Brownianmotion with drift andthe all-time minimum of branching Branching Brownian motion with absorption Journal of Functional Analysis

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift

Julien Berestycki^a,∗, Éric Brunet^b, SimonC. Harris^c, Piotr Miłoś^d

aDepartmentofStatistics,UniversityofOxford,OxfordOX13TG,UK

bLPS-ENS,UPMC,CNRS,24rueLhomond,75231ParisCedex05,France

cDepartmentofMathematicalSciences,UniversityofBath,BathBA27AY,UK

dFacultyofMathematics,InformaticsandMechanics,UniversityofWarsaw, Banacha2,02-097Warszawa,Poland

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received20September2016 Accepted2June2017 Availableonline13June2017 CommunicatedbyDanielW.

Stroock

Keywords:

Branchingprocesses Reaction–diffusionequations Dirichletproblem

Parabolicpartialdifferential equations

Westudya dyadic branchingBrownian motion on thereal line with absorptionat 0, drift μ ∈ R andstarted from a singleparticleat position x> 0.WithK(∞) the(possibly infinite)total number of individualsabsorbed at 0 over all time,weconsiderthefunctionsωs(x):=E^x[s^K(∞)] fors≥0.

Intheregime whereμislargeenough sothat K(∞)< ∞ almostsurelyandthattheprocesshasapositiveprobability ofsurvival, weshow thatωs <∞if andonlyif s ∈ [0,s0] for some s0 > 1 and we study the properties of these functions.Furthermore, ω(x):= ω0(x) =P^x(K(∞) = 0) is thecumulativedistributionfunctionofthealltimeminimum of the branching Brownian motion with drift started at 0 withoutabsorption.

We give descriptions of the family ωs,s ∈ [0,s0] through the single pair of functions ω0(x) and ωs0(x), as extremal solutions of the Kolmogorov–Petrovskii–Piskunov (KPP) travelling waveequationonthehalf-line,throughamartingale representation, and as a single explicit series expansion.

We alsoobtainapreciseresultconcerningthetailbehaviour ofK(∞).Inaddition,intheregimewhereK(∞)>0 almost surely,weshowthatu(x,t):=P^x(K(t)= 0) suitablycentred

* Correspondingauthor.

E-mailaddresses:julien.berestycki@stats.ox.ac.uk(J. Berestycki),Eric.Brunet@lps.ens.fr(É. Brunet), S.C.Harris@bath.ac.uk(S.C. Harris),pmilos@mimuw.edu.pl(P. Miłoś).

http://dx.doi.org/10.1016/j.jfa.2017.06.006 0022-1236/©2017ElsevierInc.Allrightsreserved.

converges to theKPP critical travelling waveon thewhole realline.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

ConsiderabranchingBrownianmotioninwhichparticlesmoveaccordingtoaBrow- nian motionwith drift μ∈ R and split into two particles at rate β independently one from another. CallNall(t) the population of all particles at time t and call Xu(t) the positionofagivenparticleu∈ Nall(t).Whenwestartwithasingleparticleatpositionx we writeP^x forthelaw ofthisprocess.

In a seminal paper, [21], Kesten considered the branching Brownian motion with absorption,thatis,themodeljustdescribedwiththeadditionalpropertythatparticles enteringthenegativehalf-line(−∞,0] areimmediatelyabsorbedandremoved.Wewrite Nlive(t) for theset of particles alive(notabsorbed) inthebranching Brownianmotion withabsorptionandK(t) thenumberofparticlesthathavebeenabsorbeduptotimet.

The system with absorption is said to become extinct if ∃t ≥ 0 : Nlive(t) = ∅ and to survive otherwise.WeletK(∞):= lim_t→∞K(t)∈N∪ {∞}.

Depending onthevalueofμonehasthefollowing behaviours(seeFig. 1) Regime A: ifμ≤ −√

2β,thedrifttowardsoriginissolargethatthesystemgoesextinct almostsurely.K(∞) isfinite andnon-zero.

Regime B: if−√

2β < μ<√

2β there isanon-zeroprobabilityofsurvival.Onsurvival, therewillalwaysbe particlesnear0andK(∞)=∞almost surely.

Regime C: if μ ≥√

2β there is still a non-zeroprobability of survival, butthe system isdriftingso fastawayfrom 0that,onsurvival,minu∈Nall(t)Xu(t) driftsto +∞almostsurelyast→ ∞;K(∞) isthusalmostsurelyfinite.Furthermore, thereisanon-zeroprobabilitythatK(∞)= 0.

The behaviour of K(∞) in regime A (μ ≤ −√

2β) has been the subject of very active research recently, including a conjecture by Aldous which was recently settled by P. Maillard [24] (we discuss Maillard’s results below), improving earlier results of L. Addario-Berry and N. Broutin [1] and E. Aïdékon [2]. Surprisingly, relatively little wasknownconcerningtheregimesBand C.Ourmainresultsinthepresentworkconcern thestudy ofK(∞) in regimeCandofcertainrelatedKPP-typeequations.

Fig. 1.The three regimes: The hashed region represents the cone in which one expects to find particles.

1.1. The tailbehaviour ofK(∞)

In[24],PascalMaillardhasshownthat,inregimeA(μ≤ −√

2β),thevariableK(∞) has a very fat tail. More precisely, it is shown there that, as z → ∞, there exist two constantsc,c whichdependonxsuchthat

P^x[K(∞)> z]∼

⎧⎨

⎩

c/(zlog(z)²) forμ=−√ 2β, cz^−a(μ) forμ <−√

2β wherea(μ) = ^μ+

μ²−2β μ−

μ²−2β. InregimeB(−√

2β < μ<√

2β)itisclearthatK(∞)=∞onsurvivalsoonewould essentiallyneedtoconditiononextinction tostudythetailbehaviourofK(∞).Thisis outsidethescope ofthepresentwork.

Inregime C(μ≥√

2β),however,K(∞) isalmostsurelyfinite.Weintroducefors≥0 andx≥0,

ωs(x) :=E^x s^K(∞)

, ωs(0) =s. (1)

Whens∈[0,1] this quantityis thegenerating function ofK(∞). Weshow thatω_s(x) isfiniteforsomevaluesof slargerthan1:

Theorem1.Inregime C(μ≥√

2β),thereexistsafinites0>1dependingonlyonμ/√ β suchthat

1. Fors≤s₀,ω_s(x) isfiniteforallx≥0.

2. Fors> s0,ωs(x) isinfiniteforallx>0.

The functions x → ωs(x) are increasing for any s ∈ [0,1) and decreasing for any s ∈ (1,s₀],converging to 1whenx→ ∞,andonehas ω_s0(0)= 0.

Furthermore,onehas forn large

P^x[K(∞) =n]∼ −ω_s0(x) 2sⁿ₀n³²

πβ(s₀−1). (2)

Fig. 2.Numerical determination ofs0ands0β/μ²as functions ofμ/√ β.

Afamiliarargumentusingthebranchingstructureandasimplecouplingallowsusto relatetheωs(x) withoneanother,as inthefollowing result(whichisalreadypresentin [26,24]):

Theorem 2.Inregime C (μ≥√ 2β),

1. Foreach s∈[0,1),ωs(x)=ω0(x+ω₀⁻¹(s)).

2. Foreach s∈(1,s₀],ω_s(x)=ω_s0(x+ω⁻¹_s0 (s)).

Although wedonothaveanexplicitexpression fors₀asafunctionofμ/√

β,wecan evaluate it numerically with a good precision as shown in Fig. 2. In the critical case μ=√

2β,weobtains0= 1.3486. . ..

Wealsoprovethefollowingproperty:

Proposition 3.s0 is an increasing function of μ/√

β and furthermore s0 ∼ cμ²/β for some constant cas μ/√

β→ ∞.

1.2. Distributionoftheall-timeminimum inabranching Brownianmotion

The probabilitythat K(∞)= 0 for asystem startedfrom x, is alsothe probability thattheall-timeminimumofafullbranchingBrownianmotionwithdriftμstartedfrom zerodoesnotgobelow−x:

ω(x) :=ω0(x) =P^x[K(∞) = 0] =P⁰[min

t≥0 min

u∈Nall(t)Xu(t)>−x]. (3) This quantity,ofcourse,isnon-trivialonlyinregime C(μ≥√

2β).Then, since

tlim→∞ min

u∈Nall(t)X_u(t) = +∞

Fig. 3.Solutions to(D0)forβ= 1 andμ= 2. The red (bold) curve isω.

almost surely, we see that there is an almost surely finite all-time minimum for the branchingBrownianmotionandweconcludethatlim_x→∞ω(x)= 1.

ItisnothardtoseebystandardargumentsthatωmustsatisfyaKPP-typedifferential equationwithboundaryconditions:

0 = ¹₂ω+μω+β(ω²−ω), x≥0,

ω(0) = 0, ω(∞) = 1. (D0)

In fact, ωs(x) introduced in (1), if finite, is solution to the same equation with the boundaryconditionω(0)= 0 replacedbyωs(0)=s:

0 = ¹₂ω_s+μω_s+β(ω²_s−ω_s), x≥0,

ω_s(0) =s, ω_s(∞) = 1. (D_s)

Thisis anexampleofthe deepconnectionbetween branchingBrownianmotionand theKPPequationwhichgoesbackto McKean[25]whonoticedthatonecanrepresent solutions of the KPP equation as expectations of functionals of branching Brownian motions.

Untilnowthis isveryclassical,howeverthereisoneunexpecteddifficultyhere:both (D0)and(Ds)admitinfinitelymanysolutionsandarenotsufficienttocharacterizeω(x).

Fig. 3givesseveralnumericalsolutionsto (D0).

Inthiswork,wepresentthreelargelyindependentwaystocharacterizeω(x) whichare laidoutinthethreefollowingsubsections.Thefirstapproachreliesonpartialdifferential equations,thesecondgivesω(x) astheexpectationofacertainmartingaleandthethird onegivesω(x) as apowerseries.

Onesalient propertyofω isthatitconverges to 1ratherquickly:

Proposition 4.ThereexistsB >0suchthat 1−ω(x)∼Be^{− μ+}

μ²−2β

x for largex. (4)

Furthermore, ω(x) is the only solution of (D₀) which remains in [0,1) and converges that fast to 1.

Similarly, forany s∈[0,s0),there existsBs∈Rsuchthat 1−ωs(x)∼Bse^−μ+

μ²−2β

x for large x (5)

where B1= 0,Bs>0whens<1andBs<0whens>1.

Thefollowing Corollaryisadirectconsequenceof(4)and(5):

Corollary 5.Wehave

x→∞lim E^x[s^K(∞)|K(∞)>0] = 1−B_s

B . (6)

Hence, as x→ ∞, conditionally on K(∞)>0,the variableK(∞) convergesin distri- butionand thelimithas probabilitygenerating functions→1−^B_B^s.

Remarks.

• Thefastdecayin(4)and(5)wasactuallyanunexpectedresultforreasonsexplained inSection1.2.4.Tosimplifynotation,wecallrtheexponentialdecayrateof1−ω(x) asgiven in(4):

r:=μ+

μ²−2β. (7)

Itisthelargest solutionof ¹₂r²−μr+β= 0.

• The convergence (6) can actually be thought of as a Yaglom type result. Indeed, imaginethatwe start withasingle particleat the originand thatwekill particles when they firsthit −x. We call Kx the total numberof particles absorbed at −x.

Clearly bytranslation invariance this isthe sameas starting from xand killing at zero.Thus K_x hassamedistribution asK(∞) underP^x.Indeed, withthis method we canthink of (Kx,x≥ 0) asa continuous-time Galton–Watson process indexed byx.Thisapproachisbynowclassical;itwaspioneeredbyNeveu[26]andrecently puttofruitfulusebyMaillard[24].Somewhatsurprisingly,evenwhenμ=√

2β this Galton–Watsonprocessissub-critical(withE[K_x]=e^−rx,seeLemma 7).Thusris therateatwhichthesurvivalprobability decreasesforthisGalton–Watsonprocess and(6)isthecorresponding Yaglomlimit.

1.2.1. ω(x)frompartial differentialequations

Afirstway to characterizeω(x) isto tracktheprobabilitythatnoparticlehasbeen absorbedupto timet.Define

u(t, x) :=P^x[K(t) = 0]. (8)

The function u : R²+ → [0,1] is increasing in x and decreasing in t and, clearly, for each x, u(t,x) → ω(x) as t → ∞. Furthermore, u satisfies the KPP equation with boundaryconditions

∂tu= ¹₂∂xxu+μ∂xu+β(u²−u),

u(t,0) = 0 (∀t≥0), u(0, x) = 1 (∀x >0), (9) which,byCauchy’sTheoremhasonlyonesolution.

Therefore,toobtainω(x),onecaninprinciplesolve(9)andtakethelargetimelimit.

ThisrouteleadstothefollowingTheorem:

Theorem 6.In regime C (μ ≥ √

2β), the function ω(x) defined by (3) is the maximal solution of(D0)suchthat ω(x)<1forallx≥0.

Moregenerally,whens<1,ω_s(x)isthemaximalsolution of(D_s)thatstaysbelow 1.

Whens∈(1,s₀],ω_s(x)istheminimalsolution of(D_s)that staysabove 1.

1.2.2. Martingalerepresentation forω

There is an explicit probabilistic representation of the maximum standing wave ω in regime C (μ ≥ √

2β). Recall that Nall(t) is the population of all the particles in the branching Brownian motion with no absorption and Nlive(t) is the population of particles alive at time t when we kill at 0. We now define on the same probability spacea thirdprocessbased onthe branchingBrownianmotioninwhich particles that hit 0 are stopped but not removed from the system (they neither move nor branch).

We denote by Nlive+abs.(t) the set of particles in the system at time t in this model.

WithaslightabuseofnotationswecontinuetowriteX_u(t) forthepositionsofparticles whenu∈ Nlive+abs.(t).

Letusdefinethefollowingtwo processes:

Z_live+abs.(t) :=

u∈Nlive+abs.(t)

e^−rXu(t), Z_all(t) :=

u∈Nall(t)

e^−rXu(t), (10)

where r isthe asymptoticdecay ofω(x) as givenin(7).Rewriting Xu(t)=Yu(t)+μt it is clear that

Y_u(t),u ∈ Nall(t)

is simply a standard branching Brownian motion with no drift. Therefore Z_all is nothing other than the usual exponential martingale withparameterr associated withthebranchingBrownianmotionY. Now,theprocess Zlive+abs. canbethought of as themartingale Zall stopped onthelines t∧T0 (thatis, where individual particles are stopped at the earlier ofeither time t or the time when

they first hit 0) and therefore it is also a martingale (one cancheck directly, or more generallysee Chauvin[10]).

Lemma 7.In regime C(μ≥√

2β)themartingale(Zlive+abs.(t),t≥0) convergesalmost surely andinL¹ toK(∞)andtherefore E^x[K(∞)]=Zlive+abs.(0)=e^−rx.

Following the probability tilting method pioneered by [23] and [11] we introduce a newprobabilitymeasure Q^x

dQ^x

dP^x =e^rxK(∞).

Note that since Z_live+abs. is a closed martingale we have that E^x[K(∞)|Ft] = Z_live+abs.(t).Thus

dQ^x dP^x

Ft

= Zlive+abs.(t) Z_live+abs.(0).

Underthistiltedprobabilitymeasure,thelawoftheprocessisthesameastheoriginalP^x law except for themovement and branchingrate of adistinguished particle (thespine particle ξ). The spine moves according to a Brownian motion with drift −

μ²−2β, branches atanacceleratedrateof2β andstops(i.e.sticksandstopsreproducing)upon hitting 0.

Theorem 8.Inregime C (μ≥√ 2β), 1−ω(x) =Q^x

1 K(∞)

e^−rx.

Furthermore,Q^x _K(∞)¹

convergestoafinite constant B >0whenx→ ∞andthus, as x→ ∞,

1−ω(x)∼Be^−rx. Moregenerally, forany s∈[0,s₀],one has

1−ωs(x) =Q^x

1−s^K(∞) K(∞)

e^−rx

and theexpectationQ^x(·)convergestoafinite positiveconstant asx→ ∞.

WewillseeintheproofthatwecangiveanexplicitrepresentationoftheconstantB which appearsin Proposition 4as theexpectation of K(∞)⁻¹ underthe measure Q^∞ (similar toQ^xbutwiththespineparticle“startedat infinity”).

1.2.3. ω(x)asaseriesexpansion

The functionω(x) can be understood interms of series expansion. Let {an}n≥1 be thesequencedefinedby

a₁= 1, a_n= β

1

2n²r²−nμr+β

n−1

j=1

a_ja_n−j= 1

(n−1) _2β^r2n−1ⁿ⁻¹

j=1

a_ja_n−j, n≥2, (11) (recallthatr wasdefinedin(7),that ¹₂r²−μr+β= 0 andthatr≥μ≥√

2β)andlet Φ bethefunctiondefinedbytheseries

Φ(z) =

n≥1

anzⁿ. (12)

Wehave

Proposition9.Theradiusof convergenceRofΦisnon-zero andthereexistsB∈(0,R) suchthat

ω(x) = 1−Φ(Be^−rx).

Moregenerally,forany 0≤s≤s₀,there existsanumberB_ssuchthat

ωs(x) = 1−Φ(Bse^−rx) for allx≥0such that |Bs|e^−rx<R. (13) s → Bs is decreasing, positive for s < 1, zero for s = 1, and negative for s > 1.

In particular, fors≤1,thecondition |Bs|e^−rx<R isautomaticallyfulfilled.

TheproofiscontainedinSection2.2.4. Numerically,itseemsthatRislargeenough that|B_s|e^−rx<Rforalls∈[0,s₀] andallx≥0,butwehaven’tprovedthatpoint.The representation (13) makes it very easy to compute numerically ωs by first computing Φ(z), see Fig. 4; the value s0 is then obtained as 1 minus the first minimum of Φ for negative arguments.This follows easily from thefactsthatw_s0(0)= 0,w_s0(0)<0 and thatw_s(0)<0 foralls∈(1,s₀).

1.2.4. Discussion andbackground

As we have already mentioned the Dirichlet problem (D_s) is aversion of the KPP travellingwaveequationonthehalf-linewithboundarycondition.

TheKPPpartialdifferentialequationonthewholerealline,

∂_th= 1

2∂_xxh+β(h²−h), (14)

Fig. 4.Thefunctionz→Φ(z) forβ= 1 andμ=√

2 (left)orμ= 3 (right).Numerically,onehass0≈1.35, Bs0 ≈ −0.859,B0≈0.564,R≈3.14 forμ=√

2 ands0≈14.11,Bs0 ≈ −39.86,B0 ≈0.969,R≈72.8 forμ= 3.

where 0≤h(t,x)≤ 1,h(t,−∞) = 0 and h(t,+∞) = 1, describes how astable phase (h= 0 ontheleft)invadesanunstablephase(h= 1 ontheright).Itiswell knownthat itadmits travellingwavesolutionsoftheform

h(t, x) =hμ(x−μt), 0< hμ<1, hμ(+∞) = 1, hμ(−∞) = 0, for any velocity μ greater or equal to √

2β. The travelling wave x → h_μ(x) is then solutionto

1

2h_μ+μh_μ+β(h²_μ−hμ) = 0, hμ(−∞) = 0, hμ(+∞) = 1. (15) Thesolutionto(15)isuniqueuptotranslation.

There is a large and rich literature devoted to the study of the KPP equation, its travelling wave solutions (and indeed to the convergence towards the travelling wave solutions) going all the way back to theoriginal papers of Kolmogorov et al. [22] and Fisher [13]. To cite only some of the many significant contributions, let us mention:

Aronson and Weinberger [4], Fife and McLeod [12], Uchiyama [28], Kametaka [20], as well as some more recent works of Nollen, Roquejoffre and Ryzhik [15,16]. The KPP equationcanalsobetackledfromaprobabilisticpointofviewthankstoaFeynman–Kac typeconnectionwithbranchingBrownianmotion.Thisapproachalsohasarichhistory, theseminal worksinthisdirectionbeingthoseofMcKean[25]andBramson [8].

In the present work we draw ontechniques from both partial differential equations (suchasmaximumandcomparisonprinciples,phaseplaneanalysis)andfromprobability (martingales,spinedecompositionsandprobabilitytiltingforbranchingparticlesystems) to study(D_s)andthebehaviourofK(∞).

Beforeembarkingontheproofsofourresults,letusemphasizeherethat(Ds)reallyis quitedifferentfrom(15),althoughtheymightappearsimilaratfirstglance.Inparticular,

hμ,surprisingly, does nothavethe sameasymptotic behaviouras ω for largex, inthe regionwherehμ andω arecloseto 1.Indeed,linearising(15)around 1,onegets

1

2(1−˜hμ)+μ(1−h˜μ)+β(1−h˜μ) = 0, [linearized] (16) (a term of order (1−˜hμ)² has been neglected) and thegeneral solutionto (16) is, for someconstants AandB,

1−˜hμ(x) =

Ae^{−μ−μ2−2βx}+Be^{− μ+μ2−2βx} forμ >√ 2β,

(Ax+B)e^{−√2β x} forμ=√

2β. [linearized] (17) Forxlarge,˜hμisclosetohμwiththemeaningthatforsomeconstantAandB,1−˜hμ∼ 1−hμ (seethediscussioninSection2.2.3).Of course,ifA= 0,theterminfactorofB isnegligiblecomparedto theterminfactorofA andtheAtermaloneis anequivalent to1−h_μ.Whensolving(15),itturns outthatthesolutionhasanon-zeroA termand that,therefore,

1−hμ(x)∼

Ae^{− μ−μ2−2βx} forμ >√ 2β, Axe^−√2βx forμ=√

2β, (18)

whereAdependsonμ.

Wenowconsider equation(D₀)for ω(x).Of course,theboundaryconditionof(D₀) isnotsufficienttodetermineauniquesolution,andforarangeofvaluesofcthereexists asolutionto

0 = 1

2v+μv+β(v²−v), v(+∞) = 1, v(0) = 0, v(0) =c. (19) (The difference with equation (D₀) for ω(x) is the added condition v(0) = c. Fig. 3 showsseveralsolutions.)

Onecanthendo,asabove,alargexanalysisofvand,thepartialdifferentialequation beingthe same,onefindsagain that1−v ∼1−˜hμ (xlarge)as givenin(17)forsome c and μ dependent valuesof A and B. Generically, A is non-zero and 1−v decaysas 1−hμ in(18)(upto amultiplicativeconstant;theAis usually differentand caneven benegative, seeFig. 3).

However, for a well chosen value of c (depending on μ), one has A = 0, and the asymptoticdecay of1−v isgiven bythe B term (thatis: itdecays muchfaster). The meaningofProposition 4isthatω(x) ispreciselythatveryspecialsolutionto(D₀)that decaysunlike alltheotherones andunlikethetravellingwaveh_μ.

Theabovediscussion is maderigorousinthe proof ofProposition 4where we usea phase-planeanalysisapproach.

Itis alsointeresting toremark thatanequation verysimilar to (D0)appearsinthe study[18,7]ofthe extinctionprobabilityofabranchingBrownianmotionwithabsorption

andsupercriticaldriftμ>−√

2β (regimesBandC):letθ(x)=P^x[Nlive(∞)=∅] bethe extinction probabilitywhenthesystemisstartedfrom x; thenonehas

0 = ¹₂θ+μθ+β(θ²−θ), x≥0,

θ(0) = 1, θ(∞) = 0. (D₁)

Equations(D0)and(D₁)differonlybytheirboundaryconditions;however(D₁)hasa uniquesolution,whereas(D0)hasmany.

Apossiblewaytounderstandthedifferenceisthatanasymptoticanalysisofθ(x) for large xsimilar to (17)yields onlyonepossible exponential decay: θ(x)∼Aexp[−(μ+ μ²+ 2β)x] forsome constantA, which means that,up to translations, there is only one solution which does converge to zero at infinity, whereas for ω(x) there were two possibleexponentialdecaysandinfinitelymanysolutions.Otherwisesaid,ifonewereto impose θ(0)= 1 andθ(0)=−c,there wouldonly beonevalue ofc forwhich θ would converge tozeroatinfinity.

1.3. Atravellingwave result In regimes A and B(μ <√

2β) onetrivially has ω(x) = 0 sinceK(∞)> 0 almost surely. What is more interesting in this case is the way thatu(t,x) := P^x

K(t) = 0 (as defined in (8)) converges to zero: it does so by assuming the shape of the critical travelling waveoftheKPP equation.Letus recallquicklythewell knownfactsonthis critical travellingwave.

ConsidertheKPPequation(14)withoutdriftonthewholelinewithHeavisideinitial conditions:

∂_th= ¹₂∂_xxh+β(h²−h),

h(0, x) = 0 (∀x <0), h(0, x) = 1 (∀x >0). (20) It is well knownthat h(t,x) is the probability that the leftmost particle at time t of a branching Brownian motion started at x is to the right of zero. Furthermore, this probabilityconvergesto thecriticaltravellingwaveinthefollowingsense:

h(t, mt+x)→h_∗(x) uniformly inxast→ ∞ with

mt:=

2β t− 3 2√

2β logt+ Cste (21)

and where h_∗ := h^√_2β is the travelling wave movingat the minimal possible velocity

√2β, see (15). To fix the invariance by translation, we impose the further condition h_∗(0)= 1/2:

0 = ¹₂h_∗+√

2β h_∗+β(h²_∗−h_∗),

h_∗(−∞) = 0, h_∗(0) = ¹₂, h_∗(+∞) = 1, (22) andthesolutionto(22)isnowunique.

Addingadrift+μ∂xhto(20)wouldonlyshiftthesolutionsby−μt andwouldmake (20)verysimilarto(9):theonlydifferencewouldbethatuisdefinedonR⁺andhonR. However, as both equations converge quickly to zero around the origin in regimes A and B(μ<√

2β),thisdifferenceturnsouttobe minimalandonehasthefollowing:

Theorem10. Inregimes A and B(μ<√

2β), thereexistsaconstant C depending on μ andβ suchthat

u(t, x+mt−μt+C)→h_∗(x) uniformly in xast→ ∞ wheremt isgivenby (21)andh_∗ isthesolution to(22).

Itisinterestingto comparethisresultaboutthebehaviourofu(t,x)=P^x(K(t)= 0) whenμ<√

2β tothebehaviouroftheextinction probabilityu(t,˜ x)=P^x(Nlive(t)=∅) whenμ≤ −√

2β.It is nothardto see thatu˜satisfies thesameequation (9) as uwith differentboundaryconditions, whichis 1 minustheboundaryconditionin(9); namely

˜ usolves

∂tu˜= ¹₂∂xxu˜+μ∂xu˜+β(˜u²−u),˜

˜

u(t,0) = 1 (∀t≥0), u(0, x) = 0 (∀x >0), (23) What is particularly striking is thatinthe critical case, μ= −√

2β, it is knownsince Kesten [21] (see also [6]) that to survive up to time t one must start with an initial particle at position x=ct^1/3. This meansthat ifu(t,˜ x+ ˜mt) convergesto somelimit frontshapethenthecentring termgivingthepositionofthefrontm˜thastobeoforder ct^1/3.Howevertheconvergence ofthesolutionof(23)toatravellingwaveisat present anopenproblem.

2. Proofs

Theproofssectionarepresented mostlyinthesameorderastheresults:Section2.1 contains the proofs to Theorem 1 and Theorem 2 about the structure of the func- tions ωs(x) = E^x[s^K(∞)] in regime C (μ ≥ √

2β). Section 2.2 contains the proofs of Proposition 4, Theorem 6, Lemma 7, Theorem 8and Proposition 9aboutthedifferent representations of ω(x) = ω0(x) = P^x[K(∞) = 0] in regime C (μ ≥ √

2β). Finally, Section2.3containstheproofofTheorem 10abouttheestablishmentofatravellingfor u(t,x)=P^x[K(t)= 0] in regimes Aand B(μ<√

2β)and theasymptoticbehaviour of s0 establishedinProposition 3isprovenlast.

2.1. Proofconcerningthetail behaviourof K(∞)

In this section we consider exclusively regime C (μ ≥ √

2β) and we focus on the problem of the exponential moments of K(∞). We first establish some properties of ω_s(x)=E^x[s^K(∞)] asdefinedin(1)andproceedtoproveTheorem 2andProposition 3.

Wethen provetheasymptoticbehaviour(2)tocompletetheproof ofTheorem 1.

Thefirstpropertyweneedisthatforagivens,thequantityωs(x) iseitherfinitefor allx>0 orinfiniteforallx>0:

Lemma 11.Foragivens,

∃x >0 : ωs(x)<+∞

⇔ ∀x >0 : ωs(x)<+∞ .

Proof. Fixs>0,x>0 andy >0.Thereisapositiveprobability,whichwenote(x,y), thattheinitialparticlestartingfromxreachespositionybeforeanybranchingorkilling happens. Then

ωs(x) =E^x[s^K(∞)]≥(x, y)E^y[s^K(∞)] =(x, y)ωs(y).

Therefore, ifωs(x) isfinite,thenωs(y) isalsofinite. 2

Remark:inthefollowing,wewriteωs<∞whentheconditionsofthelemmaaremet.

Clearly, thisisthecasewhens≤1.Furthermore,as s→ωs(x) is obviouslyincreasing, ifω_s0 <∞forsomes₀,thenω_s<∞foralls< s₀.

Whenω_s<∞,itisclearbystandardargumentsthatω_s(x) issolutionto 0 = ¹₂ω_s+μω_s +β(ω²_s−ωs),

ω_s(0) =s. (24)

Letusnow proveTheorem 2.Aslightlymoregeneralresultisgivenbythefollowing Lemma:

Lemma 12.If ωs<∞then,forany x≥0andh≥0, ωs(x+h) =ω_ωs(h)(x).

Using this lemma, firstsetting s = 0 and then renamingω0(h) assome generals∈ (0,1) gives the first line of Theorem 2. Once we have proved that s0 exists, similarly setting s=s₀ andthenrenamingω_s0(h) assgivesthesecondlineof theTheorem.

Proof. Insteadofstartingourbranchingprocessatpositionxandkillingparticlesat 0, it is here more convenient to think of the process as started at 0 and particles being absorbedat−x.Thisallowstocoupledifferentvaluesofthekillingposition.Inparticular,

ifHxdesignatestheparticlesstoppedwhentheyfirsthit−xandKx(∞) isthenumber ofparticlesinHx,wehavethat

ω_s(x+h) =E

u∈Hx

s^{K(u)h (∞)}

=E

u∈Hx

ω_s(h)

=E

ω_s(h)^Kx(∞)

=ω_ωs(h)(x),

where K_h^(u)(∞) isthe totalnumber ofdescendent of the particle uwhich arekilled at

−x−h (which by translation invariance of the branching Brownian motion and the branchingpropertyisanindependentcopyofK_h(∞)). 2

Wenowhaveamonotonicityresult:

Lemma13.

1. If s<1,x→ω_s(x)isanincreasing functionconverging to 1.

2. Ifs>1andω_s<∞,x→ω_s(x)isadecreasing functionconverging to 1.

Proof. Once the increasing/decreasing part is proved, the fact that the limit is 1 is obvious:fromitsdefinition,itisclearthatωs<1 ifs<1 andωs>1 ifs>1.Assuming ω_sisincreasingordecreasing(dependingons),itmusthavealimit,andfrom(24)that limitmustbe1.

From its interpretationas the distribution of the all-time minimum of a branching Brownianmotion,see(3), itisfurthermoreclearthatω0=ω is anincreasingfunction.

Then,thecouplingprovidedbyLemma 12(ormoresimplyTheorem 2)impliesthatω_s isanincreasingfunctionforalls<1.

Therefore,itonlyremainstoprovethatfors>1,ωs isdecreasingwhenitisfinite.

Assumes>1 andωs<∞.Wefirstshow thatωsismonotonous byconsideringtwo cases:

• Ifω_s(0)>0 then,forallh>0 smallenough,ωs(h)> s.But,forxfixed,s→ωs(x) is astrictly increasingfunctionsoω_ωs(h)(x)> ωs(x).ThenbyLemma 12,ωs(x+h)>

ω_s(x) forallxandallh>0 smallenough:ω_s isincreasing.

• If ω_s(0)≤ 0 then,for allh >0 smallenough, ω_s(h)< s becausein thelimit case ω_s(0) = 0, one has ω_s(0) < 0 from (24). Then, as inprevious case, ωs(x+h) = ω_ωs(h)(x)< ωs(x) forallxandallh>0 smallenough:ωs isdecreasing.

It nowremains torule outthepossibility thatω_sis increasing fors>1.Imaginethat s>1 andωsincreases.Then,from(24),ω_s(x)≤ −2β(ω²_s(x)−ωs(x))≤ −2β(s²−s) and ω_s(x)≤ω_s(0)−2β(s²−s)x,whichbecomesnegativeforxlargeenough,incontradiction withthefactthatωs increases.Soωsmustdecrease fors>1. 2

Weneednowtocharacterizethevaluesofsforwhichωs<∞.

Lemma 14.Assume s>1.If thereexistsafunctionv which solves 0 = ¹₂v+μv+β(v²−v),

v(0) =s, v(x)≥1 (∀x≥0), (D_s) then ωs<∞.

Remark:Theconverseisobvious: whenωsisfinite,itisoneofthesolutionsto(D_s).

This Lemmaallowstodefine s0= sup

s≥1 :ωs<∞ ,

= sup

s≥1 : a solution to(D_s)exists},

(25)

and becauses→ωs(x) increases,onehasωs<∞foralls< s0. Proof. Wepresenttwoproofs: oneprobabilisticandoneanalytical.

Choose s>1 suchthat(D_s)hasasolutionv.Weintroducetheprocess

M_t:=

u∈Nlive+abs.(t)

v X_u(t) ,

wherewerecallthatNlive+abs.(t) isthesetofparticlesinthebranchingBrownianmotion where particlesarefrozenattheorigin,seeSection1.2.2.

Mtisapositivelocalmartingaleandthereforeapositivesuper-martingalewhichthus converges almostsurelytoM_∞.ObservethatunderP^x

v(x) =M0≥E^x(Mt)≥E^x(M_∞).

Butsinceforallt≥0 onehas

Mt≥v(0)^K(t)=s^K(t), we seethatM_∞≥s^K(∞) andtherefore

ω_s(x) =E^x s^K(∞)

≤v(x)<∞.

The sameresult canbe proved analyticallythrough themaximum principle. Letus introduce

us(t, x) :=E^x[s^K(t)], (26)

whichisclearlysolutionto

∂_tu_s= ¹₂∂_xxu_s+μ∂_xu_s+β(u²_s−u_s), x≥0,

us(t,0) =s(∀t≥0), u_s(0, x) = 1 (∀x >0). (27) (Compareto (8).)Withv as above,oneclearlyhas∀x≥0,us(0,x)≤v(x). Therefore, bythemaximumprinciplewehavethat

u_s(t, x)≤v(x), ∀t≥0, x≥0

andasus(t,x)ωs(x) ast→ ∞wesee thatωs(x)≤v(x)<∞. 2 Itisobviousthats₀definedin(25)dependsonlyontheratioμ/√

βbyasimplescaling argument: thebranching Brownianmotion with driftμ and branchingrate β is trans- formed,whentime isscaled byλand spaceby√

λ,into abranchingBrownianmotion with drift μ√

λ and branching rate βλ. In particular, ω_s,β,μ(x) = ω_s,βλ,μ^√_λ(√ λ x) = ω_s,1,μ/^√_β(x/√

β) with the obvious new notation. What remains to be shown are the followingpropertiesofωs:s0is finite,ωs0 isfinite,ω_s0(0)= 0 ands0>1.

Lemma15. s₀<∞,that is: K(∞) doesnothave exponentialmomentsof allorders.

Proof. For the system started from x> 0, consider the following family of events for n∈N:

An=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

K(∞) =n, and

for all integers i≤n,K(i) =i, and

for all integers i≤n, there is at timeionly one particle alive and it sits in [x, x+ 1].

In words, for each i ∈ {0,1,. . . ,n−1} there is one particle alive at time i, it sits in [x,x+ 1], and during a time interval one, this particle splits exactly once, one the offspringgetsabsorbedandtheotherisagainin[x,x+ 1] attimei+ 1.Theoneparticle aliveattimengeneratesatreedriftingtoinfinitywithnomoreabsorbedparticles.

Let_y,zdz be the probabilitythata particlesitting at y has,duringa time interval one, exactly one splitting event with one offspring being absorbed and the other one endingupindz.Define furthermore

q= min

y∈[x,x+1]

x+1

x

y,zdz.

qistheminimalprobabilityforaparticlesittingsomewherein[x,x+ 1] tohave,during atime interval one, exactly onesplitting event with oneoffspring being absorbed and theotheroneending upin[x,x+ 1].Itisclearthatq >0 andthat,furthermore,

P^x(K(∞) =n)>P^x(An)> qⁿP^x(K(∞) = 0).

This impliesthatE^x[q^−K(∞)]=∞andthats0≤1/q <∞. 2 Lemma 16.ωs0 <∞.

Proof. Ifs0 = 1,thisis trivialasω1= 1.Assumenow s0>1 andletusfix x>0.For any 1< s< s0, as ωs is decreasing, onehas ωs(x)< s < s0. This implies that(using the monotone convergenceTheorem) ωs0(x)= limss0ωs(x) is finitewhich entails the result. 2

Lemma 17.ω_s0(0)= 0.

Proof. Wealreadyknowthatω_s0(0)≤0.Assumingω_s0(0)<0,onecouldcontinue the function ωs0 to negative arguments using (D_s) and one couldfind a x0 <0 such that ωs0(x0)> s0; then thefunctionx→ωs0(x0+x) satisfies(D_s)with s> s0, whichis a contradiction. 2

Theproof thats0>1 forallμ≥√

2β isdividedinto twosteps.First,we showthat s0>1 inthecriticalcaseμ=√

2β.Then weconcludebyprovingProposition 3,which statesthats0isanincreasingfunctionofμ/√

β. Lemma 18. In the critical caseμ =√

2β,for s>1small enough, there exist solutions to (D_s),that iss0>1.

Proof. Assumeμ=√

2β.Afterthechangeofvariables(x):=e^μx v(x)−1

,(D_s)reads 1

2+βe^−μx2= 0. (28)

Let us consider the solution to (28)with (0) =(0) = for some >0.We want to provethat∀x,(x)>0 ifissmall enough.Assumeotherwiseandcall x₀= inf{x≥ 0 : (x) = 0}. Then, as (x) ≤ 0, we have (x) ≤ +x and thus on [0,x0] (where (x)≥0),

(x)≥ −2β²(1 +x)²e^−μx. Weconcludethat

(x0)≥−2β²

x0

0

(1 +x)²e^−μxdx≥−2β² ∞ 0

(1 +x)²e^−μxdx,

whichisstrictlypositiveforsmallenough.Thiscontradictsthedefinitionofx0andthus wehavefoundasolutionof(28)suchthat(x)>0 forallx≥0.Thenv(x)= 1+(x)e^−μx

isasolutionto (D_s)startedfrom s= 1+; inotherwords s0≥1+intheμ=√ 2β case. 2

Wenowproceedto provethats₀isastrictlyincreasingfunctionofμ/√ β. Proof ofProposition 3. Let us fix μ ≥ √

2β₁ > √

2β₂. One can easily construct two branchingBrownianmotionswithparameters(μ,β₁) and(μ,β₂) onthesameprobability space to realise acoupling so that the particles of the second one are asubset of the particlesofthefirstone.Itisthenclearthatforany s>1 onehas

ωs,β1,μ(x)≥ωs,β2,μ(x) (29) (withtheobviousextensionofnotation)so that

s0(μ/

β1)≤s0(μ/

β2). (30)

Thisalreadygivesnon-strictmonotonicityandconcludestheproofthats0>1 forallμ, β withμ≥√

2β.

Wecannowprovethattheinequality(30)isstrict.Assumeotherwise;onewouldhave ω_s0,β1,μ(0)=ω_s0,β2,μ(0)=s₀ (wheres₀>1 would bethecommonvalue),ω_s0,β1,μ(0)= ω_s0,β2,μ(0) = 0 (from Lemma 17) and, from (24), ω_s0,β1,μ(0) = −β₁(s²₀ −s₀) <

ω_s0,β2,μ(0) = −β2(s²₀ −s0), which would imply by Taylor expansion that for x > 0 smallenoughωs0,β1,μ(x)< ωs0,β2,μ(x) incontradictionwith (29). 2

Finally,theonlyremainingpointtocompletetheproofofTheorem 1istheasymptotic behaviour(2),i.e.theassertionthatforx>0 fixed

pn(x) :=P^x[K(∞) =n]∼ −ω_s0(x) 2sⁿ₀n³²

πβ(s₀−1) asn→ ∞. (31) WriteD(z,r) fortheopen discofthecomplexplanewithcentrez∈Candradiusr.

Weextendthedefinitionofs→ω_s(x) tos∈C: ωs(x) =E^x[s^K(∞)] =

n≥0

sⁿpn(x), ∀s∈C. (32) Thisquantity isanalyticalonD(0,s₀) becausethep_n in(32)arepositiveand thefirst singularity on the real axis is at s0. Furthermore, it is finite on D(0, s0) by uniform convergencebecauseitisfinite ats0.

The arguments we use are extremely close to those used by Maillard in [24]. The key argument, which improves on usual Tauberian theorems, is an application of [14, CorollaryVI.1]relyingontheanalysisofgeneratingfunctionsneartheirsingularpoints.

Weneedtoshowthat