Math. Nrtchr. 118 (1983) 249-267

On Spatially Homogeneous Branching Processes in a Random Environment

By DONALD DAWSON~) of Ottawa and KLAUS REISCHUNN~) of Berlin

(Received August 10, 1982)

0. Introduction

Spatially honiogeneous branching processes in Euclidean space Rd are usually defined in such a way that the evolution law Ex,: of a “particle” 6, a t position x Rd a t time t doesn’t depend on x or t , of. MATTHES, KERSTAN and MECKE [8; Chapter 121 or DAWSON [2]. To model a homogeneous “random cnvironment” one can assume that E,,: is random in that {E,,t : x E Rd, t E R+} forms a random field which is invariant with respect to space and time translations. In the particular case of dimension d = 0 and discrete time such a model reduces to a GALTON-WATSON process in a random environment which was investigated by SMITE, WILKINSON, ATHREYA, KARLIN, KAPLAN, KEIDINO, NIELSEN, MOUNTFORD, TURNBAU (and probably others), cf. ATHREYA and NEY [ l ; Section 6.51 as well as JAOERS [lo; Section 3.71. If d > 0 the most interesting cp.ses are the critical processes which are candidates for the existence of a nontrivial equilibrium state (stability). Because of the richness of the model some new effects in the stability behavior can be expected. At the same time we note that this type of process can be understood as a branching process together with a random interaction with the environment which results in a process that does not have an infinitely divisible distribution.

In this paper we consider a highly simplified model in a d-dimensional lattice G = Zd. Each particle 6, a t position x E G produces a critical cluster x2. consisting of a random number of particles. The locations of the particles in the cluster are obtained by independent displacements with displacement distribution I relative to x. The number of offspring particles is random with probability generating function f, where {f,: x E G } is assumed to be a random field which is invariant under spatial translations. In other words, the random environment w = { f2 : x E G } is constant in time and for given w the particle 6, generates a random number of offspring according to f, and the displacements of the particles are independent with common distribution I . Already in this simple case a number of interesting questions arise and i t is the purpose of this paper to present some preliminary results.

1. The Deterministic Environment

Let G be the d-dimensional lattice group Zd, d 2 1, and 1 a fixed HAAR measure on G. Let N denote the set of all (non-negative) measures p on G which are integer- ~ ~ ~~

l) Research supported by NSERC grant A 7750. 2, This paper was prepared during the visit of the second author to Carleton University in the

fall of 1981.

250 Dawson/Fleischmann, Random Environment

valued on finite subsets of G. A measure H on G is called k-th order (k 2 1) if

(1.1) H(d,u) ( ~ ( x ) ) ~ < 00, for all x E G , that is, if the k-th factorial moment measure I'g' of H is locally finite.

For each g E G we introduce the translation operator T , by defining

(1.2) T,h(x) := h(x + g ) , h a function on G , x E G ,

T,,u(S) : = A x - 9)s p E N , 5 E a, !P,H(B) := EI((p: Top E B ) ) , H a measure on N , measurable B c N .

In the following the term "invariant" always refers to invariance with respect to the translation group {T,: g E G } ,

Throughout the paper il denotes a fixed displacement distra7mtion on G and

Qi(S,) := A(x), x E G ,

defines a distribution on N .

1.1. Hypothesis. The symmetrized distribution O A := A( - .) * ?. is msumed to be tran-

Let P be the set of all critical probability generating functions sient.

f (s) = z p ( n ) s", 0 5 s 5 1, f'(1) = 1 nPO

where p is a probability distribution on the non-negative integers. In this section w denotes a fixed mapping of Q into P

w : 2 E Q + fi E F. Let

1.3) D, := f,(&) for x E G , yielding the distribution of a critical cluster xz with

(1.4) E x P = I , EsX*'=f,(s), X E G , 0 5 ~ 5 1 .

For x E G, let

:= T a , . This defines a cluster field xw. The intensity measure of x;,, denoted by A+, satisfies:

so that xw is critical with respect to 1 in the sense of LIEMANT [7, I]. Thus we can form the critical branching process {OF: t = 0, 1,2, . . .} in G where @; is distributed according to the invariant POISSON distribution l& with intensity measure 1 and = (a;),. We now review some basic facts due to LIEBUNT [7].

1.2. Lemma. As t --f 00, 0: converges in distribution to a limit @& with an infinitely

Proof. Refer to LIEMANT [7, I ; Lemma 1.41. The cluster fleld xw is said to be stable of order k ( k 2 1) if there exists a distribution

P on N with intensity meaaure Ap = I which is cluster invariant with respect to xw and of k-th order.

divisz%le and cluster invariant distribution Pw and E@Z 5 1 .

Damson/Fleischmann, Random Environment 25 1

1.3 Lemma. xw i s stable (of order one) i f and only i f Apw = 1.

Proof. Refer to LIEMANT [7, I; Theorem 1.61. The k-th factorial moment measure of x;"z, is given by

= z p x ( n ) n(n - 1) ... (n - k + 1) T,1 x... x T x I n

= fhk'( 1) TxI x * * X Txl

where pz is the offspring distribution corresponding to the probability generating f unc- tion f z .

1.4. Lemma. Let P, denote the L ~ V Y measure of the infinitely divka3le and cluster invariant distm3dim P,. If x" is 8tuble, then, for k 2 2, the k-th factm-al moment measure of P, fulfils

(1.7) rf- L J z(az) fp(i) 2 TJ' x . . . x TJ' : = m)uk) 121

where I t denotes the t-fold convoldion of il with itself. I n the case k = 2, equality holds in (1.7).

Proof. Refer to LIEMANT "7, 11; Theorem 3.41.

1.6. Lemma. Let k 2 2. The cluster field xw i s stable of order k i f and only i f x" is stable and P, i s of order k. If xw i s stable of order k, then my) is locally finite. For k = 2, the converse is also true.

Proof. Refer to LIEMANT "7, 11; Satz 3.2, Theorem 3.3 and Theorem 3.41. Our next objective is to derive a sufficient condition for stability. By Lemnias 1.2

and 1.3 stability is equivalent to the weak convergence of the P m measures of the LBw measure of @y to the PALM measures if the L15w measure of @:, cf. WENBERG [5; Chapter lo]; MATTHES, KERSTAN and MEUKE [8, 10.3.31. But the PALM measures of x;"2, equal

(x&))o = 86" * TzfL(QA) 9

so that stability is equivalent to the weak convergence, as t + cy), of

/ H(Y, dyi) * - J H(yt-1, d ~ t ) T y l f l l ( Q t ) * + . * * ( Tv,fl,(&i))xo.f-l

for E G (cf. LIEMANT [7, IV; Satz 8.2]), where

H(Y, .) = T&(-.), y E G ,

and x w J is the t-fold iterate of xw. This weak convergence takes place if and only if only finitely many factors contribute points at each fixed x E G. By the BOREL- CANTELLI lemina for this it suffices to require that

z J TJ'(--dZ) ( T Z f ; ( Q A ) ) d - 1 (x(4 > 0) < 00, 2, y E Q. 121

252 Dawson/Fleisohmann, Random Environment

Now, (Tzf;(QA)*d-1 ( % ( X I = 0) = f : ( c ~ x Q l ) u w . f - l (m = 0))

=f; ( l - (TLI&n)) ,o . t - l (X(x)>O)) 2 f ; ( l - A y x - z ) ) .

This leads to the following partial generalization of the stability criterion of KALLEN- BERG [ 5 ; Theorem 3.11.

1.6. Proposition. The cluster field xu is stable i f

1.7. Hypotheses. Assume that fz( 1) is bounded in x and that the cluster field xu is stable.

1.8. Proposition. Assume that Hypotheses 1.1 and 1.7 are satisfied. Then the cluster invariant random measure distribution P,,, satisfies the following mixing property. Let h,, h, be non-negative functions on Q which vanish oudside some finite set B c a. Then for the LAPLACE functional S’p, of P, we have

(1.9) =YP,(Tglhl + Tgah2) - =YP,(TQ,hl) %$%,hZ) + 0

as g1 - g2 .+ 00.

Proof. First of all, cf. FLEISOHMANN [4, Lemma 2.2.21,

0 5 =%,,,(T& + T g, h 2 ) - %,(T,,hl) =YP,,,(T,,hZ) 5 F,J{!P: YT,lhl > 0, !PTUah2 > 0 ) ) .

Using the second moment measure A‘?’ of P, we can continue with p ,

2 1 A$b(dzi, dz2) T & I ( ~ I ) Tg,hz(zz )

= 1 q p 1 , dz2) Tglhl(Xl) T,,hz(z,) + J l(dz) T,,hl(4 T,,h2(4 *

By Lemma 1.4 (with k = 2), the first summand in the last expression equals:

1 I(&) f z ( 1 ) C T2+,,Afh1T,+,,ilfh2 5 c l(dx) A*(B - 2 - g,) It(B - z - 9,) 121 thl

where c always denotes a constant which can change from formula to formula. But then in view of Hypothesis 1.1 it follows that the last expression tends to zero as g1 - g2 + 00, cf. SPITZER [9; 8.24.41.

The second summand in the expression above equals

which also tends to zero as g1 - g, + 00. Thus the proof is complete.

2. Random Environment

In this section we assume that the environment is given by an invariant random field w = { f 2 : z E G}. Given w we employ the notation and results of the previous section. Then {@;I: t = 0, 1, . . .} is a branching process in a random environment.

DawsonlFleischmann, Random Environment 253

2.1. Lemma. As t -+ 00, @r converges in distrahdion to @: with distribution P : = E,P,.

Proof. Using LAPLACE functionals the convergence follows easily from Lemma 1.2. This P is invariant with intensity i p 5 1.

By the invariance of w we have

E,x;",,(T,x E .) = Ewx;UT,& E .) for g E G, @ E N ,

where x:,, := * ( x ~ , ) @ ' ~ ) . This together with the invariance of 111 yields the invariance in distribution of @; for t 2 1 and hence the invariance of the limiting distribution P. Finally ip 5 1 is obvious, cf. MATTHES, KERSTAN and MEUKE [8; Proposition 6.4.21.

2.2. Hypothesis. Let a2 : = E,ft( 1) be finite.

2.3. Lemma. U d e r Hypotheses 1.1 and 2.2, for almost every o, the clwter field x"

Proof . By Lemma 1.5 and 1.4, for given w , second order stability is equivalent to

is stable of second order. Hence ip = 1 (stability of the process).

z:f:(l) I f ( y - 5) I'(z - 5) < 00, y , z E G. z 121

(2.1)

But the expectation with respect to w of the last expression equals

E,ft( 1) It( -2) I'(z - y - x) = a2 OP(Z - y) 12-1 x tbl

which is finite by the transience of OI. Applying Lemma 1.3, the proof is complete.

2.4. Hypothesis. Assume that o i s an ergodic invariant random environment.

2.6. Lemma. Under Hypthesh 2.4 the set

(w : x" is stable of second order} (2.2)

i s trivial, that is , it has probability zero or one.

Proof. We have

m!&(B x B) = mE)((B - g) x ( B - g)), g E G, B c G,

so that by Lemma 1.5 the set (2.2) is invariant. Hence the claimed assertion follows by the definition of ergodicity.

{x : 121 5 n} in a. For each natural number n let 1, denote the uniform distribution on the ball

2.6. Lemma. A necessary and sufficient condition for P to be ergodic G that for h,, ha non negative fundim on G with finite support

lim Mdg) Y P ( ~ + TOh) = Y ~ ( h 1 ) Y P ( ~ ) - n-wa

Proof. Refer to ~ I S U H M A " [4, Lemma 3.811.

2.7. Proposition. Under hypotheses 1.1, 2.2 and 2.4, tibe limiting distribution P i s ergodic.

254 Daweon/Fleischmann. Random Environment

Proof. Let h,, h2 be as in Lemma 2.6. Then

J zn(dg) 9 ~ h l + ~Qh2) = J ln(dg) ~ w y P , ( h l + ~Dh2) = J L(dg)Ew[p~,(h~ + T&2) - ~ P , ( ' I ) ~ P , ( ' & Z ) I

+ J L(dg) ~ w z E P p , ( h l ) 3Pu(~0h2) - By Lemma 2.3, the cluster field x" is stable of second order as . and the first summand can be estimated as in the proof of Proposition 1.8 using Hypotheses 2.2 instead of Hypothesis 1.7. Consequently the first summand converges to zero as n -+ 60. The second summand equals

and by the ergodicity of o this term tends to (cf. FLEISUEMANN [4; Lemma 3.3.11)

Ew3p,(hi) JLpP,(hz) = ZP(~I) -LPp(h2) 88 n + 00.

By Lemma 2.6 this just means the ergodicity of P. 2.8. Hypothesis. Aseume that the random environment w i s mixing.

2.9. Proposition. Under Hypotheses 1.1, 2.2 and 2.8, P i s mixing. Proof. The proof follows in the same way as that of Proposition 2.7.

3. Particular Cases

In the following, symbols like a, m b, always mean that b, + 0 (for almost all n)

3.1. Hypotheses. For 0 < B 5 2 and 0 < y 2 1 with (1 + y ) d > /?, the series

and that the ratios an/b, converge to a positive (finite) limit as n + do.

converge for all x and z in G am? have the aqmptotics

M Iz18-(1+v)d aB 2 + do for each x E G.

tion of a symmetric stable law of index B. In the case B = 2, To get a feeling for this condition, assume that A is in the normal domain of attrac-

P(z) w t -d l z ex p (- Izla/t) as t --f 00, z + 60,

and the asymptotica in Hypotheses 3.1 follows directly.

3.2. Hypothesis. For 0 < cx 5 1

( 3 4 1 - E,fi(l - 8 ) w 8" a8 s + o . (An example for such a random environment will be given in the end of the paper.)

3.3. Proposition. Under Hyp0the.s 3.2 and 3.1 with y = cx the clwter f k M x" i8

stable with probability one if d >

Dawson/Fleischmann, Random Environment 265

Proof. By Proposition 1.6 we have stability almost surely if

E, 2 2 ( y - z ) [l - f i ( l - I f (% - z ) ) ] < 00 for x, y E U 121 I

that is

z I t ( y + z ) [l - E&(l - I t ( z ) ) ] < 00 for y E G. 121 2

From Hypotheses 3.1 we conclude

I+) - t O as [ t , 23 + 00

end by (3.1) we have to show

z ~ t ( y + z ) (A~(z) )" < 00 for y E a. t t l 2

In view of Hypotheses 3.1 this is equivalent to

d Y IzlS-(l+")d < 00, Z*O

But

(3.2) so that we get the convergence of the last series if d - 1 + ,!I - (1 + LY) d < -1, that is ad > 8. This completes the proof.

3.4. Hypotheses. Assume that the random environment w = ( f z : x E U} is given by independent (identically distrahted) variables f,. Moreover, for the probability distraWion funetion H of ft(1) we swppose

# (z : k - 112 < IzI 5 k + 1/2} m kd-' as k --f 00

(3.3) 1 - H(v) = &(j;(l) 2 v) M v-= as v -t 00

forSomeO<aL<l. Note that the last condition implies Ewf:(l) = 00 which is in contrast with Hypo-

thesis 2.2.

3.6. Propositions. Let Hypotheses 3.4 and 3.1 with y = 1 be fulfilled. Then t h cluster field x' issecond order etableurithpobabilityoneifand only if u > 112 and d > ,!Iu/(2u - 1).

Proof. Given w , the necessary and sufficient condition given by Lemma 1.6 and 1.4 is (of. (2.1))

2 /;Jl) At(y - z + x) At(%) < 00 a s . z L t 1

for all y , z in U. From Hypotheses 3.1 the criterion is

z /:-JI) IxIp-zd < 00 as . for z E U, z*o

that is

z f,"(l) IxIfl-2d < 00 as. ZS.0

Let Wz : = j,"( 1) IxIp-2d for x + 0.

256 Dewson/Fleischmenn, Random Environment

By KOLMOOOROV’S three series theorem, cf. FELLER [3; p. 3171, the series z W2 cou- verges a.s. if and only if

(3.4)

Z*O

Z 9 t ( W Z 1 1 ) < 00,

z EW~Z1{WZ<lJ < 00,

z 9 t ( W z 2 1) =

z IxI-”(Zd-@) < 00.

z*o

z v ~ ~ u W z l { W , c l J < 00. z*o 2 9 0

By (3.3) the series

(1 - H ( l 2 p - f l ) ) z*o z+o

converge if and only if

z*o (3.5)

Using again (3.2) we find the necessary and sufficient condition d - 1 - a(2d - ,8) < - 1 which is satisfied if and only if dc > 1/2 and d > ,84(2a - 1).

In view of (3.3) the series

z E,WZ1{W*<1J = 2 14fl-zd ~ U l m l ) 1(~(1)<1z1“6-@)

z lZl@-2d J as 8-’ < -. z*o 2 9 0

(3.6)

are convergent if l Z l ’ W

z+0 0

But this leads exactly to the condition (3.5). Finally, consider

(if the two series converge). The second term clearly converges if (3.6) is finite. On the other hand, by (3.3) the first term converges if

z IZ121@- ld ) i~ :88 - . < 00

z*o 0

which again lea& to (3.6). Summarizing, the three series in KOLMOQOROV’S criterion converge if the conditions

1x > 1/2 and d > ,8a/(2dc - 1) are fulfilled. Conversely, these conditione are a conse- quence of (3.4) and the proof is complete.

To finish the paper, we look at the following illustrative example.

3.6. Example. For each natural number K ;2 2 consider the critical probability generating function f ( K ; .) defined by

f ( K ; 8 ) = (1 - K-’) + K-’8K, 0 d 8 5 1, that is with the ‘small’ probability K-’ the number of offspring equals K, otherwise the cluster is empty. Then

(3.7) 1 - f ’ (K; 1 - 8) = 1 - (1 - 8)K-1, 0 5 8 5 1, and

(3.8) / ” ( K ; 1) = K - 1.

Dawson/Fleischmann, Random Environment 257

Let q be a probability distribution on the set {2, 3, . . .} satisfying

(3.9) q(k) M k-"-1 as k -+ 00

for some 0 < u < 1. Consider an invariant random field {& : z E G } where KO has distribution q and form the random environment o = { I&; .) : z E a).

By (3.8) and (3.9) the probability distribution function of f"(Ko; 1) fulfils the asymptotics (3.3) in Hypotheses 3.4. Moreover, in view of (3.7) we have

1 - Eq/'(Ko; 1 - 8) = 1 - Eq(l - p - 1

= 1 - z q ( k ) (1 -))"I for 0 5 B 5 1. k

By (3.9) and a TAuBERian theorem, for the last expression we have the asymptotics w B' as 8 + 0. Use, for instance,

2 q(k) (k - 1) (1 - Q ) ~ - Z M 8O-l as B += 0, P

of. FELLER [3 ; Theorem 13.5.51, and apply ~'HOSPITALE'S rule. Consequently, this random environment is in accordance with Hypotheses 3.2.

3.6. Remark. We have finally studied a spatially homogeneous random environ- ment model in which the number of offspring particles has always a finite variance, given the environment. Under this finiteness condition in the corresponding spataially homogeneous deterministic environment model stability, second order stability and the transience of O j l coincide, cf. MATTHES, KERSTAN and MECKE [8; 12.6.6 and 12.6.41. In our case we can find a branching system which is almost surely stable but almost surely not second order stable, for instance, consider Propositions 3.3 and 3.5 with d = 3 (getting the transience of O A ) , = 2 and u = 314.

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