK12]

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Semigroup approach to birth-and-death stochastic dynamics in continuum. J. Funct. Anal., 262(3):1274–1308, 2012.

[FKKO15] D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and M. J. Oliveira. Dynamical Widom-Rowlinson model and its mesoscopic limit. J. Stat. Phys., 158(1):57–86, 2015. [FKKZ12] D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Zhizhina. An approximative approach for construction of the Glauber dynamics in continuum. Math. Nachr., 285(2-3):223–235, 2012. [FKO13]

D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Markov evolutions and hierarchical equations in the continuum. II: Multicomponent systems. Rep. Math. Phys., 71(1):123–148, 2013.

34

[GH96]

H.-O. Georgii and O. H¨ aggstr¨om. Phase transition in continuum Potts models. Comm. Math. Phys., 181(2):507–528, 1996.

[GK06]

N. L. Garcia and T. G. Kurtz. Spatial birth and death processes as solutions of stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat., 1:281–303, 2006.

[KK02]

Y. Kondratiev and T. Kuna. Harmonic analysis on configuration space. I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5(2):201–233, 2002.

[KK16]

Y. Kondratiev and Y. Kozitsky. The evolution of states in a spatial population model. J. Dyn. Diff. Equat., 28(1):1–39, 2016.

[KKM08]

Y. Kondratiev, O. Kutoviy, and R. Minlos. On non-equilibrium stochastic dynamics for interacting particle systems in continuum. J. Funct. Anal., 255(1):200–227, 2008.

[KKP08]

Y. Kondratiev, O. Kutoviy, and S. Pirogov. Correlation functions and invariant measures in continuous contact model. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 11(2):231–258, 2008.

[KS06]

Y. Kondratiev and A. Skorokhod. On contact processes in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9(2):187–198, 2006.

[Lem10]

L. Lemle. Existence and uniqueness for C0 -semigroups on the dual of a Banach space. Carpathian J. Math., 26(1):67–76, 2010.

[Paz83]

A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied mathematical sciences ; 44. Springer, New York [u.a.], 2., corr. print edition, 1983.

[SEW05]

D. Steinsaltz, S. N. Evans, and K. W. Wachter. A generalized model of mutation-selection balance with applications to aging. Adv. in Appl. Math., 35(1):16–33, 2005.

[TV06]

H. R. Thieme and J. Voigt. Stochastic semigroups: their construction by perturbation and approximation. In Positivity IV—theory and applications, pages 135–146. Tech. Univ. Dresden, Dresden, 2006.

[WZ02]

L. Wu and Y. Zhang. Existence and uniqueness of C0 -semigroup in L8 : a new topological approach. C. R. Math. Acad. Sci. Paris, 334(8):699–704, 2002.

[WZ06]

L. Wu and Y. Zhang. A new topological approach to the L8 -uniqueness of operators and the L1 -uniqueness of Fokker-Planck equations. J. Funct. Anal., 241(2):557–610, 2006.

35