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(1)

Contents lists available atScienceDirect

## 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

a

b

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 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 rateβy2, whereβ > 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.

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

y2, where

### β >

0 andyis the spatial position of the particle.

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

Nt,Yu

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

y2

R

### )

-BBM.

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

y

pfor p

### >

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

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

Nt

### |

.

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

Nt

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).

doi:10.1016/j.spl.2010.05.011

(2)

We defineRt

maxuNtYu

t

### )

to be the right-most particle in the

y2

R

### )

-BBM. InHarris and Harris(2009) it was shown that

tlim→∞

lnRt 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

Nt

### |

is dominated by particles with spatial positions near Rt. 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 nearRtwill 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

Nt,

ln

Yu

t

p 2

t

for allt

### > τ.

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

t

e2

2β+ε

tfor 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

Nt

### }

whose positions are denoted by

Yu

t

u

Nt

. Until time

### τ

the two processesNt andN

### ¯

tcoincide and for allt

we will haveNt

### ⊆ ¯

Nt. Furthermore, we want that after time

### τ

all particles inN

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

Nt

t

. Fort

, each particleu

### ∈

Ntgives birth to an extra particle

Nt

Ntat rate

t

Yu

t

2(note that ifu

NtthenYu

t

Yu

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 inN

tbranch at rate

tfort

### ≥ τ

. It is furthermore clear that

Nt

Nt

for allt

### ≥

0. The upshot is that after time

,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

t

t

, and define

Vt

inf

s

Z s

τ

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

ZVtet

W ast

### → ∞

almost surely, whereWhas an exponential mean 1 distribution. However, sinceRVt

τ

udu

### =

twe see that this implies Ztexp

Z t

τ

sds

W ast

AsN

tfort

### ≥ τ

is composed of the union of the offspring of the

Nτ

### |

particles present at time

### τ

, we can write Mt

Nt

exp

Z t

τ

sds

|Nτ|

X

i=1

Zt(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

Nτ

### |

(which is almost surely finite), we see that

Mt

M

0

ast

### → ∞

where, more precisely,M

### ¯

is distributed as the sum of

Nτ

### |

independent exponential variables with mean 1.

Thus

ln

Nt

Z t

τ

sds

lnM

ast

(3)

and sinceRt

τ

sds

1

c

exp

ct

exp

c

withc

2

2

### β + ε

, it follows that

tlim→∞

ln ln

Nt

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

Nt

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

0

### < ∞

such that for allt

0,

Rt

e

2βε

t

(2)

Fix

0 and lettn

n

n

0

1

2

### , . . . }

. We will show that it is sufficient to consider only the offspring of the rightmost particlesRtnto 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 fromRtnduring the time interval

tn

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.

LetNs(n)be the set of descendants ofRtnat times

tn

tn+1

### ]

. As for the upper bound we introduce a coupled process

eNs(n)

s

tn

tn+1

### ] )

. This time we have that for alls

tn

tn+1

eNs(n)

Ns(n)

### ⊆

Ns. More precisely,

eNs(n)

s

tn

tn+1

is obtained from

Ns(n)

s

tn

tn+1

### ] )

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

eNs(n)

### ⊆

Ns(n)splits in the original processNs(n), it also splits ineNs(n)with probability

e

n

Yu

s

Yu

s

2

0

1

where

e

n

x

e2

2β2ε

tn ifx

e

2β2ε

tn

0 otherwise

### .

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

eNs(n)

tn

s

tn+1

### )

is thus a BBM started at timetn from a single particle at positionRtn with space- dependent branching ratee

### β

n. Observe that we trivially haveeNs(n)

Ns(n)for alls

tn

tn+1

### ]

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

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

eNs(n)

s

tn

tn+1

### ] )

. This process is denoted

bNs(n)

s

tn

tn+1

### ] )

, and is defined by adding particles to

eNs(n)

s

tn

tn+1

### ] )

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

e2

2β2ε

### )

tn, irrespective of its spatial position.

More specifically, fors

tn

tn+1

### ]

, every particleu

### ∈

eNs(n)gives birth to an extra particle

bNs(n)

eNs(n)at rate

e2

2β2ε

tn1

Yu(s)<e

2β−2ε

tn

### ,

and each particle thus created initiates an independent BBM in

bNs(n)

s

tn

tn+1

### ] )

with constant breeding rate

e2

2β2ε

### )

tn. Note that we haveeNs(n)

bNs(n)for alls

tn

tn+1

.

Now for

### v ∈

Nt(nn+)1define the event An

min

s∈[tn,tn+1]Yv

s

e

2β2ε

tn

and setAn

v∈eN(n) tn+1

An

### (v)

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

### v

ofRtn(in the modified processeN(n)) such that Yv

s

e

2β2ε

tnfor somes

tn

tn+1

. Observe that

n

eNt(nn+)1

bNt(nn+)1 o

An

### .

(3)

We also define the eventBnas Bn

n

Rtn

e

2βε

tno

and let

Ft

### }

t0be the natural filtration for the

y2

R

### )

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

(4)

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 An

An

Bn

An

Bcn

### ),

and we recall from Eq.(1)thatP

lim supBcn

### ) =

0, and hence only finitely many of the eventsAn

### ∩

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

An

Bn

Ftn

### )

.

LetPxbe the law of a driftless Brownian motionYstarted at the pointx

### ∈

R(andExthe expectation with respect to this law). Observing thatBn

Ftn, we have

1BnP

An

Ftn

1BnE

 X

v∈eN(n) tn+1

1An(v)

Ftn

1BnERtn

exp Z δ

0

e

n

Ys

ds

min

s∈[0]Ys

e

2β2ε

tn

exp

e2

2β2ε

tn

Pe

2βε

tn min

s∈[0]Ys

e

2β2ε

tn

### .

Using the reflection principle, we obtain that there existsC

0 such that

Pe

2βε

tn min

s∈[0]Ys

e

2β2ε

tn

2P0

Yδ

e

2βε

tn

eεtn

1

Cexp

1 2

e

2βε

tn eεtn

12

### .

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

An

Bn

Ftn

### )

, and so we have shown that

X

n0

P

An

Bn

Ftn

### ) < ∞

almost surely. SinceAn

Bn

### ∈

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 eventsAn

### ∩

Bnoccur. Thus only finitely many of the eventsAn occur, almost surely.

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

### < ∞

such that for alln

### >

n1, we have eNs(n)

bNs(n)for alls

tn

tn+1

### ]

. Recall also that the process

bNs(n)

s

tn

tn+1

### ] )

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

e2

2β2ε

### )

tn. Hence the population size

Ns(n)

### |

is a Yule process with constant breeding rate

e2

2β2ε

tn, fors

tn

tn+1

. Therefore

Nt(nn+)1

### |

has a geometric distribution with parameter

pn

exp

e2

2β2ε

tn

If we define

qn

l exp

e2

2β3ε

tnm

### ,

then the probability that the number of particles inbNt(nn+)1is smaller thanqnis P

Nt(nn+)1

qn

1

1

pn

qn

pnqn

o

pnqn

### ).

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

such that

Nt(nn+)1

qnfor alln

### >

n2.

Finally, for alln

max

n1

n2

and allt

tn+1

tn+2

### ]

, we have that there are at least as many particles inNtas there are descendants ofRtnat timetn+1, which is to say that

Nt

Nt(nn+)1

### |

. Hence, fornsufficiently large,

ln ln

Nt

t

ln ln

Nt(nn+)1

tn

tn t

2p

2

7

### ε,

(4)

almost surely. (Certainly we must haven

max

n1

n2

### }

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

### /

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

Nt

t

2p

2

### β,

almost surely, which completes the proof.

(5)

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 toLp-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.

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