Markov branching diffusions

martingales Girsanov type theorems

and applications to the long term

behaviour

Janos Englander and Andreas E Kyprianou

Octob er

spinetex

Abstract

Consider a spatial branching particle pro cess where the underlying mo

d

tion is a conservative diusion on D R corresp onding to the elliptic op

erator L on D and the branching is strictly binary dyadic with spatially

varying rate x  and  which is assumed to b e b ounded from

ab ove Weprove that under extremely mild circumstances the pro cess

exhibits lo cal extinction if and only if  where denotes the gen

c c

eralized principal eigenvalue for the op erator L on D This criterion

is analogous to the one obtained by Pinsky for the lo cal extinction

of sup erdiusions Furthermore we show that when the pro cess do es not

exhibit lo cal extinction every nonempty op en subset is o ccupied innitely

often with p ositive whic h can b e characterized by a solution

b ounded in to the semilinear elliptic equation Lu u  u

on D Moreover in this case there is an exp onential rate of growth on

suciently large compact domains and this rate can b e arbitrarily close

to In order to reach these conclusions we rst develop some results

c

concerning innerpro duct and multiplicative martingales and their relation

to the op erators L and L resp ectively where x x  x

In the case of the innerpro duct martingales weshow that for some cir

cumstances they can be used as changes of measure for the law of the

branching pro cess in a similar way that Girsanov densities act as changes

of measure in the context of diusions More sp ecically the change

of measure induces a drift consisten t with a certain Do obs htransform

on the path of a randomized ancestral line of descent These concepts

are essentially spatial versions of spine decomp ositions for GaltonWatson

pro cesses given in Lyons et al

AMS classication Primary J

Corresp onding author

Keywords spatial branching pro cesses branching diusions lo cal extinc

tion spine decomp osition change of measure on trees generalized prin

cipal eigenvalue

Intro duction and main results

Markov branching diusions

d

Let D R b e a domain and consider Y fY tt g the diusion pro cess

with fP x D g corresp onding to the elliptic op erator L on D

x

satisfying

d d

X X

a b on D L

ij i

x x x

i j i

ij i

where the symmetric matrix axa x is p ositive denite for x D and

ij

a C D b C D We assumed these stronger than usual smo othness

ij i

assumptions for convenience they guarantee that there will be no problem

dening the adjoint op erator We assume that Y is conservative that is that

D D

inf ft Y t D g satises P in other words

x

Y has an as innite lifetime F urthermore let C D

be b ounded from ab ove on D and Here C D denotes the usual

Holder space Then the binary L branching diusion under the measure

P is dened informally as follows A single particle starts at p osition x D

x

performs a Y motion on D This particle is killed with spatially dep endent

rate At the moment and spatial p osition of its death the particle pro duces

precisely two ospring Each of these two individuals pro ceed indep endently to

perform Y motions killed at rate at whichpoint they repro duce in the same

t each time t the branching diusion way as their parent and so on A

consists of a point pro cess Z dened on Borel sets with almost surely nite

t

total mass Z D

t

In the sequel the notation P E and Z will be used for the branching

x x

diusion and the notation P E and Y will be used for the diusion on D

x x

corresp onding to L Moreover we shall use the UlamHarris label ling notation

That is an individual u is identied by its line of decent from the initial ancestor

More precisely if u i i i then she is the i th child of the i th

n n n n

child of of the i th child of the initial ancestor Thus uv refers to the individual

who from us p ersp ective has line of descent expressed as v Further the length

juj is equal to the generation in which individual u lives We shall use N to

t

denote the set of individuals alive at time t and fY t u N g for their

u t

p ositions in D In this waywehave for example Z Acardfu N Y t

t t u

Ag where A is any Borel set

A whole array of questions can b e asked ab out the large time b ehavior of the

pro cess Z fZ g Basic questions which address the concepts of lo cal

t t

extinction and recurrence however undened as yet fo cus on whether this

pro cess will visit nonempty op en sets innitely often and if so how can this b e

quantied Surprisingly there are few results in this direction in existing liter

ature In the late fties and sixties there were a small cluster of pap ers which

considered simple prop erties of either general or sp ecic examples of branching

diusions Sp ecically we sp eak of the former Soviet and Japanese contri

butions of Sevastyanov Skoroho d Watanab e and

Ikeda et al ab

Some results can b e found amongst these references p ertaining to the kind

of problems wehave alluded to ab ove in particular Watanab e In more

recent times the numb er of articles concerning growth and spread of branching

diusions are again largely restricted to sp ecial cases for example branching

random walks and branching Brownian motions or simple variations thereof

Notably Biggins has pro duced lo cal limit theorems analogous to

the one in Watanab e which demonstrate that numb ers of particles in any

Borel set can b e appropriately rescaled over time bytheiraverage to achievea

Law of Large Numb ers typ e result Ogura also showed for branching

diusions where the motion pro cess is a with a restricted

class of space dep endent drift that in a given compact set the number of

particles will b ecome zero and remain zero or a strong law of large numb ers can

b e pro duced for the numb er of particles in that set

By comparison the recent developments and p opularity of measurevalued

diusions sup erdiusions havegiven a more thorough treatment of analogous

issues concerning concentration and migration of mass It is almost imp ossible

to give a full account of books and pap ers on measurevalued diusions We

therefore restrict ourselves here by mentioning the two basic textb o oks Daw

son and Dynkin and the recent monograph Etheridge on

measurevalued pro cesses in general and the articles Delmas Englander

and Pinsky Isco e Pinsky b Pinsky and T rib e

regarding the long term b ehaviour of these pro cesses when then underlying mo

tion pro cess is a diusion in particular

In this article wehave three main goals The rst is to formalize the rela

tionship b etween the op erators L and L where xx x and two

classes of martingales related to the Markov branching diusion Secondly to

intro duce the concept of spine decomp osition for Markov branching diusions

Thirdly we aim to show the functionality of these twoby applying them to the

fundamental question of local extinction versus recurrence

Innerpro duct and multiplicative martingales

In the mid nineties the article of Lyons et al app eared which formalized

a new approach to analyzing some fundamental problems in the GaltonWatson

pro cess It was shown that a classical martingale that has b een a p opular ob

ject of study in earlier years numb ers aliveinthe nth generation divided by

their exp ectation can serve as a RadonNikodym change of measure on the

space of GaltonWatson branching trees The eect of this change of probabil

ityistopick out a randomized line of decent called the spine and to sizebias

ospring distributions asso ciated with each no de along the spine In the case of

GaltonWatson pro cesses the spine dominates the b ehavior of the pro cess under

the new measure That is to the extent that simple and intuitively app ealing

probabilistic pro ofs of classical theorems can be achieved where the original

pro ofs were more analytical and complicated In fact this phenomenon is not

particular to GaltonWatson pro cesses Anumb er of authors haveshown that

similar constructions can b e pro duced for a variety of nonspatial and discrete

time spatial branching pro cesses see Lyons Kurtz et al Olof

sson Athreya Kyprianou and Rahimsadeh Sani Biggins

and Kyprianou In each of these cases a naturally o ccurring martingale

is used as the appropriate change of measure The natural o ccurrence of these

martingales is a consequence of the fact that they are constructed from pos

itive harmonic functions Take for example the case of a discrete time typ ed

branching pro cess Roughly sp eaking for this case the afore mentioned mar

P

n

tingales are of the form hX where the sum is typically taken over

i

i

individuals aliveinthe nth generation X is the a typ e of individual i which

i

could b e for example a birth p osition and h is an eigenfunction with resp ect

to the exp ectation op erator with eigenvalue The last of these prop erties can

P

b e written using obvious notation E hX hx where is the eigen

x i

i

value See Athreya and Ney Chapter VI Athreya and Biggins

and Kyprianou

In Section we show that the idea of martingales from harmonic functions

transfers comfortably into the context of Markov branching diusions In this

R

case the analogous class of martingales takes the form expftg hy dy

t

where is an appropriately evolving p oint pro cess emb edded within the Markov

branching pro cess and h is p ositive and harmonic with resp ect to L

for some appropriate R These martingales we refer to as innerpro duct

martingales for the case of branching Brownian motion they havealso b een

referred to in the past as additive martingales

In Section we follow the trend of the previously mentioned literature and

showthatwe can use a sp ecic class of these martingales to serveaschanges

of measure for the branching diusion The eect of the change of measure is

to p erform a Do obs htransform on the diusion along a randomized ancestral

line of descent the spine whilst doubling the rate of ssion along this path It

will turn out that this fact is a direct consequence of Girsanovs theorem for

diusions and Poisson pro cesses It is worth remarking that in the context of

sup erpro cesses there exists a path decomp osition similar in spirit to the spine

construction when one conditions a sup ercritical sup erdiusion to survive In

this case the conditioned pro cess can b e recovered by taking a particle Markov

branching diusion as a backb one pro cess along which there is a continuum

of immigration at each spacetime p oint according to the sup erdiusion condi

tioned on extinction and nally add a version of this pro cess with a random

numb er of initial particles to the pro cess conditioned on extinction This is also

referred to as the immortal particle picture see Etheridge Evans and

OConnell and Englander and Pinsky for an overview It is an

ticipated that there is a link b etween these backb one pro cesses and the spines

that we dene here We hop e to oer in future work some insightinto their true

relationship

There exists another class of martingales in the context of Markov branching

diusions These are closely related to the nonlinear op erator L recall

that xx x The works of Ikeda et al ab and Skoroho d

and McKean all demonstrate clear links between multiplicative

martingales and p ositive b ounded solutions to parab olic dierential equations

of the form u Lu u u Later other authors suchasNeveu

t

Champneys et al Harris and Kyprianou used these facts

to work with these multiplicative martingales as to ols in the contextoftyp ed

branching Brownian motion Of particular note is their use to date in analyzing

travelling wave solutions to the KolmogorovPetrovskiiPiskunov KPP equa

tion or variants of it In this article we to o shall need multiplicative martingales

as to ols in our pro ofs For this reason we sp end a little time in Section to

discuss their relationship with the semilinear op erator L It will turn out

that when evaluating the probability of lo cal extinction in certain circumstances

discussed in the following section we are essentially considering whether there

exist travelling waves solutions to a generalized version of the KPP equation

Lo cal extinction versus lo cal exp onential growth re

currence

The issue of lo cal extinction can b e understo o d in the following context Given

anynonempty op en set B D the notation B D means that the closure

of B is a b ounded subset of D what are the necessary and sucient conditions

for this set to b e visited for arbitrarily large times with p ositive probability It

turns out that the answer to this question b oils down to a simple dichotomy

concerning a sp ectral condition on the linear op erator L That is to say

whether its generalized principal eigenvalue dened shortly is p ositive or not

This fact reects a similar scenario that has b een obtained for sup erdiusions

in Pinsky and Englander and Pinsky We are also able to provide

some weak results p ertaining to characterizing the growth in the number of

individuals in the given set B when it is visited for arbitrarily large times

with p ositive probability In section we discuss several concrete examples of

branching diusions where the issues surrounding the dichotomywe demonstrate

can b e clearly seen

To formulate the question of lo cal extinction more precisely we make the

following denition

Denition lo cal extinction Fix an x D We say that Z under P

x

local extinction if for every Borel set B D there exists a random exhibits

time such that

B

P and P Z B forall t

x B x t B

Remark Since Z is a discrete particle system the ab ove denition of lo cal

extinction is tantamountto

P lim Z B

x t

t

In the sequel we will use the following notation WewriteC D todenotethe

space of twice continuously dierentiable functions with all their second order

derivatives b elonging to C D A C b oundary is dened with the help of

C maps in the usual way

Let

L D inf f R u satisfying L u in D g

c c

denote the generalized principal eigenvalue for L on D see section in

Pinsky a or App endix A for further elab oration From a probabilistic

point of view the generalized principle eigenvalue can b e equivalently expressed

as

Z

t

A

sup lim log E exp Y s ds t

c x

t

t

fA AD A is C g

A

for any x D where infft Y t Ag From the ab ove probabilistic

representation of it is clear that since is b ounded from ab ove It

c c

is standard theory see App endix A that for any there exist a function

c

C D such that L on D

The main results concerning lo cal extinctionexp onential growth are as fol

lows

exp onential growth Theorem lo cal extinction versus lo cal

i For any Borel set B D and x D

P lim sup Z B or

x t

t

ii Fix x D The branching diusion Z under P exhibits local extinction

x

if and only if there exists a function h satisfying L h on D

that is if and only if

c

iii When there exists a function in suchthat

c

lim Z B P x for all nonemptyopenB D

t x

t

and furthermore solves L on D

iv When Z exhibits local exponential growth for any there

c c

exists a large enough B D such that for al l x D

t

P lim e Z B

x t

t

The next corollary follows from part ii

Corollary The local extinction property does not depend on x D that is

either Z under P exhibits local extinction for al l x D or it does not exhibit

x

local extinction for any x D

once the martingale to ols are in place the pro of is The reader will note that

reasonably straightforward and do es not require a great deal of intricacy thus

motivating the martingale theory presented in sections and The elemen

tary nature of the arguments can also b e seen when comparing our metho d with

the techniques employed by Ogura to deduce similar conclusions for a

much less general class of branching diusions

Remark total mass In Theorem we were concerned ab out the local

behaviour of the p opulation size When considering the total mass pro cess

kZ k hZi it is easy to see that the growth rate may actually exceed

c

Indeed take for example a transient diusion corresp onding to L on D with

L D and let beconstant Then L D

c c

but since the branching rate is spatially constant a classical theorem on

t

Yules pro cesses tells us that e kZ k tends to a nontrivial random variable as

t

t that is that the growth rate of the total mass is

c

It is not clear when the function in Theorem is equal trivial solution

and when it is otherwise a nontrivial solution to the equation Lu u u

on D If there is no nontrivial solution to the semilinear elliptic equation we

obtain a zeroone law concerning the probability that a nonemptyset B D

b ecomes eventually vacant

Corollary lo cal extinction versus recurrence Assume that the equa

Lu u u has no solution in except the trivial ones u tion

and u Then either

lim sup Z B P as for all x D and nonemptyopenB D

t x

t

or

lim Z B P as for all x D B D

t x

t

according to whether or

c c

An active example of this Corollary concerning branching Brownian motion

will b e shown in Section In that case Lu u u takes the form of the

travelling wave equation to the KPP equation u cu u u

where c is the wave sp eed and is a constant

One can think of Lu u u as a generalization of the travelling

wave equation asso ciated with the KPP equation Essentially then the ques

tion of the uniqueness and nontrivialityof would seem to b e questions ab out

existence and uniqueness of travelling waves This provides then another mo

tivation for this work It emphasizes probabilistic interpretations of travelling

waves In the future we hop e to oer further insightinto these matters

Pinsky and Englander and Pinsky consider similar questions

for another typ e of spatial branching pro cess the sup erdiusion corresp ond

ing to the semilinear op erator Lu u u where and are related to

the variance of the ospring distribution and to the mass creation resp ec

tively Their conclusions are proved by considering the relationship between

the sup erdiusion and solutions to the parab olic partial dierential equation

u t Lu u u with appropriate initial and b oundary conditions The

behaviour of solutions of this class of parab olic equations together with an un

derstanding of howto express the behaviour of the sup erdiusion in terms of

these solutions is fundamental to their metho dology Given that the branch

ing diusion wehave describ ed here is asso ciated with the semilinear op erator

L one might argue that with some mild adaptations to the case

the analytical arguments of Englander and Pinsky can b e reemployed

here Indeed that is the case and in Section we follow this line although the

results obtained are weaker than what the probabilistic techniques deliver In

particular only part ii of Theorem is proved using the analytic approach

This again emphasizes the motivation for pursuing probabilistic techniques

Outline

The rest of this pap er is organized as follows In the second section we give

anumb er of results concerning innerpro duct martingales and their connection

to certain partial dierential equations This section will later be completed

by section four which treats similar questions for multiplicative martingales

Between these two sections we presenta key section Section whichcontains

the spineconstruction In section vewe utilize the preceding three sections

and prove the theorems stated in subsection An analytical pro of of Theorem

part ii will b e shown in section six Then in section seven we complete the

theory with several concrete examples Finally App endix A is intended to make

this pap er easier to read by giving the necessary background material

Natural innerpro duct martingales

As is well known the theory of diusion pro cesses lays down a clear relationship

between p ositive harmonic functions and the existence of certain martingales

leading to the sto chastic representations of solutions to certain partial dier

ential equations For an accountofthis theory one can consult Karatzas and

Shreve for example

The story is very similar for branching diusions and that forms part of

the motivation for this section In what follows we shall show that positive

functions that are harmonic with resp ect to the op erator L for R

either on the whole domain D or a compact sub domain such as a ball B are

intimately linked to certain martingales which can b e written as functionals of

the L branching diusion The results we shall prove are by no means an

exhaustive analogy of what can be proved for diusions but they suce for

our purp oses The connections we present are not exclusively new It is clear

that other authors have pursuing these connections in the past see for example

Watanab e Ikeda et al ab Champneys et al and

Harris Our exp osition if not providing more general results within

this context uses more elementary probabilisitic pro ofs In particular we will

not unlike in section six rely on the theory of evolution equations and their

connection to the Laplacefunctional of Z but rather on arguments only using

standard branching decomp osition together with Itos formula for the single L

particle Thus on b ounded domains for example one do es not havetoworry

ab out the appropriate b oundary condition for the semigroup nor ab out the

question why the solution to the integral equation b ecomes actually a classical

solution

Before continuing with the results we need to intro duce some notation Let

F be the natural ltration generated by Z For an yBorelset B D let

t t

n o

b b

us denote by Z Z t the pro cess corresp onding to the branching

t

diusion Z where particles are instantly annihilated when they meet B We

B

let N consist of those individuals who are rst in their line of descenttohit

t

B

B Recall the infft Y t B g We would liketo

B

dene similar ob jects for each u N Henceforth for sucha uwe denote the

t

B

nite hitting time of B by For technical reasons we also need to dene

u

B B B

for u N for such an individual Finally for each individual u

u t u

let b e their time of death and note that in their particular ancestral line of

u

descent if juj n then is the time of the nth arrival in a Poisson pro cess

u

with inhomogenous intensity Y t

Theorem Lo cal innerpro duct martingales Consider a bal l B D

or indeed any other compact set with a C boundary Let h C B

h on B and R Then

D E

b

B

h Z

X

t

h Y

B

u

u

t h

u

e e W B

t

h x h x

B

uN

t

is a P martingale for al l x B with respect to F if and only if

x t

L h in B

h

Pro of We b egin by assuming that W B is a martingale so that neces

t

h

sarily E W B for all x B and t By conditioning on the rst

x

t

ssion p ointwehaveforall x B

R

B

t

Y sds B



e h Y t h x E

x

B

Z

t

R

s

Y r dr



E Y s e h Y s ds

x

It follows that

B

Z

t

B

M exp Y s ds h Y t

t

B

Z

t

R

s

Y r dr



Y s e h Y s ds

is a martingale If h C B an application of Itos formula shows that when

M is written as an Ito diusion as the drift comp onent is necessarily zero we

t

are forced to haveL h h on B That is L h on B

nishing the pro of in one direction

For the other direction let us assume that L h on B and

h

hence necessarily h C B Supp ose it can b e proved that E W B

x

t

for all t andx B Then the result follows from the decomp osition

D E

b

B

h Z u

X X

s

h Y

B

u

u

t h

u

e e W B

ts

h x h x

B

b

uN

uN

t

t

B

X X

h Y B

uv

uv

t

t

uv

e e

h x

B

b

uv N

uN

t t

B

X X

h Y

B

u

u

t h

u

e W B u e

t

h x

B

b

uN

uN

t t

h h

b b

where given F Z uandW B u are indep endent copies of Z and W B

t s s

t t

under the law P resp ectively

Y t

u

h

It remains then to prove then that E W B for all t andx B

x

t

First some new notation Let L to b e those individuals who during their life

time cross the b oundary B b efore time t for the rst time their ancestral

history or are alive at time t without having an ancestor including themselves

who has met the b oundary B In the terminology of Chauvin this is a

n b e those individuals in the nth generation who are stopping line Let A

L

neither in L nor have an ancestor in L Now dene

B

X X

h Y t

B

h Y

u

u u

u

t

h

u

u

W B n e e

t

h x h x

uL

uA n

L

jujn

We shall now show that this is a mean one martingale with resp ect to G

n

the natural algebra generated by the complete life all individuals up to and

including the nth generation Note that for individuals u in the nth generation

B

or less and memb ership of L are G measurable

u n

u

Using the assumption on h and Itos formula recall that h has b ounded

derivatives on B it follows that is a martinagle and thus we have a

sto chastic representation for h given by This representation can otherwise

b e written

B

t

B

B e h Y t h x E

x

t

B e Y h Y

t

ssion time in our branching pro cess Now for all t where is the rst

consider the decomp osition

B

X

h Y t B

u

u

t

h

u

W B n e

t

h x

uL

jujn

B

X X

h Y i t

B

u

ui

t

ui

e

B

t

ui

ui

h x

i

uA n

L

h Y i

u ui

ui

e

B

t

ui

ui

h x

By taking conditional exp ectations in this decomp osition with resp ect to G and

n

applying the martingale prop ertyisproved

Now note using again that

h h

W B W B n E E

x x

t t

B

B h Y t

t

E B e

x

t

h x

Y h Y

B

e

t

h x

The branching pro cess do es not explo de in a nite time and hence A n

L

almost surely This together with monotonicityimplies

h

lim E W B n

x

t

n

X

h Y

u u

h

u

A

B limE e E W

x x

t

n

h x

uA n

L

Wewanttoprove that the limit on the right hand side is zero To this end note

that for u A n t h is b ounded on B A n fjuj n tg

L u L u

n

and the nth generation contains memb ers Wethus have the upp er b ound

X

h Y

u u

jjt k

u

A

E e Ke P t

x x k

h x

uA k

L

where is the time of the k th arrival in the Poisson pro cess fn t g whose

k t

intensity at time t conditional on Y is Y t Since fn k g is equivalentto

t

f tg it follows that

k

k

X X X

h Y B

u u t

jjt B

u t

A

e E E Ke e

x x

h x k

k k

uA k

L

jjt B

t

Ke E e

x

jjsup x t

xB

Ke

o n

R

t

Y s ds The niteness of this sum implies necessarily where B exp

t

that

X

h Y

u

u

A

lim E e

x

n

h x

uA n

L

h

hence E W B and the pro of of the Theorem is concluded

x

t

The following result is an immediate consequence of the previous theorem

Denote by the principal eigenvalue of L on B that is the supremum of the

real part of the sp ectrum Recall that see for example Theorem in Pinsky

a the corresp onding Dirichleteigenfunction b elongs to C B and

onB while on B

Corollary Dirichlet inner pro ductmartingales The process M de

nedby

X

Y t h Z i

u t

t t

e M e

t

x x

b

uN

t

is a P martingale with respect to the ltration generatedbythebranching pro

x

cess for al l x B

Here is another version of Theorem but with a little weaker assumption on h

In fact we will not need this version in the rest of the pro ofs but it shows there

are much deep er connections with Dirichlet problems for diusions

B a C B function and Theorem Let B D be a bal l h

Then

E D

b

B

h Z

X

t

h Y

B

u

u

h t

u

B W e e

t

h x h x

B

uN

t

is a P martingale with respect to the ltration generatedbythebranching pro

x

cess for al l x B if and only if

L h in B

h

Pro of Supp ose that W B is a martingale Then necessarily wehave

t

Indeed by b ounded convergence we can write

R

B

Y sds B



h x E e h Y

x

B

Z

R

s

Y r dr



h Y s ds E Y s e

x

The unique solution to the Dirichlet problem cf Karatzas and Shreve

L u h in B

u h on B

has sto chastic representation equal to the right hand side of the ab ove sto chastic

representation of h and hence u h That is L h in B Note that

we used to guarantee the p otential is p ositive and that h is continuous on

B These conditions together are sucient to guarantee that the ab ove Dirichlet

problem has a unique solution with a sto chastic representation

Now supp ose that L h in B The function h can be written

as a solution to the ab ove Dirichlet problem By uniqueness and the sto chastic

representation of the solution this implies that and hence or equivalently

holds The pro of is completed in the same way as the previous Theorem

For b oth the Theorems ab ovewehave used a technique of approximating the

exp ectation of our candidate martingale via a generational decomp osition This

technique is essentially based on a metho d used by Chauvin who used

it to showthe existence of multiplicative martingales for branching Brownian

motion Not surprisingly this metho d will b e used again when we lo ok in more

detail at multiplicative martingales

The martingales wehave considered so far are lo cal in the sense that they

concern the branching diusion up to containment in a bounded domain B

Once this containmentisremoved it is not necessarily true that wecanmake

discounted inner pro ducts which function as martingales This reects a similar

situation for diusions We nish this section with the global version of the

previous results

Corollary Innerpro duct sup ermartingale Suppose that h

L h on D Then C D solves the el liptic equation

hh Z i

t

h t

W e

t

hx

is a nonnegative rightcontinuous P supermartingale for al l x D hav

x

h

ing an almost sure limit as t Conversely if W is a martingale then

t

L h on D

S

Pro of By taking an increasing sequence of balls B suchthatD B we

n n

n

pro duce a sequence of equalities in n of the form

h

E W B forall x B t

x n n

t

B for all n Fatous Lemma together with Note that obviously h C

n

conservativeness of the underlying diusion Y implies that for all x D starting

the limit from suciently large n

h h h

W lim W B E W B E lim E

n x n x x

t t t

n n

The sup ermartingale prop erty follows from the decomp osition

X

h Y t

u

h t h

W e W u

ts s

h x

uN

t

h h

where given F W u are indep endent copies of W under the law P

t

Y t

s s

u

For the converse use conditioning on the rst ssion time to pro duce another

B

martingale similar to with t replaced by t then apply Itos formula as

b efore

Remark When considering the pro of of Theorem one can adopt the

metho dology in one of the directions there to provethatifL h in

h

D then W is a martingale providing certain growth conditions hold on h We

t

do not sp ecify general growth conditions but we will use this fact in the rst

and last examples of Section

Girsanovtyp e theorems htransforms and spines

for branching diusions

The aim here is to give a construction of a change of measure with resp ect

to P which has the eect of identifying a randomized distinguished line of

x

descent the spine and adjusting the diusion and rate of repro duction along

that line of decent according to a classical Do obs h transform This change of

measure discussed in Subsection is dened by the Dirichlet innerpro duct

martingales discussed in Section The structure of these martingales can b e

decomp osed into traditional Girsanov densities for diusions and jump pro cesses

thus rendering them contemp orary with the ubiquitous

We remind the reader that whilst the results in this section are new we

are not necessarily intro ducing new technology We will show here how the

fundamental concepts b ehind Lyons et al concerning size basing and

spine decomp ositions translate and generalize to the Markov branching diusion

setting

Let B b e a b ounded domain with C b oundary Let L Band

c

b e as in Corollary Section of Pinsky a concludes that L is a

e

critical op erator Let b e the eigenfunction corresp onding to the formal adjoint

R

e e

of L Since and are b ounded wehave x xdx and thus

c

B

by denition L is a pro ductcritical op erator see the App endix Note

also that on accountofinvariance prop erties under htransforms the op erator

r

L L a r

is also pro ductcritical which means that the corresp onding diusion is p ositive

recurrent thus ergo dizing with full supp ort on the interior of the domain B

Girsanov theorems for diusions and Poisson pro cesses

Let P be the law of the diusion Y corresp onding to the elliptic op erator L on

x

D There are two imp ortantchanges of measure asso ciated with these pro cesses

whichwe shall now discuss

Firstlywe quote a sp ecial version of Girsanovs theorem for diusions which

concerns Do obs htransforms Before stating this result we shall dene fG g

t

t

a ltration with resp ect to which the pro cess Y is adapted

e

Prop osition Girsanovs theorem for diusions Suppose that B

and compose the Dirichlet setup as above then there exists a probability mea

denedby sure P

x

Z

t

dP Y t

x

Y s ds exp Y B

t

dP x

x

G

t

where P is the law of a diusion corresponding to the operator

x

r

r L a

The new diusion does not hit the boundary and is positive recurrent in B

The second change of measure that will b e of imp ortance is the Girsanov

typ e change of measure for the Poisson pro cess n conditioned on the path of Y

We do not claim the result is new it is included merely for completeness as

it is easy to prove and not necessarily easy to nd in the literature For this we

supp ose that fH g is a ltration with resp ect to which n is adapted Further

t

t

denote by L the law conditioned on the pro cess Y of a nonhomogeneous

Y

Poisson pro cess n f i n g with instantaneous rate Y t

i t

t

Prop osition Girsanovtyp e theorem for Poisson pro cesses

Z

t

dL

n

Y

t

n exp Y s ds

dL

Y

H

t

Pro of There are a number of ways that this can b e proved The simplest

and quickest is to recall that the nonhomogeneous Poisson pro cess n being

t

a submartingale can b e characterized by its comp ensator that is the increasing

pro cess that app ears in its Do obMeyer decomp osition and therefore through

a martingale representation see for example Kallenb erg In this context

that means that n is a L Poisson pro cess with instantaneous rate Y t

Y

R

t

if and only if n Y s ds is a martingale with resp ect to H On

t t t

account of the fact that n has indep endent increments it suces to check that

has L exp ectation for all t This is a straightforward computation

t

Y

using the ab ove RadonNiko dym derivative

Spines on b ounded domains

Theorem Girsanovtyp e theorem for branching diusions Assume

e

that B and compose the Dirichlet setup as in Proposition then there

exists a probability measure Q for Z denedby

x

dQ

x

M

t

dP

x

F

t

Further under this change of measure the branching diusion has a randomized

line of descent the spine that diuses with corresponding operator

r

L a r

which does not hit the boundary B and is positive recurrent

Pro of The branching diusion Z can b e considered to b e dened on the

space T F P where T F is the appropriate measurable space of marked

x

trees Note that the marks are p oints in the path space of of Y and n See

Chauvin for a rigorous denition For any g T there exist distin

guishable genealogical lines of descent from the initial ancestor In following

such a line of descentweidentify its spatial path f g

t

t

Let T b e the enriched space of marked trees in T with distinguished line

of descent and F F T the sigma algebra it generates Write T for

t

the subspace of marked trees in T which are truncated in the obvious way

at time t Let F T where T is the space of marked trees in T with

t

t t t

distinguished line of descent

Given anyg T the measure P j for each x B can b e decomp osed

x

t

F

t

according to a particular choice of distinguished line of descent giving the path

of the spine so that

X

dP g j g j dP

x

Y t

x

t u

F F

t

t

uN

t

where P j is a nonprobability measure satisfying

x

F

t

n

t

Y

g j dP gj dP j dL nj dP

i x

x

F F F F

i t

t t t

i

i

and the tree g is decomp osed into the distinguished line of descent with path

andg the marked subtrees growing o it

i

We can dene a bivariate probability measure Q j on the measurable

x

F

t

F for each x B such that for g T space T

t t t

t

t

dQ g j e dP gj B

t

x x

F F

t t

x

B

where we understand inf ft B g Decomp osition enables us to

t

marginalize Q j to a probability measure on T F say Q j satisfying

t t x

x

F F

t

t

X

dQ g j dQ gj

x

Y t

x

t u

F F

t

t

uN

t

X

t

t

e dP gj

Y t

x

t u

F

t

x

b

uN

t

X

Y t

u

t

dP gj e

x

F

t

x

b

uN

t

M dP g j

x

t

F

t

The eect of the change of this change of measure on the branching pro cess can

b e seen though Rewrite this identityas

R

t

t

ds

s



gj dQ B e dP j

x

t

x

F F

t t

x

R

t

ds n

s t



dL nj e

F

n

t

t

n

t

Y

dP g j

i

F

i t

i

i

dP dL nj

x

F

n F

t

t

t

n

t

Y

dP g j

i

F

i t

i

i

With Prop ositions and in mind we see that this decomp osition suggests

that under Q the branching pro cess evolves as follows

x

i a particle moves as a diusion corresp onding to the op erator

r

r L L a

ii at rate this particle undergo es binary ssion

iii at the instant of ssion one of the particles is chosen with probability

iv the chosen particle rep eats steps iiii and

v the particle whichisnotchosen initiates an L branching diusion

The randomized line of descentwe refer to as the spine The spatial path

along the spine corresp onds to the op erator L Indeed for further con

rmation one can p erform the following calculation showing the distributions

of the spine p osition and the numb er of ssions along the spine Let A D be

a Borel set then

Q A n k

t t

x

Z

X

dQ g j

Y t A n k

x

u t t

F

t

uN

t

Z

X

Y t A n k

u t t

k

b

u N

t

k

Y

dP g j dL nj dP

i

x

F F

t F i

t

i

t

i

k

Z

h i

X

d P L nj

Y t A n k

x

i t t

F

k

t

i

k

Y

dP g j

i

F

t

i

i

i

k

where the indicator in the integral forces there to b e precisely no des in the

marked trees we are considering hence justifying the third equality Completing

the computation wethus have

i h

L Y A n k P t A n k Q

t t t

x x

Y

indicating that the b ehaviour of the spine is that of the sp ecied htransformed

diusion with the instantaneous rate repro duction doubled

Remark In many cases see the later sections containing Examples the

innerpro duct martingales which were shown in Corollary can b e shown to

b e martingales In these instances using a more general version of the Girsanov

Theorem for diusions it is p ossible to construct a spine which is not necessarily

conned to a compact domain by following the same pro cedure as ab ove

we are able to nd From this construction of a spine using the martingale M

t

conditions under which the martingale itself will converge in mean The im

p ortance of this as will b e seen in the next Theorem is that the measure Q

x

b ecomes absolutely continuous with resp ect to P This will eventually enable

x

statements ab out the pro cess under Q where the b ehaviour of the spine is

x

quite sp ecic to b e transferred into statements under P

x

Theorem Suppose that B and are the same as in Proposition If

then the martingale M is L P convergent for al l x B and hence

x

t

Q P

x x

Pro of First recall that M lim M exists since we are dealing with

t

t

B and hence a p ositive martingale Note that the result is trivial when x

we only consider the case that x is in the interior of B Toprove this theorem

wemake use of a standard element of measure theory For the case at hand it

says that

Q P lim sup M Q as E M

x x x x

t

t

Supp ose that S n t is the sigma algebra generated bythemove

t t

Q

ment and repro duction along the spine Let E b e the exp ectation op erator

x

asso ciated with Q Taking exp ectations under Q conditional on S wehave

x x

n

t

X X

Y t

t ui

Q Q t t

A

E M S E S e e

x x t

x x

i

b

iuN

t

n

t

X

t

Q

i

t

i

E e M S e

t

i

i

x x

i

n

t

X

t

i

t

i

e e

x x

i

n

t

X

t

i

e const e

i

where in the last inequality we have used the fact that is bounded from

ab oveon B Now note since is b ounded from ab ove the pro cess n is sto chas

tically b ounded ab ove by the Poisson pro cess with constant rate

sup x Since for this upp erb ounding homogeneous Poisson pro cess

xD

the equivalentversion of the nal sum in is almost surely convergent one can

apply the La w of Large Numbers or alternatively checkthat the righthand

side has nite exp ectation then so is Wehavethus proved that

Q

lim sup E M S Q as

t x x

t

It now follows from Fatous Lemma that

Q

E lim inf M S Q as

x t x

t

and therefore lim inf M Q almost surely Since M is a Q

t x

t x t

martingale it has a limit Q almost surely and hence we have proved that

x

lim sup M Q almost surely and L P convergence follows

x x

t t

Natural multiplicative martingales

Browsing existing literature one will again get the feeling that the results we

present in this section are already emb edded within existing knowledge The

fundamental issue is the link b etween solutions to the nonlinear elliptic equa

tion Lf f f on b oth compact domains B as well as D and certain

martingales which take the form of a multiplicative structure The reader is re

ferred to Skoroho d Ikeda et al ab McKean Neveu

and Harris Consistent with our earlier remarks ab out inner pro d

uct martingales we claim that the results here are to some extent more general

in this context than the current literature necessarily oers Further our pro ofs

are again probabilistic relying on similar techniques used for the inner pro duct

martingales Just likewe did it for innerpro duct martingales we start with a

lo cal version Again the reader will note the use of sto chastic representations of

solutions to dierential equations and generational decomp ositions

Theorem Lo cal multiplicative martingales Let B D be a bal l

and let f B and f C B Then

B

Y Y

f Y

f Y t

u

u

u

f

B

t

f x f x

B

b

uN

uN

t

t

is an P martingale for al l x B if and only if

x

Lf f f on B

f f

B B is a martingale Necessarily E Pro of First assume that

x

t t

By conditioning on the rst ssion time we obtain for all t

R

B

t

B Y sds



f x E f Y t e

x

B

Z

t

R

s

Y sds



Y s f Y s e ds

showing that

B

Z

t

R R

B

t s

B Y sds Y sds

 

e f Y t Y s f Y s e ds

is a martingale By dominated convergence it can even b e seen to b e a uniformly

integrable martingale and L P convergence implies

x

R

B

Y sds B



f x E e f Y

x

B

Z

R

s

Y sds



E Y s f Y s e ds

x

Now consider the Dirichlet problem

L u f in B

u f on B

The unique solution to this problem cf Karazas and Shreve has

sto chastic representation equal to the right hand side of and hence u f

That is to say Lf f f in B

Now supp ose that Lf f f in B Considering f as the unique

solution to the ab ove Dirichlet problem we have and hence This

latter can otherwise b e written as

B

f xE B f Y t B f Y

x

t t

Recall that denotes the rst ssion time Using the same notation as b efore

let

B

Y Y

f Y t

f Y

u

u u

u

f

B n

t

f x f x

uL

uA n

L

jujn

f

We claim that B nisaG martingale To see this note that

n

t

B

Y

f Y t

u

u

f

B n

t

f x

uL

jujn

B

Y Y

f Y t

ui

ui

B

t

ui

ui

f x

i

uA n

L

f Y

ui ui

B

t

ui

ui

f x

and apply to achieve the martingale prop erty Since f is b ounded in

f

B n is both an almost surely and L P conver it follows that

x

t

gent martingale The branching pro cess is nonexplosive which means that

lim A n These previous two facts together imply that

n L

f f

E B E B

x x

t t

B

f Y t

f Y

B E B

x

t t

f x f x

for all t Observing the decomp osition

B

Y Y

f Y

f Y t

u

u

u

f

f

B u B

s ts

f x f x

B

b

uN

uN

t t

f f

where give F B u are indep endent copies of B under the law P

t

Y s

s s

u

the exp ectation together with the strong Markovbranching prop ertyshows

f

that B is a martingale

t

An obvious consequence is the following result

Corollary Let f B be a member of C B and furthermore let

f on B The product

Y

f Y t

u

f x

b

uN

t

is a martingale if and only if f solves Lf f f on B

Finallywehave the following result concerning the whole domain

Corollary Multiplicative martingale Let f D be continu

ous The product

Y

f Y t

u

f

t

f x

uN

t

is a martingale if and only if Lf f f on D

f

Pro of If is a martingale then by conditioning on the rst ssion time

t

we obtain for all t

Z

t

R R

s t

Y sds Y sds

 

f xE Y s f Y s e ds f Y t e

x

The FeynmanKac formula cf Karatzas and Shreve now tells us that

the righthandside is the unique solution to ut L u f in D

with ux f x Hence u f and Lf f f in D

Supp ose now that Lf f f on D Let B be an increasing

n

S

f

sequence of balls such that D B Since E B for all x

n x n

t

n

B b ounded convergence and the conservativeness of the underlying diusion

n

implies that for all x D

f f f

lim E B E lim B E

x n x n x

t t t

n n

The martingale prop ertynow follows by a decomp osition similar to

In contrast to the equivalentversion of this Corollary for innerpro duct mar

tinagales note that the metho d of inating domains preserves the martingale

equality on account of b oundedness

Remark Using the Maximum Principle given in Prop osition of Pinsky

andor Prop osition of Englander and Pinsky there is at most

one nontrivial that is not identically one solution to Lf f f on

B with b oundary condition Also any function g which is not identically

solving the same equation on B is smaller or equal than f Note that in the

ab ovetwo references for the maximum principle the term u u is replaced by

u u One recovers the relevant form here by taking u in place of u

Now assume that There exists a domain B D for which

c

L B the generalized principal eigenvalue of L on B satises

c

Note that once weknowwe can nd such B then any

c c

B with a smo oth b oundary satisfying B B D also has this prop erty

b

Fix B and dene the branching diusion Z obtained from Z by killing the

particles on meeting the b oundary B whichwe can assume is C smo oth As

denote the previously remarked the op erator L is critical on B Let

corresp onding Dirichleteigenfunction In Section it is shown how to construct

an innerpro duct martingale of the form

D E

b

Z

t

t

M e

t

x

where x B The next Theorem uses the notation of this paragraph

Theorem When there exists a unique nontrivial solution to Lu

c

u u such that u when in B and u on B which can

b

be characterized as either uxp xP Z b ecomes extinct or ux

x

q xP lim M

x t

t

Pro of Note that it is automatic from their denition that on the b oundary

of B b oth probabilities are one It suces then to prove that b oth p and q

can b e used to construct pro duct martingales and are nontrivial Uniqueness

is guaranteed by the previous Theorem and its following Remark

Theorem shows that M is L P convergent for all x B This implies

x

t

necessarily that the limit M is strictly p ositive with p ositive probabilityand

b

on the complement of the extinction of Z there can be no mass in the limit

Consequently for x B wehave

p x q x with p xq x on B

This shows that the two prop osed probabilities are nontrivial

For the case of p note that by a classical branching decomp osition for all

t s

o o n n

b b b

B Z u B forall u N Z

s s t t

b b

where given F Z u B are indep endent copies of Z B under the measure

t s s

P It follows that

Y t

u

Y

b b

A

P Z B E P Z B

x ts x s

Y t

u

b

uN

t

Taking the limit as s the Dominated Convergence Theorem implies that

for all t

Y

A

p Y t p xE

u x

b

uN

t

A standard branching decomp osition again shows that this last identity guar

Q

p Y t is a pro duct martingale antees that

u

b

uN

t

Now turning to q we note that again app ealing to a standard branching

can b e written as follows for decomp osition the innerpro duct martingale M

t

t s

X

Y t

u

t

M e M u

ts s

x

b

uN

t

where given F M u are indep endent copies of M under the measure P

t

Y t

s s

u

Taking limits as s it is again clear from this equalitythat

Y

A

q Y t q xE

u x

b

uN

t

Q

and hence q Y t is a martingale

u

b

uN

t

Lo cal extinction versus lo cal exp onential growth

and recurrence probabilistic arguments

Pro of of Theorem i

Let f lim sup Z B g First note that it is sucient to

t

t

prove that for all K R

P lim sup Z B K

x t

t

Indeed from it follows that

P lim sup Z B

x t

t

e Recall that is continuous and that Therefore we now prov

is is b ounded away from zero on some ball Using this along with well

known p ositivity and continuity prop erties of the transition kernel pt x y it

is straightforward to provethat

K B inf P Z B K

x

xB

Although is intuitively clear the precise formulation of its pro of is a bit

tedious and therefore we skip here the technicalities

Then by the strong Markovprop erty

n

P lim sup Z B K lim K B

x t

n

t

nishing the pro of

Pro of of Theorem ii

Assume that This means that L is critical or sub critical whichin

c

turn means by denition see App endix A that one can pick an h with

L h on D According to Corollary Section the innerpro duct

hh Z i is a sup ermartingale b ounded ab ove in exp ectation by hx Since h is

t

b ounded b elow on compact domains it follows that for all B D Z B

t

consthh Z i and hence by comparison when x B

t

lim sup Z B P as

t x

t

It now follows from part i that

lim sup Z B limZ B P as

t t x

t

t

b

Now assume that and let B Z M p and q b e as in Theorem

c

t

Theorem shows that M can b e used as a change of measure for the law

t

of the branching diusion

dQ

x

M

t

dP

x

F

t

Further under Q there is a randomized ancestral line of descent whose path is

x

the diusion f t g corresp onding to the op erator

t

r

r L a

on B so that lives on B and is p ositive recurrent

It was shown in Theorem that

b

p xP Z b ecomes extinct P M q x

x x

and p x q x on B Clearly Q P and on fM g the two

x x

measures are equivalent with probability q p As we already know

that under Q there is one particle whose path is that of the spine which

x

ergo dizes on B with full supp ort on the interior of B then we can say that for

any Borel B B and x B

Q lim inf Z B

x t

t

Since dQ M dP it follows that

x x

P lim sup Z B P lim inf Z B

x t x t

t

t

q x

p x

Note wehave used again the fact that with probability the lim sup Z B

t

t

is either zero or innity from part i Since we can arbitrarily inate B

to enclose any x D we have proved that nonlo cal extinction o ccurs with

strictly p ositive probability for all Borel subsets of D Note generally sp eaking

the larger B is chosen the higher the value of the lower b ound Part ii of the

Theorem is nowproved

Pro of of Theorem iii

For each x D we can take a sequence of domains with smo oth b oundaries

S

g satisfying x B B B D and L B It fB

n n n n c n c

n

follows by monotonicity that in a p ointwise sense p x px where p x

n n

p xifwewould take B B Note that in the limit px Further

n

Lp p p on D

Remark shows that in fact p is the maximal solution to in Refer

ring to the last paragraph of the previous part of the pro of wenowhave that

lim sup Z B p x for all x D The function xis xP

t x

t

o see this note again from the branching also a solution to the equation T

prop erty that for t s

lim sup Z B lim sup Z u B forall u N

ts s t

s s

where given F Z u B are indep endent copies of Z B In a similar wayto

t s s

Q

previous analysis it follows that Y t is a martingale and hence by

u

uN

t

Theorem also solves

Pro of of Theorem iv

b

Denote Let B Z and b e as in the second part of the pro of

c

Obviously it is enough to show that there exists a B D such that

t

P lim inf e Z B

x t

t

b ecause then the original statement follows by replacing B with B Wenow

therefore prove Since is b ounded from ab ove wehave

t t

b

P lim inf e Z B P lim inf e Z B

x t x t

t t

t

b

P lim e h Z i

x t

t

x q

The last probability in the estimate is smaller than giving the required state

ment

Lo cal extinction criterion analytical arguments

In this section we presentananalytical pro of of the lo cal extinction criterion

Although wehave already given a probabilistic pro of for the result in the pre

vious section we feel that this pap er b ecomes more complete by showing here

how the result can be derived from an analogous one which can be found in

the sup erpro cess literature This analogous result for sup erdiusions whichwe

will utilize in the pro of has b een proved recently by Pinsky see Pinsky

Theorem and the remark afterwards using quite a bit of heavy analytical ma

chinery As far as the pro of of the condition for lo cal extinction is concerned we

will showhow to derive this from Pinskys result using a comparison argument

between branching diusions and sup erdiusions Our pro of of the condition for

no lo cal extinction will b e essentially the same as his pro of for sup erdiusions

Regarding the comparison mentioned ab ove it is likely that the deep er rea

son for it is comp ounded in the fact that one can decomp ose a sup erdiusion

using immigration and an underlying sup ercritical strictly binary branch

ing diusion see Evans and OConnell and Englander and Pinsky

Section Intuitively if the underlying branching pro cess visits a compact

again and again no lo cal extinction then so do es the sup erdiusion as well

For the rigorous pro of we will utilize a result on the weighted o ccupation time

for branching particle systems obtained by Evans and OConnell also

used for proving the immigration picture in the same pap er

Pro of of the criterion on lo cal extinction

i Assume that Let x s s x b e jointly measurable in x s

c

and let ss b e nonnegative and b ounded for each s By Evans and

h i

R

t

exp OConnell Theorem E hsZ ids ut x where u

x s

is the socalled mild solution of the evolution equation

u

sLus usu s t sus s t

s

lim us

s

Here we used the notation usus Picka C D satisfying x

c

T

for x B and x for x D n B Let u u b e the mild solution of

t

the evolution equation

u

sLus usu s T suT s s T

t

s

lim us

s

For the rest of the pro of of part i let the starting point x D be xed

Using the argument given in Isco e p wehave that Z exhibits lo cal

extinction if and only if

T

lim lim lim u T x

t

t T

Consider now X the L Dsup erdiusion that is the sup erdiusion

corresp onding to the op erator Lu u u on D see Englander and Pinsky

T

for the denition and let U U b e the mild solution of the evolution

t

equation

U

sLU sU s U s T s s T

t

s

lim U s

s

Again the argumentgiven in Isco e p shows that the supp ort of X

exhibits lo cal extinction that is the prop erty in Denition holds with X in

place of Z if and only if

T

lim lim lim U T x

t

t T

Using Pinsky has shown Theorem and the remark afterwards

that the assumption is equivalent to the lo cal extinction of the supp ort

c

of X Thus follows from We now show that implies

c

which will complete the pro of of this part Making the substitution v u

wehave that v is the mild solution of the evolution equation

v

s

s

Lv svT s v s T s v T s s T

t

lim v s

s

By Isco e pp U and v with t xed havethe following proba

bilistic representations

Z

T

U T xE exp ds h sX i

x s

t

R

T

v T x E exp ds h s v sX i

x s

t

From these equations it is clear that v U Hence

T

T x lim lim lim v

t

t T

ii Assume now that The pro of of this part is almost the same as the

c

pro of of the analogous statement for sup erdiusions Pinsky p

In that pro of it is shown that the assumption guarantees the existence

c

of a large sub domain D D and a function v dened on D whichis

not identically zero and which satises

Lv v v in D

lim v x

x D



vin D

Since f alsosolves Lf f f in D the elliptic maximum principle

see Pinsky Prop osition and Englander and Pinsky Prop osition

implies that v Let w v Then w and furthermore w

satises

Lw w w in D

w x lim

xD



win D

Let P denote the probability for the branching diusion Z obtained from Z by

killing the particles up on exiting D Obviously P Z survives P Z t D

x x

for arbitrary large t s and thus it is enough to show that

P Z survives

x

Wenow need the fact that wonD This follows from the equation

L w w in D

and the strong maximum principle Theorem in Pinsky applied to

the linear op erator L w

Now an argument similar to the one in the pro of of Lemma shows that

Z ti hlog w

w x t E e

x

Supp ose that is not true Then the lefthand side of converges to as

t On the other hand the righthand side of is indep endentoft and

is smaller than which is a contradiction Consequently is true

Examples

In this section we will presentve examples for branching diusions which will

illustrate the general results of this pap er

Branching Brownian motion with drift

This is one of the most simple branching diusion mo dels one can consider par

ticularly if D R and is a p ositive constant In this case L d dx

It is well know that the p ositive harmonic functions of L are expfxg for

R giving corresp onding to the case Here one notes that

c

wehave the luxury of innerpro duct martingales dened on the whole domain

t

Indeed the martingale with growth rate is the classical martingale N e

c t

whichconverges almost surely and in mean to its limit W

It is easy to reason without the technology wehave presented in this article

that the pro cess visits any Borel set innitely often with probability one This

follows by virtue of the fact that every line of descent diuses as a standard

Brownian motion which exhibits these prop erties

Watanab e has proved an even stronger result than this conclusion He

has shown that over and ab ove lo cal extinction with probability zero a strong

law of large numb ers exists

Z B

t

jB jW P as lim

x

t

t

e t

In fact Watanab e proved versions of this result for higher dimentional Brow

nian motion to o as well as some other sp ecial branching diusions related to

Brownian motion Supp ose nowwe consider a branching Brownian motion in

R with a constantdrift R One could reason in this case that despite the

fact that particles exhibit transient motion for a small enough the compar

ative repro duction ensures that space lls up everywhere with particles The

necessary comparison is of course captured in the sp ectral condition on in

c

Theorem In this case wehave L d dx ddx and hence again

referring to well know facts the p ositive harmonic functions are again of the

form expfxg for R This time achieved at

c

p

which is then p ositiveif jj

In Theorem we should still b e concerned ab out the probabilities x It

is not yet clear whether they will b e identically one or not The travelling wave

solutions to the KPP equation come to our rescue According to Theorem

iiiii satises

d d

dx dx

However Kolmogorovetal proved that there are no nontrivial solutions

p

boundedin to this the travelling wave KPP equation for jj and

otherwise there is a unique nontrivial solution Consistently with Corollary

we see that there is lo cal extinction with probability zero or one that is

or according to whether or resp ectively

c c

In the case of nonzero drift it can b e easily checked using metho ds similar

to those in the pro of of Theorem that innerpro duct sup ermartingale asso ci

ated with is also a martingale Further following the discussion in Section

c

it is p ossible to pro duce a spine from this martingale along which there is

a doubled rate of repro duction and the asso ciated diusion op erator is simply

that of standard Brownian motion so the drift has b een removed

Transient L and compactly supp orted

Consider the case when L corresp onds to a transient but conservative diusion

d

on D R and is a smo oth nonnegative compactly supp orted function Since

the generalized principal eigenvalue coincides with the classical principal eigen

value for smo oth b ounded domains it follows that for any nonemptybounded

B D one can pick sucha with B supp and so that L is sup er

critical on D that is all one has to do is to ensure that the inmum of

c

on a somewhat smaller smo oth domain B B is larger then the absolute

value of the principal eigenvalue on B Then a fortiori L is sup ercritical

on D as well On the other hand by the transience assumption it is clear that

the initial Lparticle wanders out to innity with p ositive probability without

ever visiting B and thus without ever branching when starting from x B

In light of Corollary this nowshows that there exists a nontrivial travel ling

wave solution to Lu u u for suchan L and To the b est of our

knowledge this is a new result concerning generalized KPP equations

The HarrisWilliams branching diusion

This is a sp ecialized example of a branching diusion studied for asp ects of its

prototypical b ehaviour and in manyways has served as a source of inspiration

for this pap er by Harris and Williams and Harris We consider

here a variant of their mo del The diusion op erator on R is

L ay y

x x y y

where R That is the op erator corresp onding to a time changed Brown

ian motion with drift where the time change is controlled by an indep endent

OrnsteinUhlenbeck pro cess op erating in the y direction Since wework with

p ositive denite op erators we should in fact replace the rst co ecient ay by

af y where f is obtained by mo difying the function y on a compact

This however will not aect the rest of the argument In the original version

of this mo del The branching rate is dep endent purely on the y variable

and takes the form

x y ry

where and r Note that is not b ounded However this is

not a problem Since as is clear from the mo del the pro cess do es not explo de

and as far as the parameter values of interest are concerned the generalized

principle eigenvalue is nite

Further note that the op erator L is symmetric with resp ect to the reference

density g x y expy Using a comparison principle for symmetric op

erators see Corollary in Pinsky a it follows that L corresp onds to

a recurrent diusion on R Consequently the generalized principal eiganvalue

of this op erator is zero see App endix and therefore

L R L

c

For the case that an application of an htransform shows that

L R L R L

c c

p

Therefore for suciently small jj the p ositivity of the generalized

principal eigenvalue is preserved The conclusion is that for a small enough

drift there is no lo cal extinction This is again consistentwithintuition that

is to say the branching rate wins against transience

Like the previous example this is another branching diusion from which

we could in principle compare the transition of lo cal extinction to nonlo cal

extinction against the event of the existence of travelling waves to the equation

u u u u

ay y ry u u

x x y y

Again one should slightly mo dify the rst co ecient Harris and Williams

haveshown there exists an asymptotic wavespeedsay for the left most

particle in the xdirection of this branching diusion when Given the

elegant probabilistic arguments in Harris linking the asymptotic sp eed

of the left most particle in a standard branching Brownian motion and the

minimal sp eed at whichtravelling waves to the KPP equation exist and then

the relationship with lo cal extinction expressed in the previous example one

might b e inclined to b elieve a similar relationship holds here That is to say is

it true that the transition in from lo cal extinction to nonlo cal extinction seen

through the p ositivityof L R coincides precisely with the transition

c

from existence to no existence of travelling waves to And further that this

transition takes place at precisely

Personal communication with Dr Harris conrms that minimal wave sp eed

for existence in is indeed For the rest some work awaits

Branching OrnsteinUhlenbeck pro cess and more gen

eral recurrent motions

d

Let L kx r on R d where k Then L corresp onds to the

ddimensional OrnsteinUhlenbeck pro cess with drift parameter k Note that

it is a p ositive recurrent pro cess Furthermore let be a p ositive constant

d

Consider nowtheL branching diusion Z on R We call Z a branching

OrnsteinUhlenbeck process By recurrence it follows that L is a critical op erator

d d

and thus L R Consequently L R By Theorem

c c c

do es not exhibit lo cal extinction and exhibits lo cal exp onential ii and iv Z

growth with rate arbitrarily close to on large compact domains

Of course nonlo cal extinction with p ositive probabilitywould immediately

follow from the obvious comparison with a single recurrent Lparticle It fol

lows that non lo cal extinction and lo cal exp onential growth hold even when

L is replaced by an arbitrary L on a domain D as far as L corresp onds to a

recurrent diusion on D In fact as Theorem i in Pinsky shows

L D whenever L corresp onds to a recurrent diusion on D and

c

the branching rate is not identically zero Therefore Z does not exhibit

ge compacts for any local extinction and exhibits local exponential growth on lar

recurrent motion and any not identical ly vanishing branching rate

Branching outward OrnsteinUhlenbeck pro cess

d

d where k Then L corresp onds to the Let L kx r on R

ddimensional outward OrnsteinUhlenbeck pro cess with drift parameter k

This pro cess is transient Furthermore let b e a p ositive constant and consider

the L branching diusion Z Following Example in Pinsky wehave

d

that L R kd From Theorem ii we conclude that if kd

c

then Z do es not exhibit lo cal extinction and exhibits lo cal exp onential growth

with rate arbitrarily close to kd on large compact domains However if

kd then Z exhibits lo cal extinction

For this case we have the luxury of having an exact form for the one di

mensional space of harmonic functions satisfying L h Indeed it

c

is easy to see that hx expfk jxj g satises L h and that

c

making an htransform with this h L transform into

c

h

L kx r

This op erator corresp onds to an inward OrnsteinUhlenbeck pro cess whichis

p ositive recurrent Thus the op erator in is critical and therefore it has

a onedimensional space of p ositive harmonic functions the p ositive constants

Using the corresp ondence b etween the p ositive solutions for the two op erators

invariance under htransforms we conclude that in fact hxexpfk jxj g

is the only up to constantmultiples p ositive harmonic function for L

c

Using the asso ciated innerpro duct martingale which can again b e shown

to be a martingale using metho ds found in Theorem we can follow the

arguments of Section to pro duce a spine with a doubled rate of repro duction

This spine is precisely the OrnsteinUhlenbeck pro cess corresp onding to the

op erator

Now irresp ectiveof the value of kd it would seem p ossible to change

measure in suchaway that there is a p ositive recurrent spine This would seem

to suggest that there is a strong case for nonlo cal extinction without a condition

on kd However in order to transfer statements of lo cal survival backto

the pro cess under the original measure wewould need mean convergence of the

innerpro duct martingale and thus the condition kd app ears

App endix A A review on criticality theory

Let L b e as in We do not assume in this general setting that L corresp onds

to a conservative diusion There exists however a corresp onding diusion pro

cess Y on D that solves the generalized martingale problem for L on D see

Chapter in Pinsky a The pro cess lives on D with playing the

role of a cemetery state Let C D again we do not assume anything

further ab out We denote by P and E the corresp onding probabilities and

x x

exp ectations and dene the transition measure pt x dy forL by

Z

t

Y ds Y B pt x B E exp

s t x

for measurable B D

Denition If

Z Z Z

t

ds Y dt pt x B dt E exp Y

s B t x

for all x D and all b ounded B D then

Z

Gx dy pt x dy dt

is called the Greens measure for L on D If the ab ove condition fails then

the Greens measure for L on D is said not to exist

In the former case Gx dy p ossesses a density Gx dy Gx y dy which

is called the Greens function for L on D

For dene

C f u C L u and uin D g

L

The op erator L on D is called subcritical if the Greens function

exists for L on D inthiscaseC If the Greens function do es

L

not exist for L on D butC then the op erator L on D

L

is called critical In this case C is onedimensional The unique function

L

on D up to a constantmultiple in C is called the ground state of L

L

The formal adjoint of the op erator L on D is also critical with ground

state If furthermore L D we call L on D product L critical

or productcritical in short For pro ductcriticality means that is

an L eigenfunction Finally if C then L on D is called

L

supercritical

If then L is not sup ercritical on D since the function f satises

Lf on D In this case L L is sub critical or critical on D according

to whether the corresp onding diusion pro cess Y is transient or recurrenton

D Pro ductcriticality in this case is equivalent to p ositive recurrence for Y If

and then L is sub critical on D

In terms of the solvability of inhomogeneous Dirichlet problems sub critical

ity guarantees that the equation L u f in D has a p ositive solution u

for every f C Here C C C If sub criticality do es not hold

c

c c

then there are no p ositive solutions for any f C

c

One of the two following p ossibilities holds

There exists a number suchthatL on D is sub critical for

c c

sup ercritical for and either sub critical or critical for

c c

L on D is sup ercritical for all inwhich case we dene

c

Denition The number is called the generalized principal

c

eigenvalue for L on D

Note that inf f C g Also if is b ounded from ab ove

c L

then case holds

The generalized principal eigenvalue coincides with the classical principal

eigenvalue that is with the supremum of the real part of the sp ectrum if D

is b ounded with a smo oth b oundary and the co ecients of L are smo oth up

to D Also if L is symmetric with resp ect to a reference measure dx

then equals the supremum of the sp ectrum of the selfadjoint op erator on

c

L Ddx obtained from L via the Friedrichs extension theorem

h

Let h C satisfy hin D The op erator L dened by

h

L hf L f

h

is called the htransform of the op erator L Written out explicitly one has

rh Lh

h

L f L a r

h h

All the prop erties dened ab ove are invariant under htransforms

For further elab oration and pro ofs see Chapter in Pinsky a

References

Athreya K Change of Measures for Markov Chains and the LlogL

Theorem for Branching Pro cesses Bernoul li

Athreya KB and NeyPE Branching Processes SpringerVerlag

BerlinHeildelb ergNew York

Biggins JD Growth rates in the branching Z

Wahrsch Verw Gebiete

Biggins JD Uniform convergence of martingales in the branching

random walk Ann Probab

Biggins JD and Kyprianou AE Measure change in multityp e

branching Preprint

Champneys A Harris SC Toland JF Warren J Williams D

Algebra analysis and probability for a coupled system of reaction

diusion equations Phil Trans Roy Soc London A

Chauvin B Multiplicative martingales and stopping lines for

branching Brownian motion Ann Probab

Dawson D A MeasureValued Markov Processes Ecole dEte Prob

abilites de Saint Flour XXI LNM

Delmas J F Some prop erties of the range of sup erBrownian mo

tion Probab Theory Relat Fields

Dynkin E B Branching particle systems and sup erpro cesses Ann

Probab

Dynkin E B Sup erpro cesses and partial dierential equations Ann

Probab

Dynkin E B An introduction to branching measurevalued pro

cesses CRM Monograph Series American Mathematical So cietyProv

idence RI pp

Etheridge A Anintroduction to superprocesses AMS lecture notes

Evans S N and OConnell N Weighted Occupation Time for

Branching Particle Systems and a Representation for the Sup ercritical Su

p erpro cess Canad Math Bul l

Englander J and Pinsky RG On the construction and supp ort

d

prop erties of measurevalued diusions on D R with spatially dep endent

branching Ann Probab

Harris SC Travellingwaves for the FKPP equation via proba

bilisitic arguments Pro c Roy Soc Edin A

Harris SC Convergence of a GibbsBoltzmann random measure

for a typ ed branching diusion SeminaredeProbabilites XXXIV

Harris SC and Williams D Largedeviations and martingales for

atyp ed branching diusion I Journal Asterisque

Ikeda N Nagasawa M and Watanab e S a Branching Markov

Pro cesses I J Math Kyoto Univ

Ikeda N Nagasawa M and Watanab e S b Branching Markov

Pro cesses I I J Math Kyoto Univ

Ikeda N Nagasawa M and Watanab e S Branching MarkovPro

cesses I I I J Math Kyoto Univ

Isco e I On the supp orts of measurevalued critical branching Brow

nian motion Ann Pr obab

Kallenb erg O Foundations of modern probability Probabilityand

its Applications SpringerVerlag New York

Karatzas I and Shreve SE Brownian motion and stochastic calcu

lus Second edition Graduate Texts in Mathematics SpringerVerlag

New York

Kurtz T Lyons R Pemantle R Peres Y A conceptual pro of of

the KestenStigum theorem for multityp e branching pro cesses In Classical

and Mo dern Branching Pro cesses KB Athreya and P Jagers eds

Springer New York

Kyprianou A E Travelling wave solutions to the KPP equation

alternatives to Simon Harris probabilistic analysis Submitted to Ann Inst

H Poincare

Kyprianou AE and Rahimzadeh Sani A Martingale convergence

in the multityp e branching random walk To app ear in Bernoulli

Lyons R A simple path to Biggins martingale convergence theo

rem In Classical and Mo dern Branching Pro cesses KB Athreya and P

Jagers eds Springer New York

Lyons R Pemantle R and Peres Y Conceptual pro ofs of L log L

criteria for mean b ehaviour of branching pro cesses Ann Probab

Neveu J Multiplicative martingales for spatial branching pro cesses

Seminar on Stochastic Processes Princeton NJ

Progress in Probability and Birkhuser Boston Boston MA

Ogura Y A limit theorem for particle numb ers in b ounded domains

of a branching diusion pro cess Appl no

Olofsson P The x log x condition for general branching pro cesses

J Appl Probab

Pinsky R Ga Positive Harmonic Functions and Diusion Cam

bridge Univ ersity Press

Pinsky R G b On the large time growth rate of the supp ort of

sup ercritical sup erBrownian motion Ann Probab

Pinsky R G Transience recurrence and lo cal extinction prop er

ties of the supp ort for sup ercritical nite measurevalued diusions Ann

Probab

Sevastyanov BA Branching Sto chastic Pro cesses for Particles Dif

fusing in a Bounded Domain with Absorbing Boundaries Th Prob Appl

Skoroho d AV Branching Diusion Pro cesses Th Prob Appl

Trib e R A representation for sup erBrownian motion Stoch Pro

cess and Appl

Watanab e S On the Branching Pro cess for Brownian Particles With

an Absorbing Boundary J Math Kyoto Univ

Watanab e S Limit Theorems for a Class of Branching Branching

Pro cesses In Markov Pro cesses and Potential Theory ed J Chover pp

Wiley New York

Janos Englander

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