Equilibrium Distributions of Branching Processes [Reprint 2021 ed.] 9783112471807, 9783112471791

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A. Liemant • K. Matthes • A. Wakolbinger

Equilibrium Distributions of Branching Processes

Mathematical Research

Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik

Band 42 Equilibrium Distributions of Branching Processes by A. Liemant K. Matthes A. Wakolbinger

Equilibrium Distributions of Branching Processes

by Alfred Liemant Klaus Matthes Anton Wakolbinger

A k a d e m i e - V e r l a g Berlin 1988

Autoren: Dr. sc. Alfred Liemant Prof. Dr. rer.nat.habil. Klaus Matthes Karl-Weierstraß-Institut

für Mathematik

der Akademie der Wissenschaften der DDR, Berlin Doz. Dr. Anton Wakolbinger Institut für Mathematik

der

Üohannes-Kepler-Universität,

Linz

Die Titel dieser Schriftenreihe werden vom Originalmanuskript Autoren

der

reproduziert

ISBN 3-05-500453-1 ISSN 0138-3019

Erschienen im Akademie-Verlag Berlin.DDR-1086 Berlin, Leipziger Str.3-4 ©

Akademie-Verlag Berlin

1988

Lizenznummer: 202 . 100/412/88 Printed in the German Democratic

Republic

Gesamtherstellung: VEB K o n g r e ß - und Werbedruck, 9273 Oberlungwitz Lektors D r . Reinhard Höppner LSV 1075 Bestellnummer: 763 827 1 (2182/42) 03400

Preface

The present monograph deals mainly with one aspect of stochastic branching models, namely their equilibrium distributions. Any model of this kind has the trivial equilibrium distribution, which is concentrated on the "void population". Unless, however, the branching model corresponds to an "independent substochastic translation" or may be reduced in a certain sense to this case (cf. section 2.9.), infinite populations occur with positive probability in any non-trivial equilibrium situation.The results presented here are thus in the intersection of two theories: stochastic infinite particle systems and branching models. From the point of view of stochastic infinite particle systems our model is elementary, because the independence hypotheses, which are characteristic of branching models, are rather restrictive. These restrictive hypotheses, however, admit a number of conclusions, which, at least partially, can serve as a "case study" for more complicated stochastic evolutions. On the other hand, from the viewpoint of classical branching models, the introduction of infinite particle systems leads to significant complications. Nevertheless the subject treated here should be regarded as a natural one in the theory of branching models. The theory of random point fields (and in particular that of infinitely divisible distributions of random point fields) is an important tool in the present investigations. The whole monograph is restricted to models in discrete time, in order to avoid questions connected with the existence and the description of stochastic branching dynamics in continuous time (cf. Ikeda, Nagasawa, Watanabe (1968,1969), Nagasawa (1977), Asmussen, Hering (1983)). Also, we do not touch upon the theory of measure valued branching processes. Parts of that theory are parallel to our setting (cf. Hermann (1981)), to which it is also connected through asymptotic considerations. From the historic viewpoint, the present investigations have two sources. The first is the theory of spatially homogeneous branching models, which has been presented in detail in Matthes, Kerstan, Mecke (1978) (quoted in the sequel by [MKM]). The second is the general theory of spatially inhomogeneous substochastic translations, especially the papers of Kerstan et al. (starting with Kerstan, Debes (1969)), Shiga and Takahashi (1974) and Liggett (1978). The spatially homogeneous branching models with different "types" of individuals (cf. eg. Prehn, Roder (1977)) can be considered as an intermediate stage between the spatially homogeneous case and the general case. The present text contains no specific chapter on the spatially homogeneous theory. This is for the sake of conciseness and also because a number of important questions in the spatially homogeneous case still 5

remain open. This case has yet to be studied from the point of view of the advanced general theory. A first impression of how the general theory developped here affects the spatially homogeneous case is provided by Fleischmann, Hermann, Matthes (1982). The results presented in this text considerably improve upon the hitherto existing publications on the subject. As a systematic presentation, it clarifies the theory, eliminates superfluous assumptions of past treatments, and creates some useful concepts such as that of a "family equilibrium distribution." Terminology and notation are modified in order to facilitate reading. Examples play an important role in the whole presentation, some of which are unbroken threads running throughout the text. In the first chapter basic facts from the theory of random point fields are reviewed, for the most part without proof, but with reference to [MKM] and to Kallenberg (1983) (cited by [K]). References and attributions of results are collected in an appendix "Comments and References", unless these results are "already standard" or are due to some of the present authors (whose relevant publications, however, are included in the list of references). We cordially thank Miss Irene Steininger for her patient and diligent work in preparing the camera-ready manuscript, and R. Siegmund-Schultze, whose contributions were essential for the development of the present theory. Berlin and Linz, March 1987 The authors

6

Contents

CHAPTER 1. An introduction to random point fields and branching dynamics 1.

9

Random

measures

10

2.

Random

point fields

15

3.

The Laplace transformation

20

4.

Clustering fields and^branching dynamics

22

5.

Intensity

kernels

25

6.

Family

structures

29

7.

Campbell

8.

Palm

9.

The backward technique

meas'ures

distributions

CHAPTER 2. Equilibrium distributions of Poisson type

34 37 .'...41 46

1.

Equilibrium distributions and stable measures

47

2.

J-invariant

50

3.

Equilibrium distributions of Poisson type

53

4.

An application of the backward technique

57

5.

A sufficient criterion for stability

59

6.

Stable and purely unstable components of J-invariant measures

62

7.

Equilibrium distributions for the family dynamics

65

8.

Poisson type equilibrium distributions with finite canonical measure

77

9.

measures

Finitely fed equilibrium distributions

10.

Finitely fed equilibrium distributions of Poisson type

11.

A characterization of the equilibrium distributions of Poisson type via random marking

CHAPTER 3. Equilibrium distributions of Cox type

86 100 106 113

1.

K-processes

114

2.

J-processes and K-processes of Cox type

117

3.

K-processes of Poisson type

120

4.

Stable

123

5.

A characterization of the K-processes of Cox type

126

6.

Once again applying the backward technique

127

J-processes

7.

J-invariant distributions and equilibrium distributions of Cox type

130

8.

J-aperiodic

133

measures

1

CHAPTER 4. Characteristic J-invariant distributions

140

1.

Extremal K-processes

141

2.

K-processes with first order representation

5.

Characteristic J-processes of Poisson type K-processes

and their characteristic J-processes

153 155

4.

Regular J-sequences and extremal K-processes of Poisson type

162

5.

Which K-processes are of Cox type?

16,7

6.

Convergence

174

7.

The stationary case: Characteristic J-invariant distributions

theorems

CHAPTER 5. Ergodic intensity distributions 1.

Extremal equilibrium distributions

2.

Equilibrium distributions with first order representation

186 187 '

and their ergodic intensity distributions 3.

178

192

Relations between ergodic intensity distribution and characteristic J-invariant distribution

197

4.

Extremal equilibrium distributions of Poisson type

200

5.

Purely non-extremal equilibrium distributions of Poisson type

201

6.

A natural factorization of Poisson type equilibrium distributions

204

7.

Cesaro clustering and J-ergodic measures

208

8.

Ergodic intensity distributions of Poisson type equilibria

210

9.

An example of a non-extremal Poisson type equilibrium distribution whose intensity measure is J-ergodic

COMMENTS

AND REMARKS

220 z27

REFERENCES

230

SYMBOL

INDEX

232

SUBJECT

INDEX

239

8

1. AN INTRODUCTION TO RANDOM POINT FIELDS AND BRANCHING DYNAMICS

Let as consider a system of particles or rather a population of individuals which are indistinguishable up to their location, and assume that the population is subject to a stoch'istic branching dynamics in discrete time in the following way: each individual in the population produces, independently of the other individuals, a random offspring, whose distribution depends on the individual's location, and the population in the subsequent generation is the superposition of all these offsprings. This monograph deals with the time evolution of such systems, in, particular investigating the structure of equilibrium distributions and establishing convergence-to-equilibrium results for certain initial distributions. As a "phase space" (i.e. the space which the particles or individuals live in) we admit an arbitrary complete, separable metric space [A, p A ], A population of individuals in a phase space [A, p A ] might be described by the set S of the individuals' locations. Thus, as a mathematical model of populations all those subsets S of A could be considered which obey the finiteness condition #(SnX)
v®k is a mapping f r o m N into the set of measures on A s the mapping V—>v® k (S) is immeasurable for all S e

w e may form, for each

measure H on iAC, the measure A%(.)

:=

jv®k(.)H(dv)

on J?®k, w h i c h is called k**1 m o m e n t m e a s u r e of H. W e say that H is o f k**1 o r d e r if is finite for all X e *B. For k = l , w e write AJJ instead of A ^ ^ and speak of the intensity m e a s u r e of H. For all f e F there holds < A H , f > = Jif(a)v(da)H(dv) . If H is interpreted as the law of a random mass allocation, then A j ^ X ) is the expectation of the random mass in X . For simplicity, w e further put

:=

H

.

L e t G denote the linear space of all finite signed measures on

Together with the

convolution operation G^GjS^GjSGjXiVj+v^O)

(GpGjeG)

,

G becomes a commutative real algebra with unit element 8 0 . For all G e G and k e Z + , w e denote the k-th convolution power of G by G k . In particular, there holds G ° = 8 0 . For abbreviation, w e put, for all X e A and all measures v o n j?, x

v

:=v((.)nX)

.

For all X e % the mapping G-»XG

:= ( ? ( x v e (.))

is a homomorphism of the algebra G into itself. W e w i l l a l w a y s write llqll for the t o t a l v a r i a t i o n of a finite signed measure q on an arbitrary measurable space. With the norm IIII, G is a complete normed algebra. Instead of XIIGII

( G e G , X e J%).

ll x GII

w e also write

G + denotes the set of all finite measures on ¡ft[,lG

stands for

the w e a k topology on G + , and => for the corresponding convergence. [ G + , T G J is again a Polish space. For all finite sequences X j , . . . , X m of sets in ® and all G e G w e put G

x

x

:= G ( [ v ( X j ) , . . . , v ( X m ) ] e (.))

.

Obviously there holds (G,*G2)Xi

Xm

= (G,)Xi

Xm

* (G2)Xi

Xm

(GL G 2 e G+)

,

where * on the right hand side means the usual convolution of finite measures on the 0-algebra (B+)® 1 " of Borel sets on R + m .

12

1.1.5. [MKM, Prop. 7.2.4.] In G + there holds G n => G iff for all finite sequences X j , . . . , ^ of sets in ®G the weak convergence = G X„...,X 1 mI„ * X11* X 'm

in the space of all finite measures on (¿J^®"1 takes place. Due to ® Gi »

= ® Gi n

there follows

1.1.6. In G + , G n => G and Hn => H imply that G^H,, => G*H . Moreover, from 1.1.5. it is easy to conclude 1.1.7. Gn => G and X e ®G imply that AG(X) < liminf AG (X). n — n Another consequence of 1.1.5. is 1.1.8. For all measures G, G p G2,... of first order in G + such that G n => G, the vague convergence A G

N

n

» A G takes place iff for all X € H the condition

sup J v(X) G (dv) » 0 n=l$,... (v:v(X)>c} " c-x» is fulfilled. Let V denote the set of all probability distributions on iA£and V be the smallest o-algebra of subsets of V with respect to which all mappings Q(Y) , YeiAC, are measurable. V is a weakly closed subset of G + , and the 8V is a homeomorphic imbedding of [N, xN] into [V, ty]. The following simple estimate will sometimes be useful: 1.1.9. For all Qj, Q 2 e V and all Xe A there holds xIIQrQ2ll

< 2 (Qj(v(X)>0) + Q2(v(X)>0))

0)

1.2.11. [MKM, Prop. 2.2.6.] For all distributions P e T of first order there holds A

(2)p

=

A

(2)_

+

A p g ) A p

.

With any measure ve N we associate via G v :=v(8 a e(.)) a measure Qy e W with intensity measure v, and call v the Poisson distribution with intensity measure v. n

1.2.12. [MKM, Prop. 2.2.14.] For all v e N , a distribution P on n v iff for any finite sequence X , X

1

Xn, of pairwise disjoint sets in ® there holds 2

^(X,)

1.2.13. [MKM, Prop. 3.3.7.] The mapping v imbedding from [N, x N ] into [P, T P ]. 18

coincides with

^(X^ II V provides a homeomorphic

With any G e G , we associate via 3(G):=Jnv(.)G(dv) a signed measure 3 ( G ) on M , the so called C o x t r a n s f o r m o f G . 1 . 2 . 1 4 . [ M K M , Prop. 7 . 1 . 2 . ] T h e C o x transform 3 is an isomorphism from the algebra G onto a subalgebra o f E . Obviously, 3 ( G ) is in E + resp. in P, if G is in G + resp. in V . In this case there holds ^ 3 ( G ) = ^ G • Distributions o f the form

3 ( G ) , G e V, are called C o x d i s t r i b u t i o n s .

1 . 2 . 1 5 . [ M K M , Prop. 7 . 2 . 1 . , 7.2.2.]

The C o x transform 3

provides a homeo-

morphism from [ G + , Tq ] into a closed subset o f [ E + , i E ).

Any counting measure e M we have G is equivalent to the validity of (feF ) . L ^ (f) LQ(S) n For all measures v on Si, let F v denote the set of all fe F with the property (v, f) < For all Ge G + and all f, ge F A q there holds I i e x p K v , f ) ) G(dv) - iexp (- l''"* m

X m (d[x 1 ,...,x m ]) = / G ( i s | n s i l x . ) .

Hence follows 1.3.6. For all G, Gj, G 2 , ...e G + and all finite sequences X 1 ,...,X m of sets in is equivalent to

20

(Gn)x,1* ' Xm \

G

X ,1 Xm m

.

Assume now that in G + the convergence relation G n => G holds true, and that, for some V G N , the inequalities AQ < V , n = 1, 2,..., are fulfilled. Subject to these n assumptions, also AQ < v holds true. Furthermore, for any fe F v and any e>0 there exists some g e F c such that (v, | f-g |) < e, and we obtain by 1.3.4. and 1.3.5. limsup | LQ (f) - L A i ) I n — n < limsup | Z^j (f) n-x~ n < limsup n-»~

n

I +

, i m

n-»~

I A3 (g)"Aj(g)l + 1 - ^ ( 0 - ^ ( 8 ) 1 n

( V , , lf-gl) + 0 + ( A r If-gl) < 2 I < 2e . n

Hence we recognise the validity of 1.3.7. If in G + the weak convergence G n => G takes place, and if for some v e N the inequalities AQ < v, n=l,2,..., are fulfilled, then there holds n AJ^O-^AJ(0

(feFv) •

In view of 1.3.6. one further gets: 1.3.8. Subject to the assumptions of 1.3.7. there holds (G_) x

x

for any finite sequence X j ,...,X m of sets in

M

The canonical representation of infinitely divisible distributions on

=> G x

x

J»-*-» M

now takes the

following form: 1.3.9. [K, Prop. 6.1.] For all v e N and all U e U there holds (f) = exp(-(v, f)-J(l-exp(-(Y,f»)U(dY))

.

Hence results 1.3.10. For all Pe T one has Lfki) = exp(-J(l-exp(-(\)/,f)))P(d\|f))

(feF) .

Particulary there hold 1.3.11.

L n j f ) = exp(-J(l-exp(-f(a)))v(da))

1.3.12.

£ 3 ( Q ) ( f ) = AJ(l-exp(-0)

(veN; f e F )

(Qe V; f e F )

.

.

In V, 1.3.2. generalizes to infinite convolution products: 1.3.13. In V, the relation Q = * Q j iel

implies

i ^ ( f ) = nXQ.(f) ( f e F ) iel 1

.

For some computations, the set F will be too large and will be replaced by the set F b of all ^-measurable bounded nonnegative real-valued functions on A with bounded support. Obviously there holds 0eM belongs to ^M iff < + £ |CM) = 0, then we form via H k :=|K w (.)H(d4>) the "clustered" measure HK on 5lf. If a distribution Pe P is interpreted as a distribution of a random population 4>0 of individuals in the phase space [A, p A ], then P K , if it exists, may be interpreted as distribution of the succeeding generation 4>,. The set {P:P€P, P ^ ) = 1} of those distributions P on iW, for which PK exists, belongs to P. There even holds 1.4.3. [MKM, Prop. 4.2.4.] Subject to the assumptions of 1.4.1. , P-* PK is a P ,P'-measurable mapping from {P: P e P , P i ^ = 1)} into P'. The clustering operation is monotone in the following sense: 1.4.4. Let Pj, P 2 be distributions in P such that Pj i P 2 and (P2)Kexists. Then also (Pj)K exists, and (P,)K