The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential

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Statistics and Probability Letters

journal homepage:www.elsevier.com/locate/stapro

The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential

J. Berestycki

^a

, É. Brunet

^b

, J.W. Harris

^c,∗

, S.C. Harris

^d

aLaboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie (UPMC Paris VI), CNRS UMR 7599, 175 rue du Chevaleret, 75013 Paris, France

bLaboratoire de Physique Statistique, École Normale Supérieure, UPMC Université Paris 6, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France

cDepartment of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

dDepartment of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

a r t i c l e i n f o

Article history:

Received 10 February 2010

Received in revised form 14 May 2010 Accepted 17 May 2010

Available online 25 May 2010 MSC:

60J80 Keyword:

Branching Brownian motion

a b s t r a c t

In this note we consider a branching Brownian motion (BBM) onRin which a particle at spatial positionysplits into two at rateβ^{y2, where}β > 0 is a constant. This is a critical breeding rate for BBM in the sense that the expected population size blows up in finite time while the population size remains finite, almost surely, for all time. We find an asymptotic for the almost-sure rate of growth of the population.

©2010 Elsevier B.V. All rights reserved.

1. Introduction

We consider a branching Brownian motion with a quadratic breeding potential. Each particle diffuses as a driftless Brownian motion onR, and splits into two particles at rate

β

^{y2, where}

β >

^{0 andy}is the spatial position of the particle.

We letN_t be the set of particles alive at timet, and then, for eachu

∈

N_t,Y_u

(

^t

)

is the spatial position of particleuat time t(and for 0

≤

s

<

^t

,

^Yu

(

^s

)

is the spatial position of the unique ancestor ofualive at times). We will call this process the

(β

^y2

;

_R

)

^-BBM.

It is known that quadratic breeding is a critical rate for population explosions. If the breeding rate were instead

β |

y

|

^pfor p

>

2, the population size would almost surely explode in a finite time. However, for the

(β

^y2

;

_R

)

^{-BBM theexpectednumber} of particles blows up in a finite time while the total number of particles alive remains finite almost surely, for all time. For p

∈ [

0

,

²

)

the expected population size remains finite for all time. SeeItô and McKean(1965,pp. 200–211) for a proof of these facts using solutions to related differential equations. In this note we prove the following result on the almost-sure rate of growth of

|

N_t

|

.

Theorem 1. Suppose that the initial configuration consists of a finite number of particles at arbitrary positions inR. Then, almost surely,

lim

t→∞

ln ln

|

N_t

|

t

=

2p

2

β.

∗Corresponding author. Tel.: +44 0 7812 107318.

E-mail addresses:julien.berestycki@upmc.fr(J. Berestycki),eric.brunet@lps.ens.fr(É. Brunet),john.harris@bristol.ac.uk(J.W. Harris), s.c.harris@bath.ac.uk(S.C. Harris).

0167-7152/$ – see front matter©2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.spl.2010.05.011

We defineR_t

:=

max_u∈NtY_u

(

^t

)

to be the right-most particle in the

(β

^y2

;

_R

)

^{-BBM. In}Harris and Harris(2009) it was shown that

tlim→∞

lnR_t t

=

p

2

β

⁽¹⁾

almost surely, and this result is crucial in our proof ofTheorem 1because it allows us both some control over the maximum breeding rate in the BBM, and also to show that the growth rate of

|

N_t

|

is dominated by particles with spatial positions near R_t. Indeed, a BBM in which every particle has branching rate

β

^R2t would see its population grow as inTheorem 1, and this point will provide our upper bound. Furthermore we shall see that, for any

δ >

0, at all sufficiently large timest, a single particle located nearR_twill single-handedly build during time interval

[

t

,

^t

+ δ ]

a progeny so large that it is again of the magnitude given byTheorem 1, which will prove the lower bound.

2. Proof of the growth rate

To proveTheorem 1, it is sufficient to consider the case of the initial configuration of particles being a single particle at positionx

∈

_R.

Upper bound. Since the breeding potential is symmetric about the origin, we have from Eq.(1)that for

ε >

0 fixed, there exists, almost surely, a random time

τ < ∞

such that for allu

∈

N_t,

ln

|

Y_u

(

^t

) | <

p 2

β + ε

t

,

^{for allt}

> τ.

As a consequence of this, the breeding rate of any particle in the population is bounded by

β ¯

t

:= β

^e2

(

^√2β+ε

)

^t_{for allt}

> τ

^. We now introduce a coupled branching process

(

_N

¯

t

,

^t

≥

0

)

as follows. For eacht

≥

0 it consists in a population of particles

{

u

:

u

∈ ¯

N_t

}

whose positions are denoted by

{ ¯

Y_u

(

^t

) :

u

∈ ¯

N_t

}

. Until time

τ

the two processesN_t and_N

¯

tcoincide and for allt

≥ τ

we will haveN_t

⊆ ¯

N_t. Furthermore, we want that after time

τ

all particles in_N

¯

tbranch at rate

β ¯

t. In order for this to make sense we must construct

(

_N

¯

t

,

^t

≥

0

)

conditionally on

(

^Nt

,

^t

≥

0

)

^{. Given}

(

^Nt

,

^t

≥

0

)

^{, we let}

(

_N

¯

t

,

^t

≤ τ) = (

^Nt

,

^t

≤ τ)

^{. Fort}

≥ τ

, each particleu

∈

N_tgives birth to an extra particle

v ∈ ¯

N_t

\

N_tat rate

β ¯

t

− β

^Yu

(

^t

)

^2(note that ifu

∈

N_tthenY_u

(

^t

) = ¯

Y_u

(

^t

)

). Each particle thus created starts an independent BBM in

(

_N

¯

t

,

^t

≥

0

)

with time-dependent branching rate

β ¯

tat timet. Thus it is clear that all particles in_N

¯

tbranch at rate

β ¯

tfort

≥ τ

^. It is furthermore clear that

|

N_t

| ≤ | ¯

N_t

|

for allt

≥

0. The upshot is that after time

τ

^,_N

¯

tis a pure birth process, and as such is very well studied.

Consider

(

^Zt

,

^t

≥ τ)

a pure birth process starting with a single particle at time

τ

with inhomogeneous rate

( β ¯

t

,

^t

≥ τ)

^, and define

V_t

:=

inf

s

≥ τ :

Z s

τ

β ¯

udu

≥

t

.

It is easily seen that

(

^ZVt

,

^t

≥

0

)

is simply a Yule process (i.e. a pure birth process where particles split into two at rate 1) and so we know that

Z_Vte^−t

→

W∞ ast

→ ∞

almost surely, whereW∞has an exponential mean 1 distribution. However, sinceRVt

τ

β ¯

udu

=

twe see that this implies Z_texp

−

Z t

τ

β ¯

sds

→

W∞ ast

→ ∞ .

As_N

¯

tfort

≥ τ

is composed of the union of the offspring of the

|

N_τ

|

particles present at time

τ

, we can write M_t

:= | ¯

N_t

|

exp

−

Z t

τ

β ¯

sds

=

|Nτ|

X

i=1

Z_t⁽ⁱ⁾exp

−

Z t

τ

β ¯

sds

where theZ⁽ⁱ⁾are independent, identically distributed, copies of a pure birth process with birth rate

β ¯

t. By conditioning on the value of

|

N_τ

|

(which is almost surely finite), we see that

M_t

→ ¯

M∞

∈ (

⁰

, ∞ )

^ast

→ ∞

where, more precisely,_M

¯

∞is distributed as the sum of

|

N_τ

|

independent exponential variables with mean 1.

Thus

ln

| ¯

N_t

| −

Z t

τ

β ¯

sds

→

ln_M

¯

∞ ast

→ ∞

and sinceRt

τ

β ¯

sds

=

¹

c

[

exp

(

^ct

) −

exp

(

^c

τ) ]

withc

=

2

√

2

β + ε

, it follows that

tlim→∞

ln ln

| ¯

N_t

|

t

=

2p

2

β + ε ,

almost surely. Finally, since

ε

may be arbitrarily small, using the coupling constructed above, we have that lim sup

t→∞

ln ln

|

N_t

|

t

≤

2p

2

β,

almost surely, as required.

Lower bound. Fix

ε >

^{0. From}(1), we know that there exists, almost surely, a random time

τ

⁰

< ∞

such that for allt

> τ

^0,

R_t

>

^e

(

^√2β−ε

)

^t

.

⁽²⁾

Fix

δ >

^{0 and lett}n

:=

n

δ,

ⁿ

∈ {

0

,

¹

,

²

, . . . }

. We will show that it is sufficient to consider only the offspring of the rightmost particlesR_tnto prove the lower bound for the global population growth rate. There are two steps to the argument.

First, we will couple the sub-population descended fromR_tnduring the time interval

[

t_n

,

^tn+1

]

to a BBM in which the expected population size remains finite. Second, we will show that, asntends to infinity, the population sizes of the coupled processes grow sufficiently quickly. For this, we use Eq.(2)to give us a lower bound on the breeding rate in the coupled processes.

LetN_s⁽ⁿ⁾be the set of descendants ofR_tnat times

∈ [

t_n

,

^tn+1

]

. As for the upper bound we introduce a coupled process

(

eN_s⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

. This time we have that for alls

∈ [

t_n

,

^tn+1

] ,

eN_s⁽ⁿ⁾

⊆

N_s⁽ⁿ⁾

⊆

N_s. More precisely,

(

eN_s⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

^is obtained from

(

^Ns⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

by cancelling some of the split events inN_s⁽ⁿ⁾, in the following way: if at timesa particle u

∈

eN_s⁽ⁿ⁾

⊆

N_s⁽ⁿ⁾splits in the original processN_s⁽ⁿ⁾, it also splits ineN_s⁽ⁿ⁾with probability

e

β

n

(

^Yu

(

^s

))

β

^Yu

(

^s

)

²

∈ [

0

,

¹

] ,

where

e

β

n

(

^x

) :=

β

^e2

(

^√2β−2ε

)

^tn _ifx

≥

e

(

^√2β−2ε

)

^tn

,

0 otherwise

.

If the split event is rejected, one of the two offspring inN_s⁽ⁿ⁾is chosen at random to be the one which we keep ineN_s⁽ⁿ⁾. The process

(

eN_s⁽ⁿ⁾

,

^tn

≤

s

≤

t_n+1

)

is thus a BBM started at timet_n from a single particle at positionR_tn with space- dependent branching ratee

β

n. Observe that we trivially haveeN_s⁽ⁿ⁾

⊆

N_s⁽ⁿ⁾for alls

∈ [

t_n

,

^tn+1

]

, as announced, since we obtain eN_s⁽ⁿ⁾fromN_s⁽ⁿ⁾merely by erasing some particles fromN⁽ⁿ⁾.

We also require another process on the same probability space, coupled to

(

eN_s⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

. This process is denoted

(

bN_s⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

, and is defined by adding particles to

(

eN_s⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

in such a way that every particle inbN⁽ⁿ⁾breeds at constant rate

β

^e2

(

^√2β−2ε

)

^tn, irrespective of its spatial position.

More specifically, fors

∈ [

t_n

,

^tn+1

]

, every particleu

∈

eN_s⁽ⁿ⁾gives birth to an extra particle

v ∈

bN_s⁽ⁿ⁾

\

eN_s⁽ⁿ⁾at rate

β

^e2

(

^√2β−2ε

)

^tn₁

Yu(s)<e

(

^√2β−2ε

)

^tn

,

and each particle thus created initiates an independent BBM in

(

bN_s⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

with constant breeding rate

β

^e2

(

^√2β−2ε

)

^tn. Note that we haveeN_s⁽ⁿ⁾

⊆

bN_s⁽ⁿ⁾for alls

∈ [

t_n

,

^tn+1

]

.

Now for

v ∈

N_t⁽_nⁿ₊⁾₁define the event A_n

(v) :=

min

s∈[tn,tn+1]Y_v

(

^s

) <

^e

(

^√2β−2ε

)

^tn

,

and setA_n

= ∪

v∈eN(n) tn+1

A_n

(v)

, i.e., the event that there exists a descendant

v

^ofRtn(in the modified processeN⁽ⁿ⁾) such that Y_v

(

^s

) <

^e

(

^√2β−2ε

)

^tnfor somes

∈ [

t_n

,

^tn+1

]

. Observe that

n

eN_t⁽_nⁿ₊⁾₁

6=

bN_t⁽_nⁿ₊⁾₁ o

⊆

A_n

.

⁽³⁾

We also define the eventB_nas B_n

:=

n

R_tn

>

^e

(

^√2β−ε

)

^tno

,

and let

{

_Ft

}

_t≥0be the natural filtration for the

(β

^y2

;

_R

)

-BBM. Our aim is to show that, almost surely, only finitely many of the eventsA_noccur, from which it follows that the populations of the coupled subprocesseseN⁽ⁿ⁾are equal to the populations

of thebN⁽ⁿ⁾processes for all sufficiently largen, almost surely. Then the final step is to show that the populationsbN⁽ⁿ⁾grow sufficiently quickly to imply the desired lower bound on the size of the original population.

We start by writing the event of interest as A_n

= (

^An

∩

B_n

) ∪ (

^An

∩

B^c_n

),

and we recall from Eq.(1)thatP

(

^{lim supBc}n

) =

0, and hence only finitely many of the eventsA_n

∩

B^c_noccur. We now use a standard ‘many-to-one’ argument (see, for example,Hardy and Harris, 2009) to bound the probabilitiesP

(

^An

∩

B_n

|

_Ftn

)

^.

LetP^xbe the law of a driftless Brownian motionYstarted at the pointx

∈

_R(andE^xthe expectation with respect to this law). Observing thatB_n

∈

_Ftn, we have

1_BnP

(

^An

|

_Ftn

) ≤

1_BnE





 X

v∈eN(n) tn+1

1_An(v)

Ftn







=

1_BnE^Rtn

exp Z _δ

0

e

β

n

(

^Ys

)

^ds

;

min

s∈[0,δ]Y_s

<

^e

(

^√2β−2ε

)

^tn

≤

exp

β

^e2

(

^√2β−2ε

)

^tn

δ

P^e

(

^√2β−ε

)

^tn min

s∈[0,δ]Y_s

<

^e

(

^√2β−2ε

)

^tn

.

Using the reflection principle, we obtain that there existsC

>

0 such that

P^e

(

^√2β−ε

)

^tn min

s∈[0,δ]Y_s

<

^e

(

^√2β−2ε

)

^tn

=

2P⁰

Y_δ

<

^e

(

^√2β−ε

)

^tn

(

^e−εtn

−

1

)

≤

Cexp

−

¹ 2

δ

e

(

^√2β−ε

)

^tn _e^−εtn

−

12

.

Combining this series of inequalities gives, almost surely, a faster than exponentially decaying upper bound onP

(

^An

∩

B_n

|

_Ftn

)

, and so we have shown that

X

n≥0

P

(

^An

∩

B_n

|

_Ftn

) < ∞

almost surely. SinceA_n

∩

B_n

∈

_Ftn+1, Lévy’s extension of the Borel–Cantelli lemmas (seeWilliams(1991,Theorem 12.15)) lets us conclude that, almost surely, only finitely many of the eventsA_n

∩

B_noccur. Thus only finitely many of the eventsA_n occur, almost surely.

Recalling Eq.(3)we see that, almost surely, there exists a random integern₁

< ∞

such that for alln

>

ⁿ1, we have eN_s⁽ⁿ⁾

=

bN_s⁽ⁿ⁾for alls

∈ [

t_n

,

^tn+1

]

. Recall also that the process

(

bN_s⁽ⁿ⁾

,

^s

∈ [

t_n

,

^tn+1

] )

is initiated with a single particle at time t_n, and every particle in this process breeds at constant rate

β

^e2

(

^√2β−2ε

)

^tn. Hence the population size

|b

N_s⁽ⁿ⁾

|

is a Yule process with constant breeding rate

β

^e2

(

^√2β−2ε

)

^tn_{, fors}

∈ [

t_n

,

^tn+1

]

. Therefore

|b

N_t⁽_nⁿ₊⁾₁

|

has a geometric distribution with parameter

p_n

:=

exp

− δβ

^e2

(

^√2β−2ε

)

^tn

.

If we define

q_n

:=

l exp

e²

(

^√2β−3ε

)

^tnm

,

then the probability that the number of particles inbN_t⁽_nⁿ₊⁾₁is smaller thanq_nis P

( |b

N_t⁽_nⁿ₊⁾₁

| ≤

q_n

) =

1

− (

¹

−

p_n

)

^qn

=

p_nq_n

+

o

(

^pnq_n

).

Using the Borel–Cantelli lemmas again, we see that there exists almost surely a random integern₂

< ∞

such that

|b

N_t⁽_nⁿ₊⁾₁

| >

^qnfor alln

>

ⁿ2.

Finally, for alln

>

^max

{

n₁

,

ⁿ2

}

and allt

∈ [

t_n+1

,

^tn+2

]

, we have that there are at least as many particles inN_tas there are descendants ofR_tnat timet_n+1, which is to say that

|

N_t

| ≥ |e

N_t⁽_nⁿ₊⁾₁

|

. Hence, fornsufficiently large,

ln ln

|

N_t

|

t

≥

^{ln ln}

|e

N_t⁽_nⁿ₊⁾₁

|

t_n

t_n t

≥

2p

2

β −

7

ε,

⁽⁴⁾

almost surely. (Certainly we must haven

>

^max

{

n₁

,

ⁿ2

}

, but we may require thatnbe larger still in order that the multiplicative factort_n

/

^tis close enough to 1 for Eq.(4)to hold.)

We can take

ε

to be arbitrarily small, and so obtain lim inf

t→∞

ln ln

|

N_t

|

t

≥

2p

2

β,

almost surely, which completes the proof.

Acknowledgement

The third author was supported by the Heilbronn Institute for Mathematical Research.

References

Hardy, R., Harris, S.C., 2009. A spine approach to branching diffusions with applications toL^p-convergence of martingales. In: Séminaire de Probabilités XLII. In: Lecture Notes in Math., vol. 1979. Springer, Berlin, pp. 281–330.

Harris, J.W., Harris, S.C., 2009. Branching Brownian motion with an inhomogeneous breeding potential. Ann. Inst. H. Poincaré Probab. Statist. 45 (3), 793–801.

Itô, K., McKean, H.P., 1965. Diffusion Processes and their Sample Paths. In: Die Grundlehren der Mathematischen Wissenschaften, Band 125, Academic Press Inc., Publishers, New York.

Williams, D., 1991. Probability with Martingales. Cambridge University Press.