Almost sure central limit theorem for branching random walks in random environment

0.26 MB·English

by  Makoto Nakashima

TheAnnalsofAppliedProbability
2011,Vol.21,No.1,351–373
DOI:10.1214/10-AAP699
(cid:13)c InstituteofMathematicalStatistics,2011
ALMOST SURE CENTRAL LIMIT THEOREM FOR BRANCHING
RANDOM WALKS IN RANDOM ENVIRONMENT
1
1 By Makoto Nakashima
0
2 Kyoto University

n
a We consider the branching random walks in d-dimensional in-
J teger lattice with time–space i.i.d. offspring distributions. Then the
6 normalizationofthetotalpopulationisanonnegativemartingaleand
it almost surely converges to a certain random variable. When d≥3

] andthefluctuationofenvironmentsatisfiesacertain uniformsquare
R
integrability then it is nondegenerate and we prove a central limit
P theorem for the density of the population in terms of almost sure
. convergence.
h
t
a
m 1. Introduction. We write N= 0,1,2,..., , N∗ = 1,2,..., and Z=
[ x:x N . For x = (x1,...,xd) {Rd, x sta}nds for{the ℓ1-n}orm: x =
{± ∈ } ∈ | | | |
d x . For ξ=(ξ ) RZd, ξ = ξ . Let (Ω, ,P) be a prob-
1 i=1| i| x x∈Zd ∈ | | x∈Zd| x| F
v abilityspace.WewriteE[X]= XdP andE[X:A]= XdP forarandom
6 P P A
variable X and an event A. We denote the constants by C,C .
i
7 R R
Weconsiderthebranchingrandomwalksinrandomenvironment.Branch-
1
1 ing random walks have been much studied [1, 2] and a central limit theorem
1. forthedensity ofthepopulationhasbeenprovedinthenonrandomenviron-
0 ment case [2]. Also, in the random environment case, one has been proved
1
in the sense of “convergence in probability” [20] when d 3 and the fluctu-
1 ≥
ation of environment is well moderated by the random walk. In this article
:
v
we prove a central limit theorem in the sense of “almost sure convergence”
i
X under the same condition as in [20]. The time–space continuous counterpart
r isthebranchingBrownianmotioninrandomenvironmentforwhichthecen-
a
tral limit theorem has been proved in [15]. On the other hand, a localization
property has been proved in [10] for the branching random walks in random
environment if the randomness of the environment dominates.
Received March 2009; revised September2009.
AMS 2000 subject classifications. Primary 60K37; secondary 60K35, 60F05, 60J80,
60K35, 82D30.
Key words and phrases. Branching random walk, random environment, central limit
theorem, linear stochastic evolutions, phase transition.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2011,Vol. 21, No. 1, 351–373. This reprint differs from the original in pagination
and typographic detail.
1
2 M. NAKASHIMA
It has been mentioned that the branching random walks in random en-
vironment (BRWRE) have a similar structure to the directed polymers in
random environment (DPRE) [3, 5, 7, 20]. Also, we will see the relation
between BRWRE and DPRE in Section 1.3. A central limit theorem has
been proved for a Markov-chain-generalization of the directed polymers in
random environment [4, 6, 12, 13] assuming a certain square integrability.
Since we use an analogy to [13], we extend the framework to contain the
branching random walks in random environment.
1.1. Branching random walks in random environment. We consider par-
ticlesinZd,performingrandomwalksandbranchingintoindependentcopies
at each step of the random walk:
(i) At time t=0 there is one particle at the origin x=0.
(ii) When a particle is located at site x Zd at time t N, it moves to a
∈ ∈
uniformly chosen nearest neighbor site and is replaced at time t+1 by
k-particles with probability q (k)(k N),
t,x
∈
where we assume that the offspring distributions qt,x=(qt,x(k))k∈N are i.i.d.
in time–space (t,x). This model is investigated in [3] and we call it the
branching random walks in random environment (BRWRE). Let N be
t,y
the number of the particles which occupy the site y Zd at time t. Let N
t
∈
bethetotal populationat time t.Inthis article westudythebehavior of the
densityρ (y)= Nt,y 1 .Welookatthebranchingprocesstogiveamore
t Nt {Nt>0}
precise definition of the branching random walks in random environment.
First, we define Vn,n N, N by
∈ V
V = 1 , V =(N∗)2,...,V =(N∗)n+1 for n 1,
0 1 n
{ } ≥
N= Vn.
V
n∈N
[
Then we label all particles as follows:
(i) At time t=0 there exists just one particle which we call 1 V .
0
∈
(ii) A particle which lives at time t is identified with a genealogical chart
y=(1,y ,...,y ) V .Iftheparticleygivesbirthtok particlesattime
1 t t y
∈
t,thenthechildrenarelabeledby(1,y ,...,y ,1),...,(1,y ,...,y ,k )
1 t 1 t y
∈
V .
t+1
By using this naming procedure we rigorously define the branching random
walks in random environment. This definition is based on the one in [20].
Note that the particle with name x can be located at x anywhere in Zd.
As both information genealogy and place are usually necessary together, it
is convenient to combine them to =(x,x); think of x and x written very
x
closely together.
ALMOSTSURECLT FOR BRANCHINGRW IN RANDOMENVIRONMENT 3
Spatial motion. A particle at time–space location (t,x) is supposed to
•
jump to some other location (t+1,y) and is replaced there by its children.
Therefore, the spatial motion should be described by assigning a destina-
tion for each particle at each time–space location (t,x). So we are guided
to the following definition. Let the measurable space (Ω , ) be the set
X X
F
(Zd)N×Zd×VN with the product σ-field and Ω X =(Xy ) .
X ∋ t,x (t,x,y)∈N×Zd×VN
We define P (Ω , ) as the product measure such that
X X X
∈P F
1
y , if e =1,
PX(Xt,x=e)= 2d | |
(0, if e =1
| |6
for e Zd and (t,x,y) N Zd N. Here we interpret Xy as the step at
∈ ∈ × ×V t,x
time t+1 of the particle y at time–space location (t,x).
Offspring distribution. We set Ω = (N)N×Zd where (N) denotes the
q
set•of probability measure of N: P P
(N)= q=(q(k))k∈N [0,1]N; q(k)=1 .
P ∈
(cid:26) k∈N (cid:27)
X
Thus each q Ωq is a function (t,x) qt,x =(qt,x(k))k∈N from N Zd to
(N). We int∈erpret q as the offspri7→ng distribution for every part×icle oc-
t,x
cPupying the time–space location (t,x). The set (N) is equipped with the
P N
natural Borel σ-field introduced from that of [0,1] . We denote by the
q
F
product σ-field on Ω .
q
We define the measurable space (Ω , ) as the set NN×Zd×VN with the
K K
y F
product σ-field and Ω K =(K ) . For each fixed q Ω
K ∋ t,x (t,x,y)∈N×Zd×VN ∈ q
q
we define P (Ω , ) as the product measure such that
K ∈P K FK
PKq(Kty,x=k)=qt,x(k) for all (t,x,y)∈N×Zd×VN and k∈N.
We interpret Ky as the number of the children born from the particle y at
t,x
time–space location (t,x).
We now define the branching random walks in random environment. We
fix a product measure Q (Ω , ), which describes the i.i.d. offspring
q q
∈P F
distribution assigned to each time–space location. In the following we also
use Q as Q-expectation, that is, we write Q[Y]= Y dQ and Q[Y :A]=
Y dQ for a q-random variable Y and an -measurable set A. Finally, we
A Fq R
define (Ω, ) by
R F
Ω=Ω Ω Ω , =
X K q X K q
× × F F ⊗F ⊗F
and Pq,P (Ω, ) for q Ω by
q
∈P F ∈
Pq =P Pq δ , P = Q(dq)Pq.
X ⊗ K ⊗ q
Z
4 M. NAKASHIMA
We want to look at N but here we investigate more detailed information.
t,y
y
We define N by
t,x
(1.1) Ny =1 the particle y is located at time–space location (t,x)
t,x { }
for (t,x,y) N Zd N. Here we set Ny = δ1,y where δ is the Dirac
∈ × ×V 0,x 0,x
function such that
δ1,y = 1, if x=0,y=1∈V0,
0,x 0, otherwise.
(cid:26)
y
Then we can describe N inductively by
t,x
(1.2) Ny = Nx 1 y x=Xx ,1 y/x Kx for t 1,
t,y t−1,x { − t−1,x ≤ ≤ t−1,x} ≥
xX∈Zd
x∈VN
where y/x is given for x,y N as follows:
∈V
k, if x=(1,x ,...,x ) V ,y=(1,x ,...,x ,k) V
1 n n 1 n n+1
y/x= for some n N, ∈ ∈
( , otherwise. ∈
∞
Moreover, N and N can be rewritten respectively as
t,y t
y y
(1.3) N = N and N = N
t,y t,y t t,y
yX∈VN yX∈Zd
y∈VN
for t N,y Zd. We remark that the total population is exactly the clas-
sical ∈Galton∈–Watson process if q q, where q (N) is nonrandom. For
t,x
≡ ∈P
simplicity we write (1.2) as
(1.4) Ny = Nx Ax,y for t 1,
t,y t−1,x t,x,y ≥
xX∈Zd
x∈VN
where we set
Ax,y =1 y x=Xx ,1 y/x Kx .
t,x,y { − t−1,x ≤ ≤ t−1,x}
This formula is similar to the one for directed polymers in random environ-
ment and linear stochastic evolutions. For p>0, we write
m(p)=Q[m(p)] with m(p)= kpq (k),
t,x t,x t,x
k∈N
X
m=m(1), m =m(1).
t,x t,x
We set
(1.5) Ny =Ny /mt, N =N /mt and N =N /mt
t,y t,y t,y t,y t t
ALMOSTSURECLT FOR BRANCHINGRW IN RANDOMENVIRONMENT 5
for(t,y,y) N Zd N.WeprovelaterthatNt isamartingalewithrespect
to Ft =σ(A∈s:s×≤t)×wVhere At ={Axt,,xy,y:(x,x),(y,y)∈N×VN}. Therefore,
the following limit always exists (see Theorem 1.2):
N = lim N , P-a.s.
∞ t
t→∞
It is easy to see that
y/x def x,y
a =a = E[A ]
/ y−x 1,x,y
(1.6) y x
1
q(j), if x y =1,y/x=k,k N∗,
= 2d | − | ∈
 j≥k
X
0, otherwise,
and

ayy−/xx=m for x∈Zd,x∈VN,
y∈ZXd,y∈VN
where q(j) is the Q-expectation of q (j).
t,x
1.2. Properties. In this section we look through the properties of BR-
y
WRE. First we introduce an important Markov chain and represent N by
t,y
using it. We define the Markov chain (S,PSx)=((S1,S2),PSx1,x,S2) on Zd×VN
for =(x,x) Zd N, independent of At t≥1, by
x ∈ ×V { }
PSx(S0=x)=PSx1,x,S2(S0=(x,x))=1, (x,x)∈Zd×VN
and for each x,y Zd,x,y N,
∈ ∈V
P (S =(y,y)S =(x,x))
S t+1 t
|
1 q(j)
j≥k , if x y =1,y/x=k N∗,
=
2d m | − | ∈
 P
0, otherwise.

We remark that we can regard S1 and S2 as independent Markov chains

on Zd and N, respectively, and that S1 is a simple random walk on Zd.
V
Here we introduce a certain martingale which is essential to the proof of our
results:
ASs
ζ =1 and for t 1 ζ = s,Ss−1 ,
0 t
≥ a
1≤s≤t Ss/Ss−1
Y
where ASt =ASt2−1,St2 and a =aSt2/St2−1 . In fact, this is a martin-
t,St−1 t,St1−1,St1 St/St−1 St1−St1−1
gale with respect to the filtration defined by =σ(A ,S ;u t) as in the
t u u
H ≤
following lemma where for t=0, =σ(S ).
0 o
H
6 M. NAKASHIMA
Lemma 1.1. ζ is a martingale with respect to . Moreover, we have
t t
H
that
(1.7) Nty,y =mtES0[ζt;St=(y,y)], P-a.s. for t∈N,(y,y)∈Zd×VN,
where E0[] denotes the expectation with respect to P0d=efP0,1.
S · S S
Proof. Since A are i.i.d. random variables, it follows from the
t t≥1
{ }
independence of A and S that
t t≥1 t t≥1
{ } { }
AS1
E0 [ζ ]=E0 [ζ ,S ,...,S ]ESt−1 t,S0
A,S t|Ht−1 A,S t−1|Ft−1 1 t−1 A,S a
(cid:20) S1/S0(cid:21)
=ζ , P0 -a.s.,
t−1 A,S
where P is the product probability measure of P and P and where
Ax,S Sx
E denotes the expectation with respect to P . We now prove (1.7) by
Ax,S Ax,S
induction. It is easy to see that (1.7) holds for t=0. If (1.7) holds for t 0,
≥
then
Ny = Nx Ax,y
t+1,y t,x t+1,x,y
x∈ZXd,x∈VN
= mtE0[ζ ;S =(x,x)]Ax,y .
S t t t+1,x,y
x∈ZXd,x∈VN
Since we have that
AS1
Ax,y =mEx,x t+1,S0;S =(y,y) ,
t+1,x,y S a 1
(cid:20) S1/S0 (cid:21)
(1.7) holds for t+1 and the proof is complete. (cid:3)
We remark that N =mtE0[ζ ;S1=y]. From this lemma we obtain an
t,y S t t
important result. The following theorem means that a phase transition oc-
curs for the growth rate of the total population.
Theorem 1.2. N is a martingale with respect to =σ(A :s t) and
t t s
F ≤
there exists the limit
(1.8) N = lim N , P-a.s.,
∞ t
t→∞
and
(1.9) E[N ]=1 or 0.
∞
Moreover, E[N ]=1 if and only if the limit (1.8) is convergent in L1(P).
∞
ALMOSTSURECLT FOR BRANCHINGRW IN RANDOMENVIRONMENT 7
BeforeweproveTheorem1.2weintroducesomenotationsanddefinitions.
For(s,z,z)∈N×Zd×VN,wedefineNts,z,z=(Nts,,yz,,yz)(y,y)∈Zd×VN andNst,z,z=
(Ns,z,z) , t N, respectively by
t,y,y (y,y)∈Zd×VN ∈
Ns,z,z=δy,z= 1, if y=z, and y=z,
0,y,y y,z 0, otherwise,
(cid:26)
s,z,z s,z,z x,y
(1.10) N = N A and
t+1,y,y t,x,x s+t+1,x,y
x∈ZXd,x∈VN
Ns,z,z=m−tNs,z,z.
t,y,y t,y,y
s,z,z s,z,z
We remark that we can regard N = N as the state of
t { s,y,y}(y,y)∈Zd×VN
the branching random walks starting from particle z at time–space (s,z)
observed at time s+t.
Proof of Theorem 1.2. The limit (1.8) exists by the martingale con-
def
vergence theorem since N is a nonnegative martingale and ℓ = E[N ] 1
t ∞
≤
by Fatou’s lemma. To show (1.9) we will prove that ℓ=ℓ2 using the argu-
ment in [11]. With the notation (1.10) we write
z s,z,z
N = N N ,
s+t s,z t
z∈ZXd,z∈VN
Nwh×erZedN×zsV,zNi,sNdestfi,zn,zedisbaym(1a.5r)tifnogra(lse,wz,izth)∈reNsp×ecZtdt×oVFNts.=Siσnc(Aees+veur:yu(≤s,zt),za)n∈d
has the same distribution as N , the limit
t
s,z,z s,z,z
N = lim N
∞ t
t→∞
exists almost surely and is identically distributed as N . Moreover, by let-
∞
ting t , we have that
→∞
z s,z,z
N = N N
∞ s,z ∞
z∈ZXd,z∈VN
and hence, by Jensen’s inequality, that
E[exp( N ) ] exp( E[N ])=exp( N ℓ) exp( N ).
∞ s ∞ s s s
− |F ≥ − |F − ≥ −
By letting s in the above inequality, we obtain
→∞
a.s.
exp( N ) exp( N ℓ) exp( N )
∞ ∞ ∞
− ≥ − ≥ −
and thus N a=.s.N ℓ. By taking expectation we get ℓ=ℓ2. Once we know
∞ ∞
(1.9), the final statement of the theorem is standard (e.g., [9], formula (5.2),
pages 257–258). (cid:3)
8 M. NAKASHIMA
WerefertothecaseE[N ]=1astheregular growth phase andtotheone
∞
E[N ]=0 as the slow growth phase. The regular growth phase means that
∞
the growth rate of the total population is the same order as its expectation
mt and the slow growth phase means that, almost surely, the growth rate is
slower than the growth rate of its expectation.
We discuss the case of the regular growth phase in this article. The slow
growth phaseispartially studiedin[10].If N isuniformlysquareintegrable
t
then it is the regular growth phase since N is a martingale.
t
Here we give the main theorem in this article.
Theorem 1.3. Suppose that d 3 and
≥
Q[(m )2] 1
(1.11) m>1, m(2)< and t,x < ,
∞ m2 π
d
where π is the return probability of a simple random walk in Zd. Then for
d
all f C (Rd),
b
∈
x
(1.12) lim f N =N f(x)dν(x), P-a.s.,
t,x ∞
t→∞xX∈Zd (cid:18)√t(cid:19) ZRd
where C (Rd) stands for the set of bounded continuous functions on Rd and
b
ν is the Gaussian measure with mean 0 and covariance matrix 1I.
d
The proof of Theorem 1.3 will be given in the next section.
Remark. From Lemma 1.1 we can rewrite (1.12) as
S1
(1.13) lim E0 f t ζ =N f(x)dν(x), P-a.s.
t→∞ S(cid:20) (cid:18)√t(cid:19) t(cid:21) ∞ZRd
In Lemma 2.2 we see that (1.11) is equivalent to sup E[(N )2]< so
t≥1 t
∞
that we have E[N ]=1, that is, P(N >0)>0. Also, if we set ρ (x)=
∞ ∞ t
Nt,x1 N >0 , we can interpret ρ (x) as the density of the particles. From
Nt { t } t
this observation we can regard Theorem 1.3 as a central limit theorem for
probability measures with the density ρ (√tx) on N >0 .
t ∞
{ }
1.3. Relation to directed polymers in random environment. Inthe end of
this section we discuss the relation between BRWRE and DPRE (see [20],
pages 1631–1634, for more detailed information).
Random walk.(S′,Px)isasimplerandomwalkond-dimensionallattice
• t S′
defined on the canonical path space (Ω , ). Px is the unique probability
S′ FS′ S′
measure on (Ω , ) such that S′ S′,...,S′ S′ are independent and
S′ FS′ 1− 0 t− t−1
1
, if e =1,
PSx′(S0′ =x)=1, PSx′(St′+1−St′=e)= 2d | |
(0, if e =1,
| |6
ALMOSTSURECLT FOR BRANCHINGRW IN RANDOMENVIRONMENT 9
where e Zd. For x = 0 we write simply P as P0 . We denote by Ex
∈ S′ S′ S′
the Px-expectation. We can regard (S′,Px) as an independent copy of
S′ t S′
(S1,Px ).
t S1
Random environment. η= η :(t,x) N Zd are R-valued i.i.d. ran-
t,x
• { ∈ × }
dom variables which are nonconstant and defined on a probability space
(Ω , ,Q′) such that
η η
F
eλ(β)d=efQ′[exp(βη )]< for all β R.
t,x
∞ ∈
Polymer measure. For any t N, define the polymer measure µ on
t
• ∈
(Ω , ) by
S′ S′
F
1
dµ = exp(βH tλ(β))dP ,
t t S′
Z −
t
where β>0 is a parameter and
t−1
H = η and Z =E0 [exp(βH tλ(β))]
t u,Su′ t S′ t−
u=0
X
are the Hamiltonian and the partition functions. We call this system the
directed polymers in random environment.
Coming back to BRWRE, fix the environment q= q ;(t,x) N Zd
t,x
with m >0, Q-a.s. Set exp(βη )=m for each (t,x{) N Zd∈. Th×en w}e
t,x t,x t,x
∈ ×
have from Lemma 1.1 that
Eq[N ]=E0 [exp(βH tλ(β)):S′=x] and Eq[N ]=Z ,
t,x S′ t− t t t
where Eq[] denotes the expectation with respect to Pq since we have that
·
1
q (j), if x y =1,y/x=k,
Eq[Ay,y ]= 2d t−1,x | − |
t,x,x  Xj≥k
0, otherwise.
Here we remark thatλ(β)=log(m) so we can construct DPRE from BR-
WRE. In [20] we find the converse, that is, how to construct i.i.d. random
offspring distributions q from the environment η .
t,x t,x
2. Proof of Theorem 1.3.
2.1. Preparation. First we introduce some useful notations. We define
w( ,˜, ,˜) for =(x1,x2),˜=(x˜1,x˜2), =(y1,y2),˜=(y˜1,y˜2) N N by
x x y y x x y y ∈ ×V
E[Ax2,y2 Ax˜2,y˜2 ]
w( ,˜, ,˜)= 1,x1,y1 1,x˜1,y˜1
x x y y a a
/ ˜/˜
(2.1) y x y x
10 M. NAKASHIMA
0, if a a =0,
/ ˜/˜
1, if xy =x xy˜x,a a =0,
1 1 / ˜/˜
0, if =6 ˜,y =yy˜x ,yax6 a =0,
 1 x x 16 −11 y/x ˜y/˜x6

 q(j)
 2d
(cid:20) j≥min{k,l} (cid:21)
 X
= if xy2=/x˜x,2y=1=k,yy˜˜12,/x˜2=l,ay/xa˜y/˜x6=0,
E[ q (i) q (j)]
i≥k 0,0 j≥l 0,0


 P i≥kq(i) Pj≥lq(j)

 x =x˜ ,x =x˜ ,
 Pif y12/xP21=k2,y˜6 2/x˜22=l,ay/xa˜y/˜x6=0.



Let(S˜t,PS˜)beanindependentcopyof(St,PS)andPSx,,S˜x˜ betheproductmea-
sureof PSx and PS˜˜x for x,˜x∈Zd×VN.Thenwehave thefollowing Feynmann–
Kac formula:
Lemma 2.1. For t N,(y,y),(y˜,y˜) Zd N,
∈ ∈ ×V
(2.2) E[Ny Ny˜ ]=E0,0[e ;(S ,S˜)=((y,y),(y˜,y˜))],
t,y t,y˜ S,S˜ t t t
where e is defined by
t
t
e = w(S ,S˜ ,S ,S˜ ).
t u−1 u−1 u u
u=1
Y
Proof. From Lemma 1.1 we can write that
Ny Ny˜ =E0[ζ ;S =(y,y)]E0[ζ ;S =(y˜,y˜)]
t,y t,y˜ S t t S t t
=E0,0[ζ ζ˜;(S ,S˜)=((y,y),(y˜,y˜))],
S,S˜ t t t t
where ζ˜ is ζ defined by S˜ . It is easy to see that the P-expectation of
t t t
the right-hand side coincides with the right-hand side of (2.2) from Fubini’s
theorem. (cid:3)
By using this formula we can represent the uniform square integrability
of N in terms of the environment, that is, q ;(t,x) N Zd . This is the
t t,x
{ ∈ × }
same condition as in [20].
Lemma 2.2. Suppose d 3. Then the following are equivalent.
≥
(i) sup E[(N )2]< .
t≥1 t ∞