TThe Λ-lookdown model with selection
B. Bah and E. PardouxNovember 14, 2018
Abstract
The goal of this paper is to study the lookdown model with selection in the caseof a population containing two types of individuals, with a reproduction model whichis dual to the Λ-coalescent. In particular we formulate the infinite population “Λ-lookdown model with selection”. When the measure Λ gives no mass to 0, we showthat the proportion of one of the two types converges, as the population size N tendsto infinity, towards the solution to a stochastic differential equation driven by a Poissonpoint process. We show that one of the two types fixates in finite time if and only ifthe Λ-coalescent comes down from infinity. We give precise asymptotic results in thecase of the Bolthausen–Sznitman coalescent. We also consider the general case of acombination of the Kingman and the Λ-lookdown model. Subject classification
Keywords
Look-down with selection, Lambda coalescent, Fixation and non fixa-tion.
In this paper we consider the lookdown (which is in fact usually called the “modifiedlookdown”) model with selection where we replace the usual reproduction model by apopulation model dual to the Λ-coalescent. We first recall the models from [20] and [9],and then we will describe the variant which will be the subject of the present paper.Pitman [20] and Sagitov [21] have pointed at an important class of exchangeablecoalescents whose laws can be characterized by an arbitrary finite measure Λ on [0, 1].Specifically, a Λ-coalescent is a Markov process (Π t , t ≥
0) on P ∞ (the set of partitionof N ) started from the partition 0 ∞ := {{ } , { } , . . . } and such that, for each integer n ≥
2, its restriction (Π [ n ] t , t ≥
0) to P n (the set of partitions of { , , . . . , n } ) is acontinuous time Markov chain that evolves by coalescence events, and whose evolutioncan be described as follows.Consider the rates λ k,(cid:96) = (cid:90) p (cid:96) − (1 − p ) k − (cid:96) Λ( dp ) , ≤ (cid:96) ≤ k . (1.1) a r X i v : . [ m a t h . P R ] O c t tarting from a partition in P n with k non-empty blocks, for each (cid:96) = 2 , . . . , k, everypossible merging of (cid:96) blocks (the other k − (cid:96) blocks remaining unchanged) occurs at rate λ k,(cid:96) , and no other transition is possible. This description of the restricted processesΠ n determines the law of the Λ-coalescent Π.Note that if Λ( { } ) = Λ([0 , >
0, then only pairwise merging occurs, and the cor-responding Λ-coalescent is just a time rescaling (by Λ(0)) of the Kingman coalescent.When Λ( { } ) = 0 which we will assume except in the very last section of this paper,a realization of the Λ-coalescent can be constructed (as in [20]) using a Poisson pointprocess m = ∞ (cid:88) i =1 δ t i ,p i (1.2)on R + × (0 ,
1] with intensity measure dt ⊗ ν ( dp ) where ν ( dp ) = p − Λ( dp ). We willassume that the measure ν ( dp ) has infinite total mass. Each atom ( t, p ) of m influencesthe evolution as follows : • for each block of Π( t − ) run an independent Bernoulli ( p ) random variable; • all the blocks for which the Bernoulli outcome equals 1 merge immediatelyinto one single block, while all the other blocks remain unchanged.In order to obtain a construction for a general measure Λ, one can superimpose ontothe Λ-coalescent independent pairwise mergers at rate Λ( { } ).The lookdown construction was first introduced by Donnelly and Kurtz in 1996 [9].Their goal was to give a construction of the Fleming-Viot superprocess that providesan explicit description of the genealogy of the individuals in a population. Donnellyand Kurtz subsequently modified their construction in [10] to include more generalmeasure-valued processes. Those authors extended their construction to the selectiveand recombination case [11].We are going to present our model which we call Λ-lookdown model with selection.An important feature of our model is that we will describe it for a population of infinitesize, thus retaining the great power of the lookdown construction. As far as we know,this has not yet been done in the case of models with selection except in our previouspublication [4], where we considered a model dual to Kingman’s coalescent.We consider the case of two alleles b and B , where B has a selective advantage over b . This selective advantage is modelled by a death rate α for the type b individuals.We will consider the proportion of b individuals. The type b individuals are coded by 1,and the type B individuals by 0. We assume that the individuals are placed at time 0on levels 1 , , . . . , each one being, independently from the others, 1 with probability x ,0 with probability 1 − x , for some 0 < x <
1. For each i ≥ t ≥
0, let η t ( i ) ∈ { , } denote the type of the individual sitting on level i at time t . The evolution of ( η t ( i )) i ≥ is governed by the two following mechanisms.1. Births
Each atom ( t, p ) of the Poisson point process m corresponds to a birthevent. To each ( t, p ) ∈ m , we associate a sequence of i.i.d Bernoulli randomvariables ( Z i , i ≥
1) with parameter p . Let I t,p = { i ≥ Z i = 1 } . nd (cid:96) t,p = inf { i ∈ I t,p : i > min I t,p } At time t , those levels with Z i =1 and i ≥ (cid:96) t,p modify their label to η t − (min I t,p ).In other words, each level in I t,p immediately adopts the type of the smallest levelparticipating in this birth event. For the remaining levels, we reassign the typesso that their relative order immediately prior to this birth event is preserved.More precisely η t ( i ) = η t − ( i ) , if i < (cid:96) t,p η t − (min I t,p ) , if i ∈ I t,p \ { min I t,p } η t − ( i − ( { I t,p ∩ [1 , . . . , i ] } − , otherwiseWe refer to the set I t,p as a multi-arrow at time t , originating from min I t,p ,and with tips at all other points of I t,p . This procedure is usually referred toas the modified lookdown construction of Donnelly and Kurtz. In the originalconstruction, the types of the levels in the complement of I t,p remained unchangedat time t , hence the types η t − ( i ), for i ∈ I t,p \ { min I t,p } got erased from thepopulation at time t .2. Deaths
Any type 1 individual dies at rate α , his vacant level being occupied by hisright neighbor, who himself is replaced by his right neighbor, etc. In other words,independently of the above arrows, crosses are placed on all levels according tomutually independent rate α Poisson processes. Suppose there is a cross at level i at time t . If η t − ( i ) = 0, nothing happens. If η t − ( i ) = 1, then η t ( k ) = (cid:40) η t − ( k ) , if k < i ; η t − ( k + 1) , if k ≥ i. We refer the reader to Figure 1 for a pictural representation of our model. Note thatthe type of the newborn individuals are found by “looking down”, while the type ofthe individual who replaces a dead individual is found by looking up. So maybe ourmodel could be called “look-down, look-up”.Since we have modelled selection by death events, the evolution of the N firstindividuals η t (1) , . . . , η t ( N ) depends upon the next ones, and X Nt = N − ( η t (1) + · · · + η t ( N )), the proportion of type b individuals among the N lowest levels, is not a Markovprocess. We will show however that for each t > { η t ( k ) , k ≥ } is well defined (which is not obvious in our setup) and constitutes an exchangeablesequence of { , } –valued random variables. We can then apply de Finetti’s theorem,and prove that X Nt → X t a.s for any fixed t ≥
0, where ( X t ) t ≥ is a [0 , X t = x − α (cid:90) t X s (1 − X s ) ds + (cid:90) [0 ,t ] × ]0 , p ( u ≤ X s − − X s − ) ¯ M ( ds, du, dp ) , (1.3) X t23456789 ● ● ●
Figure 1: The graphical representation of the Λ-lookdown model with selection of size N = 9.Solid lines represent type B individuals, while dotted lines represent type b individuals. where ¯ M ( ds, du, dp ) = M ( ds, du, dp ) − p − dsdu Λ( dp ), and M is a Poisson point mea-sure on R + × ]0 , × ]0 ,
1] with intensity dsdup − Λ( dp ). The process ( X t ) t ≥ representsthe proportion of type b individuals at time t in the infinite size population. Note thatuniqueness of a solution to (1.3) is proved in [8].The paper is organized as follows. We both construct our process, and establishthe crucial exchangeability property satisfied by the Λ-lookdown model with selectionin section 2. In section 3 we establish the convergence of X N to the solution to (1.3).In section 4 we show that one of the two types fixates in finite time if and only ifthe Λ-coalescent comes down from infinity. Moreover, in the case of no fixation, weshow that X t → X ∞ ∈ { , } as t → ∞ , and discuss when X ∞ = 0 a.s and when P ( X ∞ = 1) >
0. In the case of the Bolthausen–Sznitman coalescent (which does notcome down from infinity), we precise the law of X ∞ , and study the speed at whicheither of the two types invades the whole population. Finally, we extend our results tothe case Λ( { } ) > N to denote the set of positive integers { , , . . . } , and [ n ] todenote the set { , . . . , n } . We suppose that the measure Λ fulfills the condition0 < Λ((0 , < ∞ , Λ( { } ) = 0 , (1.4)and in all the paper except in section 5, we assume that Λ( { } ) = 0. The lookdown process, exchangeability Λ Throughout the paper, the notation µ r := (cid:90) [0 , p r Λ( dp )is used for the r th moment of the finite measure Λ on [0, 1] for arbitrary real r . Notethat µ r is a decreasing function of r with ∞ > µ ≥ µ r > r ≥
0, while µ r may be either finite or infinite for r <
0. For r = 0 , , · · · observe from (1.1) that µ r = λ r +2 ,r +2 is the rate at which Π n jumps to its absorbing state { [n] } from anystate with r + 2 blocks. Let X denote a random variable with distribution µ − Λ,defined on some background probability space (Ω , F , P ) with expectation operator E , so E ( X r ) = µ r /µ . Recall the formula (1.1) for the transition rates λ k,(cid:96) of theΛ-coalescent, which we rewrite as λ k,(cid:96) = µ E ( X (cid:96) − (1 − X ) k − (cid:96) ) for all 2 ≤ (cid:96) ≤ k. For any partition with a finite number n ≥ λ n : = n (cid:88) (cid:96) =2 (cid:18) n(cid:96) (cid:19) λ n,(cid:96) = (cid:90) − (1 − p ) n − np (1 − p ) n − p Λ( dp )= µ E (cid:20) − (1 − X ) n − nX (1 − X ) n − X (cid:21) . By monotone convergence, λ n ↑ µ − = (cid:90) [0 , p − Λ( dp ) as n ↑ ∞ . In this section, we will construct the process { η t ( i ) , i ≥ , t ≥ } corresponding to agiven initial condition ( η ( i ) , i ≥
1) defined in the Introduction.Recall the Poisson point process m defined in (1.2). For each n ≥ t ≥ I ( n, t ) = { k ≥ t k ∈ [0 , t ] and { I t k ,p k ∩ [ n ] } ≥ } . We have
Lemma 2.1.
For each n ≥ and t ≥ , I ( n, t ) < ∞ a.s. roof : Each atom ( t, p ) of m affects at least 2 of the n first individuals with proba-bility 1 − (1 − p ) n − np (1 − p ) n − ≤ (cid:18) n (cid:19) p . Consequently E ( I ( n, t )) ≤ (cid:18) n (cid:19) t (cid:90) Λ( dp ) < ∞ . The result follows. (cid:3) Λ -lookdown model without selection In this subsection, we essentially follow [10]. For each N ≥
1, one can define the vector ξ Nt = ( ξ Nt (1) , . . . , ξ Nt ( N )) , t ≥ { , } N , by1. ξ N ( i ) := η ( i ) for all i ≥ t, p ) ∈ m and such that { I t,p ∩ [ N ] } ≥
2, for each i ∈ [ N ], ξ Nt ( i ) evolves as follows ξ Nt ( i ) = ξ Nt − ( i ) , if i < (cid:96) t,p ξ Nt − (min I t,p ) , if i ∈ I t,p \ { min I t,p } ξ Nt − ( i − ( { I t,p ∩ [1 , . . . , i ] } − , otherwise . Using the above lemma, we see that the process ξ Nt has finitely many jumps on [0 , t ]for all t >
0, hence its evolution is well defined. From this definition, one can easilydeduce that the evolution of the type at levels 1 up to i depends only upon the typesat levels up to i . Consequently, if 1 ≤ N < M , the restriction of ξ M to the N firstlevels yields ξ N , in other words : { ξ Mt (1) , . . . , ξ Mt ( N ) , t ≥ } ≡ { ξ Nt (1) , . . . , ξ Nt ( N ) , t ≥ } . Hence, the process η = ξ ∞ is easily defined by a projective limit argument as a { , } ∞ -valued process. Λ -lookdown model with selection This section is devoted to the construction of the infinite population lookdown modelwith selection.For each M ≥
1, we consider the process ( η Mt ( i ) , i ≥ , t ≥
0) obtained by applyingall the arrows between 1 ≤ i < j < ∞ , and only the crosses on levels 1 to M . Usingthe fact that we have a finite number of crosses on any finite time interval, it is nothard to see that the process ( η Mt , t ≥
0) is well defined by applying the model withoutselection between two consecutive crosses, and applying the recipe described in theIntroduction at a death time. More generally, our model is well defined if we suppressall the crosses above a curve which is bounded on any time interval [0 , T ]. Note alsothat, if we remove or modify the arrows and or the crosses above the evolution curve f a type B individual, this does not affect her evolution as well as that of those sittingbelow her.At any time t ≥
0, let K t denote the lowest level occupied by a B individual. Ofcourse, if K = 1, then K t = 1, for all t ≥
0. If for any T , sup ≤ t ≤ T K t < ∞ a.s, thenthe process { K t , t ≥ } is well defined by taking into account only those crosses belowthe curve K t , and evolves as follows. When in state n > K t jumps to1. n + k at rate (cid:0) n + k − k +1 (cid:1) λ n + k,k +1 , k ≥ n − α ( n − α > λ k,(cid:96) defined by (1.1). In other words, the infinitesimalgenerator of the Markov process { K t , t ≥ } is given by: L g ( n ) = ∞ (cid:88) k =1 (cid:18) n + k − k + 1 (cid:19) λ n + k,k +1 [ g ( n + k ) − g ( n )] + α ( n − g ( n − − g ( n )] . (2.1)Now, we are going to show that the process { η t ( i ) , i ≥ , t ≥ } is well defined. Forthis, we study two cases. Case 1: K t → ∞ as t → ∞ . For each N ≥ , t ≥
0, we define K Nt = the level of the N − th individual of type B at time t ;and T N ∞ = inf { t ≥ K Nt = ∞} . We have T ∞ ≥ T ∞ ≥ · · · >
0. For each N ≥
1, we define H N = { ( s, k ); k ≤ K Ns } . Consider first the event A = { T N ∞ = ∞ , ∀ N ≥ } . Recall the Poisson point measure m defined in (1.2). Now, for each N ≥
1, we definethe process( η Nt ( i ) , i ≥ , t ≥ { , } ∞ , by1. η N ( i ) := η ( i ) for all i ≥ t, p ) ∈ m, η Nt evolves as follows η Nt ( i ) = η Nt − ( i ) , if i < (cid:96) t,p η Nt − (min I t,p ) , if i ∈ I t,p \ { min I t,p } η Nt − ( i − ( { I t,p ∩ [1 , . . . , i ] } − , otherwise ,
3. Suppose there is a cross on level j at time s . If ( s, j ) / ∈ H N or ( s, j ) ∈ H N and η s − ( j ) = 0, nothing happens. If ( s, j ) ∈ H N and η s − ( j ) = 1, then η Ns ( i ) = (cid:40) η Ns − ( i ) , if i < j ; η Ns − ( i + 1) , if i ≥ j. We shall see below that P ( A ) = 1 if the Λ-coalescent does not come down from infinity and P ( A ) = P (type B fixates) otherwise. n other words, the process { η Nt ( i ) , i ≥ , t ≥ } is obtained by applying all the arrowsbetween 1 ≤ i < j < ∞ , and only the crosses on levels 1 to K Nt . On the event A , wehave a finite number of such crosses on any finite time interval, and ( η Nt ( i ) , i ≥ , t ≥ H = ∪ N H N . By a projective limit argument, we can easily deduce that the process { η t ( i ) , i ≥ , t ≥ } is well defined on the set H . Our model is defined on the event A .Now we consider the event A c . We first work on the event { T ∞ < ∞} . This meansthat the allele b fixates in finite time. It implies that for each N ≥ , T N ∞ is finite aswell. Consider first the process { η t ( i ) , i ≥ , t ≥ } defined on H , i.e we take intoaccount all the arrows between 1 ≤ i < j ≤ K t , and only the crosses on levels 1 to K t . This process is well defined on the time interval [0 , T ∞ ). However, on the interval[ T ∞ , ∞ ) , η t ( i ) = 1 , ∀ i ≥ , hence the process is well defined in H . We next considerthe process { η t ( i ) , i ≥ , t ≥ } defined on H . This process is well defined on the timeinterval [0 , T ∞ ). But on the interval [ T ∞ , ∞ ), there is at most one B , whose position iscompletely specified from the previous step. Iterating that procedure, and using againa projective limit argument, we define the full Λ-lookdown model with selection.If T ∞ = + ∞ , but T N ∞ < + ∞ for some N , the construction is easily adapted to thatcase. In fact some arguments in section 4 below show that this cannot happen withpositive probability. Case 2 : K t (cid:57) ∞ , t → ∞ . Let T = inf { t ≥ K t = 1 } . We now show that { T < ∞} a.s. on the set { K t (cid:57) ∞ , t → ∞} . Indeed, for anystopping time T and M >
1, define D T,M to be the event that there is at least onecross on each of the levels 1 , , . . . , M − T, T + 1), and B T,M tobe the event that no birth arrow points to a level less than or equal to M on the timeinterval ( T, T + 1). It is plain that the quantity p α,M = P ( D T,M ∩ B T,M |F T )is deterministic, independent of T , and that p α,M >
0. Now clearly { K T ≤ M } ∩ D T,M ∩ B T,M ⊂ { K T +1 = 1 } . Hence P ( K T +1 = 1 | K T ≤ M ) ≥ p α,M , or equivalently P ( K T +1 > | K T ≤ M ) ≤ − p α,M . Let now A M := (cid:40) there exists an infinite sequence of stopping times T kM such that T k +1 M ≥ T kM + 1 and K T kM ≤ M, for all k ≥ . (cid:41) e deduce from the last inequality and the strong Markov property that for any n ≥ P ( A M ∩ { K T n +1 M > } ) ≤ (1 − p α,M ) n . consequently P ( A M ∩ { T = + ∞} ) = 0. This being true for all M >
0, the claimfollows.If T < ∞ , the idea is to show that there exists an increasing mapping ψ : N → N such that a.s. for N large enough, any individual sitting on level ψ ( N ) at any timenever visits a level below N , with the convention that if that individual dies, we replacehim by his neighbor below. Once this is true, the evolution of the individuals sittingon levels 1 , , . . . , N is not affected by deleting the crosses above level ψ ( N ). Hence itis well defined. If this holds for all N large enough, the whole model is well defined.Let M = sup ≤ t
2. At any time t > T , we shall have η t ( i ) = 0 for i < ¯ K t , and η t ( i ) = 1 for i ≥ ¯ K t . Again all crosses are kept, and we keep only thosearrows whose tip hits a level j ≤ ¯ K t .This model is well defined. For each N ≥
1, we define S N as the first time whereall the N first individuals of this model are of type B . We have Lemma 2.2. If T < ∞ , then for each N ≥ , S N < ∞ a.s Proof : The result follows from T < ∞ and the fact that the process of arrows from1 to 2 is a Poisson process with rate λ , = Λ((0 , (cid:3) Now, let ϕ ( N ) = N e αS N ( N e αS N + 1) + K and { ξ ϕ ( N ) t , t ≥ } denote the processwhich describes the position at time t of the individual sitting on level ϕ ( N ) at time 0in the present model.We will prove below that the individual who sits on level ϕ ( N ) at time 0 will remainbelow the level ϕ ( N ) + N on the time interval [0 , S N ]. If she does not visit any levelbelow N before time S N , she will never visit any level below N at any time, andmoreover any individual who visits level ϕ ( N ) + N before time S N will remain abovethe individual who was sitting at level ϕ ( N ) at time 0 until S N , hence will never visitany level below N . ince the “true” model has more arrows and less “active crosses” than the presentmodel, if we show that in the present model a.s. there exists N such that the individualwho starts from level ϕ ( N ) at time 0 never visits a level below N , we will have that inthe true model a.s. for N large enough the evolution within the box ( t, i ) ∈ [0 , ∞ ) ×{ , , . . . , N } is not altered by removing all the crosses above ϕ ( N ) + N . A projectivelimiting argument allows us then to conclude that the full model is well defined.The result will follow from the Borel-Cantelli lemma and the following lemma. Lemma 2.3. If T < ∞ , then for each N ≥ M , (cid:98) P N ( ∃ < t ≤ S N such that ξ ϕ ( N ) t ≤ N ) ≤ N , where (cid:98) P N [ . ] = P ( . | S N ) Proof : It is clear from the definition of ξ ϕ ( N ) t that there exists a death process( D t , t ≥ K t , t ≥
0) conditionally upon D = ϕ ( N ) − K ,and such that ξ ϕ ( N ) t = ˜ K t + D t , ∀ t ≥ , where ˜ K t = (cid:40) K t , ≤ t ≤ T ;¯ K t − , t > T . On the other hand, we have { inf ≤ t ≤ S N ξ ϕ ( N ) t > N } ⊃ { inf ≤ t ≤ S N D t > N } ⊃ { D S N > N } . All we need to prove is that (cid:98) P N ( D S N ≤ N ) ≤ N . The process ( D t , t ≥
0) is a jump Markov death process which takes values in the space { , , . . . , ϕ ( N ) − K } . When in state n , D t jumps to n − αn (recall that allcrosses are kept in the present model). In other words the infinitesimal generator of { D t , t ≥ } is given by Qf ( n ) = αn [ f ( n − − f ( n )] . Let f : N → R . The process ( M ft ) t ≥ given by M ft = f ( D t ) − f ( D ) − α (cid:90) t D s [ f ( D s − − f ( D s )] ds (2.2)is a martingale. Applying (2.2) with the particular choice f ( n ) = n , there exists amartingale ( M t ) t ≥ such that M = 0 and D t = D − α (cid:90) t D s ds + M t , t ≥ . (2.3) e note that { M t , t ≥ } is a martingale under (cid:98) P N [ . ]. This is due to the fact that thePoisson process of crosses above K t is independent of K t . We first deduce from (2.3)that (cid:98) E N ( D s ) = D e − αs . Using the fact that D t is a pure death process, we obtain the identity[ M ] t = D − D t , which, together with (2.3), implies < M > t = α (cid:90) t D s ds. From (2.3), it is easy to deduce that (recall that ϕ ( N ) = N e αS N ( N e αS N + 1) + K ) D t = e − αt ( ϕ ( N ) − K ) + (cid:90) t e − α ( t − s ) dM s , which implies that (cid:98) P N ( D S N ≤ N ) ≤ (cid:98) P N (cid:16) | (cid:90) S N e − α ( S N − s ) dM s | ≥ N e αS N (cid:17) = (cid:98) P N (cid:16) | (cid:90) S N e αs dM s | ≥ N e αS N (cid:17) ≤ N e αS N (cid:90) S N αe αs (cid:98) E N ( D s ) ds ≤ N . The result is proved . (cid:3)
From now on, we equip the probability space (Ω , F , P ) with the filtration definedby F t = ∩ ε> ˚ F t + ε , where ˚ F t = σ { η s ( i ) , i ≥ , ≤ s ≤ t } ∨ N , and N stands forthe class of P –null sets of F . Any stopping time will be defined with respect to thatfiltration. In this subsection, we will show that the Λ-lookdown model with selection preservesthe exchangeability property, by an argument similar to that which we developed in[4]. Let S n denote the group of permutations of the set { , , . . . , n } . For all π ∈ S n and a [ n ] = ( a i ) ≤ i ≤ n ∈ { , } n , we define the vectors π − ( a [ n ] ) = ( a π − (1) , . . . , a π − ( n ) ) = ( a πi ) ≤ i ≤ n ,π ( ξ [ n ] t ) = ( ξ t ( π (1)) , . . . , ξ t ( π ( n ))) . e should point out that π ( ξ [ n ] t ) is a permutation of ( ξ t (1) , . . . , ξ t ( n )) and it is clearfrom the definitions that { π ( ξ [ n ] t ) = a [ n ] } = { ξ [ n ] t = π − ( a [ n ] ) } , for any π ∈ S n . (2.4)The main result of this subsection is Theorem 2.4. If ( η ( i )) i ≥ are exchangeable random variables, then for all t > , ( η t ( i )) i ≥ are exchangeable. We first establish two lemmas, which treat repectively the case of resampling and ofdeath events (we refer the reader to (1.2) for the definition of the collection { t i , i ≥ } ). Lemma 2.5.
For any finite stopping time τ , any N –valued F τ –measurable randomvariable n ∗ , if the random vector η [ n ∗ ] τ = ( η τ (1) , . . . , η τ ( n ∗ )) is exchangeable, and T isthe first time after τ of an arrow pointing to a level ≤ n ∗ or a death at a level ≤ n ∗ , thenconditionally upon the fact that T = t i , for some i ≥ and I t i ,p i ∩ [ n ∗ ]) = k , where k ≥ , the random vector η [ n ∗ − k ] t i = (cid:16) η t i (1) , . . . , η t i ( n ∗ − k ) , η t i ( n ∗ − k ) (cid:17) is exchangeable. Note that η [ n ∗ − k ] t i is the list of the types of the individuals sitting on levels1 , . . . , n ∗ − k just after a birth event during which one of the individuals sitting ona level between 1 and n ∗ has put k − n ∗ . Proof : For the sake of simplifying the notations, we condition upon n ∗ = n, t i = t , p t i = p and I t i ,p i ∩ [ n ∗ ]) = k . We start with some notation. A j ,...,j k − t := { the k levels selected by the point ( t, p ) between levels 1 and n are j , j , . . . , j k − } . We define (cid:98) P t,n [ . ] = P ( . | t i = t, n ∗ = n, I t,p ∩ [ n ]) = k ) . Thanks to (2.4), we deduce that, for π ∈ S n − k , a [ n − k ] ∈ { , } n − k , (cid:98) P t,n ( π ( η [ n − k ] t ) = a [ n − k ] )= (cid:88) ≤ j 1) = a πn − , B it (cid:1) . Define c π,ni = ( a π , . . . , a πi − , , a πi , . . . , a πn − ) , c ni = ( a , . . . , a i − , , a i , . . . , a n − ) . The last term in the previous relation is equal to (cid:88) ≤ i ≤ n (cid:98) P t,n (cid:16) η [ n ] t − = c π,ni , B it (cid:17) = (cid:88) ≤ i ≤ n P (cid:16) η [ n ] t − = c π,ni (cid:17) (cid:98) P t,n (cid:16) B it | η [ n ] t − = c π,ni (cid:17) = 11 + (cid:80) n − j =1 a πj (cid:88) ≤ i ≤ n P (cid:16) η [ n ] t − = c π,ni (cid:17) . Thanks to the exchangeability of ( η t − (1) , . . . , η t − ( n )), we have (cid:98) P t,n ( π ( η [ n − t ) = a [ n − ) = 11 + (cid:80) n − j =1 a j (cid:88) ≤ i ≤ n P (cid:16) η [ n ] t − = c ni (cid:17) , since (cid:80) n − j =1 a πj = (cid:80) n − j =1 a j and c π,ni is a permutation of c πi . The result follows. (cid:3) We can now proceed with the Proof of Theorem 2.4 For each N ≥ 1, let { V Nt , t ≥ } denote the N –valuedprocess which describes the position at time t of the individual sitting on level N attime 0, with the convention that, if that individual dies, we replace him by his neighborbelow. The construction of our process { η t ( i ) , i ≥ , t ≥ } in section 2.2 shows thatinf t ≥ V Nt → ∞ , as N → ∞ .It follows from Lemma 2.5 and 2.6 that for each t > N ≥ 1, ( η t (1) , . . . , η t ( V Nt ))is an exchangeable random vector.Consequently, for any t > n ≥ π ∈ S n , a [ n ] ∈ { , } , | P ( η [ n ] t = a [ n ] ) − P ( η [ n ] t = π − ( a [ n ] )) | ≤ P ( V Nt < n ) , hich goes to zero, as N → ∞ . The result follows. (cid:3) For each N ≥ t ≥ 0, denote by X Nt the proportion of type b individuals attime t among the first N individuals, i.e. X Nt = 1 N N (cid:88) i =1 η t ( i ) . (2.6)We are interested in the limit of ( X Nt ) t ≥ as N tends to infinity. The following Corollaryis a consequence of the well–known de Finetti’s theorem (see e. g. [2]), which says thatsince they are exchangeable, the r.v.’s { η t ( i ) , i ≥ } are i.i.d., conditionally upon theirtail σ –field. Corollary 2.7. For each t ≥ , X t = lim N →∞ X Nt exists a.s. (2.7) Remark 2.8. Since the r.v.’s η t ( i ) take their values in { , } , their tail σ –field isexactly σ ( X t ) . This fact will be used below. Λ -W-F SDEwith selection { X N , N ≥ , t ≥ } In this part, we will prove the tightness of ( X N ) N ≥ in D ([0 , ∞ [), where for each N ≥ t ≥ X Nt is defined by (2.6). For that sake, we shall write an integral equationfor X Nt . We start with some notation.For any N, n, r, p such that N ≥ , N r ∈ N , r ∈ ]0 , , p ∈ [0 , Y ( · , N, p ) to be the binomial distribution function with parameters N and p ; H ( · , N, n, r )the hypergeometric distribution function with parameters ( N − ,n − , Nr − N − ); ¯ H ( · ,N, n, r )the hypergeometric distribution function with parameters ( N − , n − , NrN − ). For every v, w ∈ [0 , F Np ( v ) = inf { s ; Y ( s, N, p ) ≥ v } ,G N,n,r ( w ) = inf { s ; H ( s, N, n, r ) ≥ w } , ¯ G N,n,r ( w ) = inf { s ; ¯ H ( s, N, n, r ) ≥ w } . It follows that if V, W are U ([0 , F Np ( V ) is binomial withparameters N, p . G N,n,r ( W )(resp ¯ G N,n,r ( W )) is hypergeometric with parameters N − , n − , Nr − N − (resp N − , n − , NrN − ). Note that F Np ( · ) = Y − ( · , N, p ) , G N,n,r ( · ) = H − ( · , N, n, r ) and ¯ G N,n,r ( · ) = ¯ H − ( · , N, n, r ). We recall that if X is hypergeometricwith parameters ( N, n, p ) such that N p ∈ N and p ∈ [0 , E ( X ) = np and V ar ( X ) = N − nN − np (1 − p ) . ow, for every r, u, p, v, w ∈ [0 , ψ N ( r, u, p, v, w ) = 1 N FNp ( v ) ≥ (cid:104) u ≤ r (cid:16) F Np ( v ) − − G N,F Np ( v ) ,r ( w ) (cid:17) − u>r ¯ G N,F Np ( v ) ,r ( w ) (cid:105) . (3.1)From the identity r ( n − − E [ G N,n,r ( W )]) = (1 − r ) E [ ¯ G N,n,r ( W )], we deduce the Lemma 3.1. For each N ≥ , r, p, v ∈ [0 , and t ≥ , (cid:90) ]0 , ψ N ( r, u, p, v, w ) dudw = 0 . Using the definition of the model, one deduces that X Nt = X N + (cid:90) [0 ,t ] × ]0 , ψ N ( X Ns − , u, p, v, w ) M ( ds, du, dp, dv, dw ) − N (cid:90) [0 ,t ] × [0 , u ≤ X Ns − η s − ( N +1)=0 M N ( ds, du )where M and M N are two mutually independent Poisson point processes. M isa Poisson point process on R + × [0 , × [0 , × [0 , × [0 , 1] with intensity measure µ ( ds, du, dp, dv, dw ) = dsdup − Λ( dp ) dvdw , M N is a Poisson point process on R + × [0 , αN λ ( ds, du ) = αN dsdu . The reason why X Nt follows the aboveSDE is as follows. Births events happen according to the PPP m . With probability X Ns − , the individual which is copied (if at all) is of type 1. It is copied in a number whichequals ( F Np ( V ) − + , where F Np ( V ) follows the binomial law ( N, p ). The increase in thenumber of 1’s is that number, minus the number of ones which get pushed over level N ,and that umber is the hypergeometric r.v. G N,F Np ( V ) ,X Ns − ( W ). In case the individualwho is copied is a 0, the decrease in the number of ones is the hypergeometric r.v.¯ G N,F Np ( V ) ,X Ns − ( W ). Concerning the deaths, they happen according to a PPP with rate αN X Ns − , and a death at time s decreases the number of 1’s by 1 iff η s − ( N + 1) = 0.Now let ¯ M = M − µ, ¯ M N = M N − αN λ. (3.2)Using Lemma 3.1, we have X Nt = X N + (cid:90) [0 ,t ] × ]0 , ψ N ( X Ns − , u, p, v, w ) ¯ M ( ds, du, dp, dv, dw ) − N (cid:90) [0 ,t ] × [0 , u ≤ X Ns − η s − ( N +1)=0 ¯ M N ( ds, du ) − α (cid:90) t X Ns η s ( N +1)=0 ds. (3.3) or each N ≥ , t ≥ 0, we define M Nt = (cid:90) [0 ,t ] × ]0 , ψ N ( X Ns − , u, p, v, w ) ¯ M ( ds, du, dp, dv, dw ) N Nt = 1 N (cid:90) [0 ,t ] × ]0 , u ≤ X Ns − η s − ( N +1)=0 ¯ M N ( ds, du ) V Nt = − α (cid:90) t X Ns η s ( N +1)=0 ds. M Nt and N Nt are two orthogonal martingales. We have X Nt = X N + V Nt + M Nt − N Nt . ∀ N ≥ , X N ∈ [0 , Proposition 3.2. The sequence ( X N , N ≥ is tight in D ([0 , ∞ ]) . We first establish the lemma : Lemma 3.3. For each N ≥ and t ≥ , (cid:104)M N (cid:105) t = Λ((0 , (cid:90) t X Ns (1 − X Ns ) ds (cid:104)N N (cid:105) t = αN (cid:90) t X Ns η s ( N +1)=0 ds Proof : Using the fact that M N and N N are pure-jump martingales, we deduce that (cid:104)M N (cid:105) t = (cid:90) [0 ,t ] × ]0 , ( ψ N ( X Ns , u, p, v, w )) dsdup − Λ( dp ) dvdw. Let A N ( X Ns , p ) = (cid:90) ]0 , ( ψ N ( X Ns , u, p, v, w )) dudvdw = 1 N (cid:90) ]0 , FNp ( v ) ≥ (cid:104) X Ns (cid:16) F Np ( v ) − − G N,F Np ( v ) ,X Ns − ( w ) (cid:17) + (1 − X Ns )( ¯ G N,F Np ( v ) ,X Ns ( w )) (cid:105) dvdw. Tedious but standard calculations yield (cid:90) [0 , (cid:104) r (cid:16) F Np ( v ) − − G N,F Np ( v ) ,r ( w ) (cid:17) dw + (1 − r ) (cid:16) ¯ G N,F Np ( v ) ,r ( w ) (cid:17) (cid:105) dw = NN − r (1 − r ) F Np ( v )( F Np ( v ) − , or every v, r ∈ [0 , . Consequently A N ( X Ns , p ) = X Ns (1 − X Ns ) N ( N − (cid:90) [0 , F Np ( v ) ≥ F Np ( v )( F Np ( v ) − dv = p X Ns (1 − X Ns ) . We deduce that (cid:104)M N (cid:105) t = (cid:90) [0 ,t ] × [0 , A N ( X Ns , p ) dsp − Λ( dp )= Λ((0 , (cid:90) [0 ,t ] X Ns (1 − X Ns ) ds. Similarly, we have (cid:104)N N (cid:105) t = αN (cid:90) [0 ,t ] × [0 , u ≤ X Ns η s ( N +1)=0 dsdu = αN (cid:90) [0 ,t ] X Ns η s ( N +1)=0 ds. The lemma has been established. (cid:3) We can now proceed with the Proof of Proposition 3.2 We have X Nt = X N + V Nt + M Nt − N Nt and (cid:104)M N − N N (cid:105) t = (cid:104)M N (cid:105) t + (cid:104)N N (cid:105) t . Moreover, from Lemma 3.3 (cid:12)(cid:12)(cid:12)(cid:12) dV Nt dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ α, ≤ d (cid:104)M N (cid:105) t dt ≤ Λ((0 , , ≤ d (cid:104)N N (cid:105) t dt ≤ αN . Aldous’ tightness criterion (see Aldous [1]) is an easy consequence of those estimates. (cid:3) Now, from Proposition 3.2 and (2.7), it is not hard to show there exists a process X ∈ D ([0 , ∞ )), such that for all t ≥ ,X Nt → X t a.s, (3.4)and X N ⇒ X weakly in D ([0 , ∞ )) . .2 Convergence to the Λ -Wright-Fisher SDE with selec-tion Our goal is to get a representation of the process ( X t ) t ≥ defined in (3.4) as the uniqueweak solution to the stochastic differential equation (1.3).Let (Ω , F , P ) be a fixed probability space, on which the above Poisson measuresare defined, which is equipped with the filtration described at the end of section 2.2.Recall the Poisson point measure M = (cid:80) ∞ i =1 δ t i ,u i ,p i defined in the Introduction, andfor every u ∈ ]0 , 1[ and r ∈ [0 , u, r ) = u ≤ r − r. We rewrite equation (1.3) as X t = x − α (cid:90) t X s (1 − X s ) ds + (cid:90) [0 ,t ] × ]0 , p Ψ( u, X s − ) ¯ M ( ds, du, dp ) , t > , < x < , (3.5)which we call the Λ-Wright-Fisher SDE with selection. Without loss of generality,we shall assume that α > 0, which means that X t represents the proportion of non-advantageous alleles.The proof of the following identity is standard and left to the reader. Lemma 3.4. For each r ∈ [0 , , (cid:90) [0 , (cid:0) ψ N ( r, u, p, v, w ) − p Ψ( u, r ) (cid:1) dup − Λ( dp ) dvdw = 2 r (1 − r ) (cid:104) NN − (cid:90) [0 , (1 − up ) N − du Λ( dp ) − Λ([0 , N − (cid:105) . Let us now prove the main result of this section. Theorem 3.5. Suppose that X N → x a.s., as N → ∞ . Then the [0 , − valued process { X t , t ≥ } defined by (3.4) is the unique solution to the Λ -W-F SDE with selection (3.5) . Proof : Strong uniqueness of the solution to (3.5) follows from Theorem 4.1 in [8].We now prove that ( X t ) t ≥ defined by (3.4) is a solution to the Λ-Wright-Fisher (3.5).We know that X Nt → X t a.s. for all t ≥ X N ⇒ X weakly in D ([0 , + ∞ ))as N → ∞ . Recall the decomposition X Nt = X N + M Nt − N Nt + V Nt . (3.6)It follows from Lemma 3.3 that N Nt → N → ∞ . We next showthat M Nt → (cid:90) [0 ,t ] × ]0 , × ]0 , p Ψ( X t − , u ) ¯ M ( ds, du, dp ) in probability , as N → ∞ , (3.7) here M = (cid:82) [0 , ¯ M ( ., ., ., dv, dw ). For each N ≥ t ≥ 0, let h N ( t ) = (cid:90) [0 ,t ] × [0 , (cid:0) ψ N ( X Ns − , u, p, v, w ) − p Ψ( X s − , u ) (cid:1) ¯ M ( ds, du, dp, dv, dw ) , where ¯ M is defined by (3.2). { h N ( t ) , t ≥ } is a martingale, and (cid:104) h N (cid:105) t = (cid:90) [0 ,t ] × [0 , (cid:0) ψ N ( X Ns , u, p, v, w ) − p Ψ( X s , u ) (cid:1) dsdup − Λ( dp ) dvdw. We have (cid:104) h N (cid:105) t ≤ (cid:90) [0 ,t ] × [0 , (cid:0) p Ψ( X Ns , u ) − p Ψ( X s , u ) (cid:1) dsdup − Λ( dp )+ 2 t sup ≤ s ≤ t (cid:90) [0 , (cid:0) ψ N ( X Ns , u, p, v, w ) − p Ψ( X Ns , u ) (cid:1) dup − Λ( dp ) dvdw ≤ (cid:90) [0 ,t ] × [0 , (cid:0) p Ψ( X Ns , u ) − p Ψ( X s , u ) (cid:1) dsdup − Λ( dp )+ 2 t sup ≤ r ≤ (cid:90) [0 , (cid:0) ψ N ( r, u, p, v, w ) − p Ψ( r, u ) (cid:1) dup − Λ( dp ) dvdw. Using the fact that X Ns → X s a.s., it is not hard to show by the dominated convergencetheorem that as N → ∞ , (cid:90) [0 ,t ] × [0 , (cid:0) Ψ( X Ns , u ) − Ψ( X s , u ) (cid:1) dsdu Λ( dp ) → . (3.8)Now from lemma 3.4 , it is easy to show that as N → ∞ ,sup ≤ r ≤ (cid:90) [0 , (cid:0) ψ N ( r, u, p, v, w ) − p Ψ( r, u ) (cid:1) dup − Λ( dp ) dvdw → ∀ t ≥ , (cid:104) h N (cid:105) t → a.s, as N → ∞ . On the other hand, from the bound | ψ ( r, u ) | ≤ (cid:104) h N (cid:105) t ≤ Ct Λ([0 , , ∀ N ≥ . Hence from the dominated convergence theoremlim N →∞ E [ h N ( t ) ] = 0 ∀ t ≥ M Nt = (cid:90) [0 ,t ] × [0 , ψ N ( X Ns − , u, p, v, w ) ¯ M ( ds, du, dp, dv, dw ) L −→ (cid:90) [0 ,t ] × [0 , p Ψ( X s − , u ) ¯ M ( ds, du, dp ) s N → ∞ , in particular M Nt → (cid:90) [0 ,t ] × [0 , p Ψ( X s − , u ) ¯ M ( ds, du, dp ) in probability , as N → ∞ . (3.7) is established.From (3.6), we deduce that1 N N (cid:88) k =1 V kt = 1 N N (cid:88) k =1 X kt − N N (cid:88) k =1 X k − N N (cid:88) k =1 M kt + 1 N N (cid:88) k =1 N kt . It follows from the above arguments and our assumption on the initial condition thatfor all t ≥ 0, as N → ∞ , the right–hand side converges in probability towards X t − x − (cid:90) [0 ,t ] × [0 , p Ψ( X s − , u ) ¯ M ( ds, du, dp ) . But1 N N (cid:88) k =1 V kt = − α (cid:90) t N N (cid:88) k =1 X ks η s ( k +1)=0 ds = − α (cid:90) t N N (cid:88) k =1 (cid:16) X ks − X s (cid:17) η s ( k +1)=0 ds − α (cid:90) t X s (cid:32) N N (cid:88) k =1 η s ( k +1)=0 (cid:33) ds → − α (cid:90) t X s (1 − X s ) ds a.s., as N → ∞ . The result follows clearly from the above facts. (cid:3) Remark 3.6. Our proof establishes in fact that for all t > , as N → ∞ , (cid:90) t X Ns η s ( N +1)=0 ds → (cid:90) t X s (1 − X s ) ds in probability. This does not mean that η s ( N +1)=0 converges, but it seems intuitivelyclear that for any ≤ r < t , (cid:90) tr η s ( N +1)=0 ds → (cid:90) tr (1 − X s ) ds. However, that convergence is not really easy to establish. Remark 3.7. Suppose we know a priori that ( X t ) t ≥ defined by (3.4) is a Markovprocess. Then we can prove that ( X t ) t ≥ is a solution to the Λ -Wright-Fisher SDE (3.5) as follows. Let us look backwards from time t to time . For each ≤ s ≤ t ,we denote by Z n,ts the highest level occupied by the ancestors at time s of the n firstindividuals at time t . We know that conditionally upon X t , the { η t ( i ) , i ≥ } are i.i.dBernoulli with parameter X t . Consequently, for any n ≥ , X nt = P ( η t (1) = · · · = η t ( n ) = 1 | X t ) , his implies that E x [ X nt ] = E x [ P ( η t (1) = · · · = η t ( n ) = 1 | X t )]= P x ( η t (1) = · · · = η t ( n ) = 1)= P x ( the . . . Z n,t individuals at time are all b )= E n [ x Z n,t ] . It is plain that the conditional law of Z n,t , given that ( η t (1) = · · · = η t ( n ) = 1) equalsthe conditional law of R t , given that R = n . Consequently, for each n ≥ E [ X nt | X = x ] = E [ Y nt | Y = x ] , where ( Y t ) t ≥ is a solution to (3.5) . But for all t > , r ∈ [0 , , the conditional lawof X t , given that X = x is determined by its moments, since X t is a bounded r. v.So ( X t ) t ≥ and ( Y t ) t ≥ have the same transition densities, that is { X t , t ≥ } is theunique weak solution to (3.5) . Uniqueness in law could also by proved as in [5] (where the case α = 0 is treated) bya duality argument, which we now sketch .Recall the notation Ψ( u, y ) = u ≤ y − y . For every y ∈ [0 , 1] and every function g : [0 , → R of class C , we set L g ( y ) = (cid:90) [0 , × [0 , (cid:104) g ( y + p Ψ( u, y )) − g ( y ) − p Ψ( u, y ) g (cid:48) ( y ) (cid:105) p − Λ( dp ) du − αg (cid:48) ( y )(1 − y ) y. A solution ( Y t ) t ≥ of (3.5) is a Markov process with generator L . Hence for every g : [0 , → R of class C , the process g ( Y t ) − (cid:90) t ds L g ( Y s ) , t ≥ g ( z ) = z n L g ( z ) = n (cid:88) k =2 (cid:18) nk (cid:19) λ n,k ( z n − k +1 − z n ) + αn ( z n +1 − z n ) . (3.10)Let { R t , t ≥ } be a N -valued jump Markov process which, when in state k , jumpsto 1. k − (cid:96) + 1 at rate (cid:0) k(cid:96) (cid:1) λ k,(cid:96) , ≤ (cid:96) ≤ k ;2. k + 1 at rate αk , α > n other words, the infinitesimal generator of { R t , t ≥ } is given by: L ∗ f ( k ) = k (cid:88) (cid:96) =2 (cid:18) k(cid:96) (cid:19) λ k,(cid:96) [ f ( k − (cid:96) + 1) − f ( k )] + αk [ f ( k + 1) − f ( k )] . For every z ∈ [0 , 1] and every r ∈ N , we set P ( z, r ) = z r . (3.11)Viewing P ( z, r ) as a function of r , we have L ∗ P ( z, r ) = r (cid:88) k =2 (cid:18) rk (cid:19) λ r,k [ z r − k +1 − z r ] + αr [ z r +1 − z r ]On the other hand, viewing P ( z, r ) as a function of z we can easily evaluate L P ( z, r )from formula (3.10), and we deduce that L P ( z, r ) = L ∗ P ( z, r ) . (3.12)Now suppose that ( Y t ) t ≥ is a solution to (3.5), and let R = n . By a standardargument (see Section 4.4 in [12]) we deduce from (3.12) that E [ P ( Y t , R )] = E [ P ( Y , R t )] , i.e E [ Y nt | Y = x ] = E [ x R t | R = n ] . Since this is true for each n ≥ Y t take values in the compact set [0, 1], this isenough to identify the conditional law of Y t , given that Y = x , for all 0 ≤ x ≤ Y t ) t ≥ is a homogeneous Markov process, this implies that the law of ( Y t ) t ≥ isuniquely determined. Λ -W-F SDE Λ -coalescent In this subsection, we recall a remarkable property of the Λ-coalescent (Π t ) t ≥ defined inthe introduction. For each n ≥ 1, let [ n ] t denote the number of blocks in the partitionΠ [ n ] t (Π [ n ] t is the restriction of Π t to [ n ] ). Then let T n = inf { t ≥ [ n ] t = 1 } . Asstated in (31) of [20], we have0 = T < T ≤ T ≤ . . . ↑ T ∞ ≤ ∞ . We say the Λ-coalescent comes down from infinity (Λ ∈ CDI ) if P ( t < ∞ ) = 1for all t > 0, and we say it stays infinite if P ( t = ∞ ) = 1 for all t > 0. Thecoalescent comes down from infinity if and only if T ∞ < ∞ a.s. We will show that his is equivalent to fixation. Kingman showed that the δ -coalescent comes down frominfinity.A necessary and sufficient condition for a Λ-coalescent to come down from infinitywas given by Schweinsberg [22]. Define φ ( n ) = n (cid:88) k =2 ( k − (cid:18) nk (cid:19) λ n,k , and ν ( dp ) = p − Λ( dp ) . It is not hard to deduce from the binomial formula that φ ( n ) = (cid:90) [ np − − p ) n ] ν ( dp ) . Schweinsberg’s result [22] says that the Λ-coalescent comes down from infinity if andonly if ∞ (cid:88) n =2 φ ( n ) < ∞ . (4.1)We shall see below that the convergence of this series is also necessary and sufficientfor fixation in finite time. Using the fact that the function f n ( p ) = (1 − p ) n − n , we have (cid:90) ( np − ν ( dp ) ≤ φ ( n ) ≤ n (cid:90) pν ( dp ) , ∀ n ≥ . The last assertion together with (4.1), implies that if (cid:82) pν ( dp ) < ∞ then the Λ-coalescent stays infinite. This result has been proved by Pitman (see lemma 25 in[20]).Theorem 3.5 shows that ( X t ) t ≥ is a bounded supermartingale. Indeed, if ( X t ) t ≥ is a solution to (3.5) , then for all 0 ≤ t ≤ s , E ( X t | F s ) ≤ x − α (cid:90) s X r (1 − X r ) dr + E (cid:104) (cid:90) [0 ,t ] × ]0 , × ]0 , p Ψ( u, X s − ) ¯ M ( ds, du, dp ) | F s (cid:105) = X s . Consequently the following limit exists a.s X ∞ = lim t →∞ X t ∈ { , } . (4.2)Indeed, 0 and 1 are the only possible limit values. .2 Fixation and non-fixation in the Λ -W-F SDE We assume that the initial proportion x of type B individuals satisfies 0 < x < Lemma 4.1. φ ( n ) n ↑ (cid:90) pν ( dp ) as n ↑ ∞ , where ν ( dp ) = p − Λ( dp ) . Proof : φ ( n ) = (cid:90) [ np − − p ) n ] ν ( dp )= (cid:90) (cid:20) np (cid:18) − (cid:90) (1 − up ) n − du (cid:19)(cid:21) Λ( dp ) . On the last line, we have made use of the identity(1 − p ) n − (cid:90) − np (1 − up ) n − du. For each p ∈ ]0 , f n ( p ) = 1 p (cid:18) − (cid:90) (1 − up ) n − du (cid:19) . We have, n − φ ( n ) = (cid:90) f n ( p )Λ( dp ) . The result follows from the monotone convergence theorem. (cid:3) We now deduce that Lemma 4.2. The function φ increases, and ∞ (cid:88) n =2 φ ( n ) < ∞ ⇒ ∞ (cid:88) n =2 φ ( n ) − αn < ∞ . Proof : We have φ ( n + 1) − φ ( n ) = (cid:90) [ p + (1 − p ) n +1 − (1 − p ) n ] ν ( dp )= (cid:90) p (1 − (1 − p ) n ) ν ( dp ) ≥ . hich implies the first claim. Now, we already know that if (cid:80) ∞ n =2 1 φ ( n ) < ∞ , then (cid:82) pν ( dp ) = ∞ . Thus, the second assertion is a consequence of the last lemma and thefollowing relation ∞ (cid:88) n =2 φ ( n ) − αn = ∞ (cid:88) n =2 αφ ( n )( n − φ ( n ) − α ) + ∞ (cid:88) n =2 φ ( n ) . The lemma is proved. (cid:3) For each t ≥ 0, we define again K t = inf { i ≥ η t ( i ) = 0 } . and T = inf { t ≥ K t = 1 } . We have the following Theorem 4.3. If Λ ∈ CDI , then one of the two types ( b or B ) fixates in finite time,i.e. ∃ ζ < ∞ a.s : X ζ = X ∞ ∈ { , } If Λ / ∈ CDI , then ∀ t ≥ , < X t < a.s. Proof : The proof has been inspired by [7] (see Section 4 ). Step 1 : Suppose that Λ ∈ CDI . We consider two cases. Case 1 : K = 1.In this case, the allele B fixates in the population. Indeed, the individual at level 1never dies and he cannot be pushed to an upper level. Let ζ = inf { t > η t ( i ) = 0 , ∀ i ≥ } .ζ is the time of fixation of allele B . We are going to show that ζ < ∞ a.s .We couple our original population process with the following N –valued process { Y t , t ≥ } , which describes the growth of a population which we denote “the B –population”, and whose dynamics we now describe. Y = 1, at time zero the B –population consists of a unique individual who occupies site 1, while all other sites k ≥ m on R + × [0 , 1] (see (1.2)) and of the sets I t,p as presented in the Introduction.At each time t corresponding to an atom ( t, p ) of the Poisson point process m , weassociate the set I t,p . We put a cross at time t on all levels i ∈ I t,p , except the lowestone. If there is at least one cross on the interval [2 , Y t − + 1], we modify the populationas follows (otherwise we do nothing). All individuals sitting at time t − below the lowestcross don’t move. All others are displaced upwards in such a way that all sites witha cross become free, and the respective orders of the individuals remain unchanged.Finally, individuals are added on all sites with a cross which lie below or immediately bove an occupied site. Clearly, as long as the growing number of individuals of the B –population remains below any given value k , the number of atoms of the Poissonprocess m which modify the size of the population on any given finite time intervalremains finite, and each jump in the population size is finite. However, we will nowshow that as a consequence of the CDI property of the associated coalescent process,the jumps of Y t accumulate in such a way that Y t = + ∞ , for some finite (random) t . Since it is plain that Y t is less than the total number of type B individuals in thepopulation, this will show that ζ < ∞ a.s.Indeed, looking backward in time, starting from any t > B –population” is the Lambda–coalescent. More precisely,as a time–reversal of our B –population process, it is the Lambda–coalescent startingfrom the random value Y t , and conditioned upon the fact that all the partitions havecoalesced into one single partition by time 0.This claim is justified as follows. Let { U s = Y t − s , ≤ s ≤ t } . At each time s of a point ( s, p ) of the PPP m where (cid:93) ( I s,p ∩ [1 , U s − ]) ≥ 2, all lineages of the set I s,p ∩ [1 , U s − ] coalesce. Would we describe the evolution of { U s , s ≥ } using copies of m and the I s,p ’s which would be independent of those used to describe the growth of Y · , then U · , starting from U = Y t = N , would be an instance of the N –Λ–coalescent.Here and below we make a slight abuse of terminology, calling Λ–coalescent the processwhich describes the number of blocks in a Λ–coalescent.For each N ≥ 2, we define ξ N = inf { t > , Y t ≥ N } , and by θ N the time taken by the N –Λ–coalescent to reach 1. It follows from an obviouscoupling that N → θ N is increasing. In fact we shall only use the fact that N → E θ N is increasing. Since Y ξ N − < N , it is plain that E ξ N ≤ E ξ + E θ N , (4.3)and moreover the law of ξ is exponential with parameter (cid:82) p ν ( dp ). Let us admit fora moment the Lemma 4.4. For any N > , E θ N ≤ ∞ (cid:88) k =2 φ ( k ) − , and this bound is finite since Λ ∈ CDI . Since ζ ≤ lim N →∞ ξ N , it follows from (4.3) and Lemma 4.4 that E ζ < ∞ .In order to conclude Case 1 of the first step of the proof of our Theorem, let usproceed with the Proof of Lemma 4.4 The Markov process which describes the number of ancestorsin a Λ–coalescent jumps from n to n − (cid:96) + 1 (2 ≤ (cid:96) ≤ n ) at rate (cid:18) n(cid:96) (cid:19) λ n,(cid:96) . In otherwords, its infinitesimal generator Q is given by Qf ( n ) = n (cid:88) (cid:96) =2 (cid:18) n(cid:96) (cid:19) λ n,(cid:96) [ f ( n − (cid:96) + 1) − f ( n )] . et us define for each n ≥ f ( n ) = ∞ (cid:88) k = n +1 φ ( k ) . We have for 2 ≤ (cid:96) ≤ n f ( n − (cid:96) + 1) − f ( n ) = n (cid:88) k = n − (cid:96) +2 φ ( k ) . Recall Lemma 4.2. Since 1 /φ is decreasing, we have for 2 ≤ (cid:96) ≤ n , f ( n − (cid:96) + 1) − f ( n ) ≥ ( (cid:96) − 1) 1 φ ( n ) , and therefore Qf ( n ) ≥ φ ( n ) n (cid:88) (cid:96) =2 (cid:18) n(cid:96) (cid:19) ( (cid:96) − λ n,(cid:96) = 1 . Using the fact that the process f ( U t ) − f ( U ) − (cid:90) t Qf ( U s ) ds, t ≥ E ( θ N ) ≤ E (cid:18)(cid:90) θ N Qf ( U s ) ds (cid:19) ≤ f (1) (cid:3) Case 2 : K > T < ∞ then type B fixates in finite time. Indeed, wait until T which is astopping time at which the Markov process { η t ( i ) , i ≥ } t ≥ starts afresh, and then usethe argument from Case 1.We suppose now that T = ∞ , which implies that K t → ∞ as t → ∞ , as alreadynoted in section 2.2.2. In other words, if T = ∞ , then the allele B does not fixate inthe population. Let n = inf { n ≥ φ ( n ) − αn ≥ } . Such an n exists because since Λ ∈ CDI , (cid:82) pν ( dp ) = + ∞ , hence by Lemma 4.1, wehave lim n →∞ n − φ ( n ) = + ∞ .We define a “ b –population” { Y t , t ≥ } , which again starts from a unique ancestorsitting on level 1. The novelty is that now each individual dies at rate α . It thenmay happen that the “ b –population” gets empty. In that case, we immediately startafresh with a new unique ancestor sitting at level 1. The fact that eventually the“ b –population” grows and become larger than any N is a consequence of the fact that K t → ∞ as t → ∞ . ote that the process describing the number of ancestors of the present individualsin that population is now a jump–Markov process with generator Q α given by Q α f ( n ) = n (cid:88) (cid:96) =2 (cid:18) n(cid:96) (cid:19) λ n,(cid:96) [ f ( n − (cid:96) + 1) − f ( n )] + αn ( f ( n + 1) − f ( n )) , conditioned upon hitting 1 before time t .Let N > n denote a fixed integer, ξ N the time taken by the “ b –population” toreach the value N , i.e. ξ N = inf { t > , Y t ≥ N } , and by θ n N the time taken by the process with generator Q α to come down below n ,starting from N . Similarly as in (4.3), we have E ξ N ≤ E ξ n + E θ n N . (4.4)In order to show that the allele b fixates in finite time, it remains to establish the Lemma 4.5. There exists a constant C < ∞ such that E θ n N ≤ C, for all N > n . Proof : For each n ≥ 1, we define f ( n ) = ∞ (cid:88) k = n +1 φ ( k ) − αk ) ∨ . By Lemma 4.2, for each n ≥ f ( n ) is finite. We have for 2 ≤ (cid:96) ≤ nf ( n − (cid:96) + 1) − f ( n ) = n (cid:88) k = n − (cid:96) +2 φ ( k ) − αk ) ∨ . Since k → / ( φ ( k ) − αk ) ∨ f ( n − (cid:96) + 1) − f ( n ) ≥ ( (cid:96) − 1) 1( φ ( n ) − αn ) ∨ , and therefore Qf ( n ) ≥ φ ( n ) − αn ) ∨ n (cid:88) (cid:96) =2 (cid:18) n(cid:96) (cid:19) ( (cid:96) − λ n,(cid:96) − αn ( φ ( n + 1) − α ( n + 1)) ∨ φ ( n )( φ ( n ) − αn ) ∨ − αn ( φ ( n + 1) − α ( n + 1)) ∨ ≥ φ ( n )( φ ( n ) − αn ) ∨ − αn ( φ ( n ) − αn ) ∨ ence Qf ( n ) ≥ , for each n ≥ n . Since the process f ( U t ) − f ( U ) − (cid:90) t Qf ( U s ) ds, t ≥ U t remains bounded while 0 ≤ t ≤ θ n N , E ( θ n N ) ≤ E (cid:32)(cid:90) θ n N Qf ( U s ) ds (cid:33) = f ( U θ n N ) − f ( U ) ≤ f (1) (cid:3) Step 2 : Suppose Λ / ∈ CDI , that is the Λ-coalescent does not come down frominfinity. We have ∞ (cid:88) n =2 φ ( n ) = + ∞ . (4.5)We claim that ( K t , t ≥ 0) does not reach ∞ in finite time. The contrary would implythat ∃ T < ∞ such that K T = ∞ a.s., so the number of ancestors at tiome 0 ofthe infinite population at time T in the Λ-lookdown model would be finite, whichcontradicts the fact that Λ / ∈ CDI . Hence K t < ∞ a.s. This implies that X t < t ≥ 0. Indeed if X t = 1, for some t > 0, by applying de Finetti’s Theorem, wededuce that η t ( i ) = 1 , ∀ i ≥ 1, which contradicts the fact that K t < ∞ . It remains toshow that X t > t ≥ m ≥ , t > 0, we define the event A mt = { The m first individuals of type b at time 0 are dead at time t } We have P ( A mt ) = (1 − e − αt ) m , and then P ( ∩ m A mt ) = 0 ∀ t > . From this, we deduce that ∃ i ≥ η t ( i ) = 1. The same argument used forthe proof of X t < X t > 0, for all t ≥ (cid:3) X ∞ Let x be the proportion of type b individuals at time 0, where 0 < x < 1. As theindividual at level 1 cannot be pushed to an upper level, we have { η (1) = 0 } ⊂ { X ∞ = 0 } , hence P ( X ∞ = 0) ≥ − x. If α = 0, ( X t ) t ≥ is a bounded martingale, so P ( X ∞ = 1) = E ( X ∞ ) = E ( X ) = x. f α > 0, by using (3.5) together with (4.2), we deduce that P ( X ∞ = 1) = E X ∞ < x. In this subsection we want to describe those cases where can we decide whether P ( X ∞ = 1) > P ( X ∞ = 1) = 0. We first prove Proposition 4.6. If Λ ∈ CDI , then P ( X ∞ = 1) > . Proof : Since Λ ∈ CDI , if all individuals at time 0 would be of type b , there wouldbe a (random) level J such that the individual sitting on level J at time 0 reaches + ∞ in finite time. Now P ( X ∞ = 1) > P ( K > J ) > 0, where K denotes the lowest level occupied by a type B individual at time 0. (cid:3) In the case Λ (cid:54)∈ CDI , since selection has infinite time to act, one may wonderwhether or not P ( X ∞ = 1) = 0. Some partial results have been obtained in thatdirection in Bah [3], but since then the question has been completely settled by Foucart[13] and Griffiths [15], who prove Theorem 4.7. Suppose that < x < , and let α ∗ := (cid:90) log (cid:18) − p (cid:19) ν ( dp ) . 1. If α < α ∗ , then < P ( X ∞ = 0 | X = x ) < .2. If α ≥ α ∗ , then X ∞ = 0 a.s. Needless to say, if α ∗ = + ∞ , which is in particular the case when Λ ∈ CDI , we arein the first case. Note that [13] settles the two cases α < α ∗ and α > α ∗ , while [15]treats the case α = α ∗ as well, assuming α ∗ < ∞ in the first case. We refer to [13] and[15] for references to earlier partial results on this problem in the biological literature. The aim of this section is to connect our model and results with the recent work ofH´enard [17], and to compute the law of X ∞ and the speed at which either type invadesthe whole population, in the case of the Bolthausen–Sznitman coalescent.H´enard’s definition of the fixation line is as follows. Consider the levels of theoffsprings at time t > N , the connected component containing 1 of which is of the form { , . . . , L t } .This defines the fixation line L t . In our case (in contradiction with H´enard’s situation),there may be no such connected component containing 1, if η (1) = 1 and η t (1) = 0 forsome t > 0, in which case we define L t to be 0. H´enard’s fixation line is an increasingprocess. Our is increasing if the individual sitting on level 1 at time 0 is of type B (i.e.is a 0), but this is not the case if that individual is of type b (i. e. is a 1). e are only interested in this second case, which is the only one where conditionallyupon the value of η (1), X ∞ is random. However, we will not necessarily assume that L = 1. We prefer to define the fixation line as follows.For all t ≥ 0, let L t = max { k ≥ η ( t ) = η ( t ) = · · · = η k ( t ) = 1 } , and this defines also L . Equivalently, L t = K t − 1, where K t is the lowest leveloccupied at time t by an individual of type 0, see the discussion in subsection 2.2.2. L t is clearly a Z + –valued continuous time Markov process. L t does not evolve as discussed in [17], since those individuals sitting on levels { , , . . . , L } , as well as their offsprings, are type b individuals, who die at rate α ,each death inducing a jump of L t of size − 1. The process { L t , t ≥ } is a Z + –valuedMarkov process, whose jump rates are given byΓ i,j = (cid:32) jj − i + 1 (cid:33) (cid:82) x j − i − (1 − x ) i Λ( dx ) , if 1 ≤ i < j < ∞ ; αi, if j = i − i ≥ 1, and the process is absorbed at 0. Indeed, Γ i,j is the rate at which K t jumps from i + 1 to j + 1. As was shown in subsection 2.2.2, either L t → ∞ , as t → ∞ ,in which case X ∞ = 1, or else L t hits zero in finite time, in which case X ∞ = 0. Inthe first case, L t explodes in finite time iff Λ ∈ CDI . In the case where Λ (cid:54)∈ CDI , itis of interest to describe the speed at which L t → ∞ , whenever this happens. This isdone in the case without selection (and it applies in our situation to the case where K = 1) in [17], in the situation Λ( dx ) = dx , i.e. the case of the Bolthausen–Sznitmancoalescent. We will show that the same result applies in our case, i.e. the slow–downdue to the death essentially does not modify that speed.Recall that the Bolthausen–Sznitman coalescent belongs to the family of the Beta(2 − α, α ) (0 < α < 2) coalescents, it corresponds to the case α = 1. Note that theBeta(2 − α, α ) coalescent comes down from infinity iff 1 < α < 2. The Bolthausen–Sznitman coalescent is the border case. One may expect that in this model, on theevent { L t → ∞} , L t → ∞ very fast, as t → ∞ .Before going to that, let us compute explicitly the law of X ∞ , in that case ofthe Bolthausen–Sznitman coalescent. The possibility of that computation is due tothe remark that in this particular case (and only in that one), the process L t is acontinuous time branching process. Indeed in the case Λ( dx ) = dx , we haveΓ i,i + j = (cid:40) ij ( j +1) , if j ≥ αi, if j = − L t is a Markov continuous time branching process, with life timeexponential with parameter 1 + α , and family size distribution { p j , j = 0 , , , . . . } given by p = α α , p j = 1 j ( j − α ) . ote that the generating function of that probability distribution is given by h ( s ) = s + α α + 1 − s α log(1 − s ) . We have Proposition 4.8. Conditionally upon L = k ( k ≥ ), P ( L t = 0) = (cid:2) − exp {− α (1 − e − t ) } (cid:3) k , P ( lim t →∞ L t = 0) = (cid:2) − e − α (cid:3) k . Proof : It suffices to consider the case L = 1, which we now do. In that case, itfollows from general results on continuous time branching processes, see e.g. chapterV in [16], that the collection of generating functions f t ( s ) = E [ s L t ] satisfies the ODE ∂ t f t ( s ) = Φ( f t ( s )) ,f ( s ) = s, where Φ( z ) = (1 + α )( h ( z ) − z ) = (1 − z )[ α + log(1 − z )] is the so–called infinitesimalgenerating function. It is not too hard to check that the solution of that ODE is f t ( s ) = 1 − exp (cid:2) α ( e − t − 1) + e − t log(1 − s ) (cid:3) . Hence P ( L t = 0) = f t (0) = 1 − exp (cid:2) α ( e − t − (cid:3) , from which the result follows. (cid:3) We can now conclude Corollary 4.9. Again in the case Λ( dx ) = dx , P ( X ∞ = 0 | X = x ) = 1 − x − x (1 − e − α ) . Proof : Recall that L = K − B individual attime 0, and that at time 0 individuals placed at levels 1 , , . . . are choosen in an i.i.d.manner, each one being of type b (i.e. 1) with probability x , and of type B (i.e. 0)with probability 1 − x . We have P ( X ∞ = 0 | X = x ) = ∞ (cid:88) k =0 (1 − e − α ) k P ( L = k )= (1 − x ) ∞ (cid:88) k =0 [ x (1 − e − α )] k = 1 − x − x (1 − e − α ) , were we have used Proposition 4.8 for the first equality. The result follows. (cid:3) Note that in the case Λ( dx ) = dx , Theorem 4.7 tells us that 0 < P ( X ∞ = 0) < α > α ∗ = + ∞ , which is consistent with the last result. Note also that[15] gives, for a general Λ coalescent, an expression for the above quantity in termsof the sum of an infinite series. It does not seem easy to deduce our result from thatformula. emark 4.10. The proportion of advantageous alleles is Y t = 1 − X t . Our formulasays (here “BS” refers to the Bolthausen–Sznitman coalescent) P BS ( Y ∞ = 1 | Y = y ) = yy + (1 − y ) e − α . If we replace the Bolthausen–Sznitman by Kingman’s coalescent, it is well–known (seee. g. [15]) that the formula reads P K ( Y ∞ = 1 | Y = y ) = 1 − e − αy − e − α . We note that these two formulae coincide, and are equal to (1 + e − α ) − , in the case y = 1 / . The following comparison holds : for all α > , P BS ( Y ∞ = 1 | Y = y ) > P K ( Y ∞ = 1 | Y = y ) if < y < / , while P BS ( Y ∞ = 1 | Y = y ) < P K ( Y ∞ = 1 | Y = y ) if / < y < . Indeed the difference P BS ( Y ∞ = 1 | Y = y ) − P K ( Y ∞ = 1 | Y = y ) hasthe same sign as Φ( y ) = e − αy ( e − α + y (1 − e − α )) + ( e − α − e − α ) y − e − α . Now Φ(0) = Φ(1 / 2) = Φ(1) = 0 , Φ (cid:48) (0) > , Φ (cid:48) (1 / < and Φ (cid:48) (1) > for all α > ,while Φ (cid:48)(cid:48) ( y ) vanishes at the unique point < y α = 1 − e − α − αe − α α (1 − e − α ) < . We now establish Theorem 4.11. In the case Λ( dx ) = dx , if L = 1 , then conditionally upon L t → ∞ as t → ∞ , e − t log L t → e a.s. , where e is a standard exponential r.v. Proof : Recalling the infinitesimal generating function Φ specified in Proposition 4.8,it is not hard to see that the function1Φ(1 − x ) − x log x is integrable near zero (one way to see that is to make the change of variable y = 1 /x ,and note that the resulting integral, say from 2 to ∞ , converges, by comparison with aBertrand series). Hence condition (3) of Theorem 3 from Grey [14] is satisfied, whichimplies the stated convergence, but is remains to specify the law of e .We follow the strategy of proof of Proposition 3.8 in [17]. For each t > s (cid:55)→ f t ( s )is a bijection from [0 , 1] onto [1 − exp {− α (1 − e − t ) } , g t ( s ) = 1 − exp (cid:2) α ( e t − 1) + e t log(1 − s ) (cid:3) . t is a bijection from [1 − exp {− α (1 − e − t ) } , 1] onto [0 , − exp {− α } ≤ s ≤ < g t ( s ) ≤ 1, and the process { g t ( s ) L t , t ≥ } is Markov and has constant expectation.Indeed E (cid:2) g t ( s ) L t (cid:3) = f t ( g t ( s )) = s. Hence it is a [0 , t → ∞ to a r.v. V ( s ).Moreover, by dominated convergence and explicit computation, for any β > E [ V ( s ) β ] = lim t →∞ E (cid:104) g t ( s ) βL t (cid:105) = lim t →∞ f t [ g t ( s ) β ] = s. This implies that V ( s ) takes values in { , } , and P ( V ( s ) = 1) = E ( V ( s )) = s .Let us now define the r.v. U = inf { − e − α < s ≤ , V ( s ) = 1 } It is plain that { U ≤ s } = { V ( s ) = 1 } , hence P ( U ≤ s ) = s , for s ∈ (1 − e − α , g (1 − e − α ) = 1 − e − α , we have that { V (1 − e − α ) = 1 } = { L ( t ) → } ,and we see that the law of U has a Dirac measure of mass 1 − e − α at 1 − e − α , and hasdensity 1 on the interval (1 − e − α , ≤ s ≤ e − α , we have that log[ g t (1 − s )] (cid:39) − ( ρs ) e t as t → ∞ , where ρ = e α . If s < − U , then g t (1 − s ) L t → , hence L t log[ g t (1 − s )] → , or equivalently( ρs ) e t L t → , as t → ∞ , while if s > − U , g t (1 − s ) L t → , hence L t log[ g t (1 − s )] → −∞ , or equivalently( ρs ) e t L t → ∞ , as t → ∞ . Let Θ := 1 − U . For any ε > ρ (Θ − ε )] e t L t → , and [ ρ (Θ + ε )] e t L t → + ∞ , as t → ∞ . Taking again the logarithm, we deduce that as t → ∞ , αe t + log(Θ − ε ) e t + log( L t ) → −∞ ,αe t + log(Θ + ε ) e t + log( L t ) → + ∞ . Consequently − log(Θ + ε ) − α ≤ lim inf t →∞ e − t log( L t ) ≤ lim sup t →∞ e − t log( L t ) ≤ − log(Θ − ε ) − α. This being true for any ε > 0, we have proved that, as t → ∞ , e − t log( L t ) → − log(Θ) − α. e now define e := − log(Θ) − α . Conditionally upon L t → ∞ as t → ∞ , the law of U is uniform on [1 − e − α , , e − α ]. Then for r > P ( e > r | L t → ∞ ) = P (Θ < exp[ − ( α + r )] | L t → ∞ ) = exp[ α ] exp[ − ( α + r )] = e − r . (cid:3) Let β n the time taken by the fixation line L t , starting from L = 1, to exceed thevalue n . As noted in [17], a consequence of Theorem 4.11 is that β n − log log( n ) → − log( e ) a.s. as n → ∞ . In the situation treated in [17], β n has the same law as τ n , the time taken by the n –Λcoalescent to hit the value 1, i.e. the time taken for n individuals to find their mostrecent common ancestor.In our case, the Λ–coalescent must be replaced by the Λ – Ancestral SelectionGraph. Indeed, since in the forward time direction individuals die, in the backwardtime direction we have birth of lineages.The n –Λ–ASG is defined as follows. Starting from n lineages, the lineages coalesceaccording to the Λ–coalescent, while new lineages are born according to the followingrule. While there are k ≥ αk , this lineagebeing placed on a level chosen uniformly among the levels { , , . . . , k + 1 } . If the level (cid:96) ≤ k is chosen, the lineages located on levels (cid:96), (cid:96) + 1 , . . . , k just before the birth eventget pushed one level up. We refer to [18] and [19] for the description of the ASG,where the coalescent is Kingman’s coalescent. We note that here we consider only type b individuals, type B individuals occupying possibly some of the higher levels.Define τ n to be the time for the n –Λ–ASG to find a common ancestor, i.e. the timefor the number of lineages to reduce to 1. It follows from Theorem 4.11 that, in thecase Λ( dx ) = dx , as n gets large, the decrease of the number of lineages due to thecoalescence events is much faster than the creation of new lineages, hence τ n < ∞ a.s.[17] shows that in the case α = 0, the law of τ n coincides with that of β n , the timetaken by the fixation line starting from 1 to reach a value greater than or equal to n .This is no longer true in the case α > 0, since the process of the number of lineages inthe n –Λ–ASG is no longer decreasing. Here β n has rather the law of the time elapsedbetween the last time when there are at least n lineages in the n –Λ–ASG, and the timewhen there is one lineage. However for large n this does not make a real difference, asfollows from the following result. Lemma 4.12. Fix an arbitrary h > . On the event that L t → ∞ as t → ∞ , for n large enough, L τ n + s > n , for all s ≥ h . Proof : Choose ε > e − ε e + ε > e − h . It follows from Theorem 4.11 that there exists t ε such that for any t ≥ t ε , e − ε ≤ e − t log L t ≤ e + ε, nd n ε such that whenever n ≥ n ε ,log log n − log( e + ε ) ≤ τ n ≤ log log n − log( e − ε ) . Choose n ≥ n ε such that moreover τ n ≥ t ε . Consequently e τ n ≥ log n e + ε , and whenever s ≥ h , L τ n + s ≥ exp (cid:0) e τ n + s ( e − ε ) (cid:1) ≥ exp (cid:0) e τ n e s ( e − ε ) (cid:1) > exp (cid:0) e τ n ( e + ε ) (cid:1) ≥ n. (cid:3) Consequently, the time elapsed between the first visit of a level above n by L t , andthe last visit below n after that time (if any) tends to zero in probability, as n → ∞ .As a result, we can conclude as in [17] Proposition 4.13. Suppose we are again in the case Λ( dx ) = dx , and define τ n asabove. Then, as n → ∞ , τ n − log log n ⇒ − log e . Remark 4.14. We expect that our look–down construction, and the duality with the Λ –ASG can produce new results beyond the case of the Bolthausen–Sznitman coalescent,at least in the case of the Beta–coalescents, in particular concerning the law of thenumber of blocks implied in the last coalescence in the Beta (2 − α, α ) –ASG, and theexpectation of the depth of the Beta (2 − α, α ) –ASG in case < α < . Λ –coalescent In this last section we suppose that the measure Λ is general (i.e Λ( { } ) > ν is infinite. Note that we could have Λ((0 , X t of type b individuals at time t in the population of infinite size is a solution to the stochasticdifferential equation with selection X t = x − α (cid:90) t X s (1 − X s ) ds + (cid:90) t (cid:112) cX s (1 − X s ) dB s + (cid:90) [0 ,t ] × ]0 , × ]0 , p ( u ≤ X s − − X s − ) ¯ M ( ds, du, dp ) , (5.1) here c = Λ( { } ) , ¯ M is the compensated measure M defined in section 3.2, and B is astandard Brownian motion. Let { W ( ds, du ) } be a white noise on (0 , ∞ ) × (0 , 1] basedon the Lebesgue measure dsdu . We remark that if ( X t ) t ≥ satisfies (5.1), then X t is asolution in law of the following stochastic differential equation X t = x − α (cid:90) t X s (1 − X s ) ds + √ c (cid:90) [0 ,t ] × ]0 , ( u ≤ X s − X s ) W ( ds, du )+ (cid:90) [0 ,t ] × ]0 , p ( u ≤ X s − − X s − ) ¯ M ( ds, du, dp ) . We first define the model. Recall the process { η t ( i ) , i ≥ , t ≥ } defined in theintroduction. The evolution of the population is the same as that described in the caseΛ( { } ) = 0 except that we superimpose single births, which are described as followsFor any 1 ≤ i < j , arrows are placed from i to j according to a rate Λ( { } ) Poissonprocess, independently of the other pairs i (cid:48) < j (cid:48) . Suppose there is an arrow from i to j at time t . Then a descendent (of the same type) of the individual sitting onlevel i at time t − occupies the level j at time t , while for any k ≥ j , the individualoccupying the level k at time t − is shifted to level k + 1 at time t . In other words, η t ( k ) = η t − ( k ) for k < j , η t ( j ) = η t − ( i ), η t ( k ) = η t − ( k − 1) for k > j .By coupling our model with the simplest lookdown model with selection defined in[4], it is not hard to show that for N large enough, the individual sitting on level 2 N at time 0 never visits a level below N , that is the evolution within the box ( t, i ) ∈ [0 , ∞ ) × { , , . . . , N } is not altered by removing all crosses above 2 N . The process { η t ( i ) , i ≥ , t ≥ } is well-defined.For each N ≥ t ≥ 0, denote by X Nt the proportion of type b individuals attime t among the first N individuals, i.e. X Nt = 1 N N (cid:88) i =1 η t ( i ) . (5.2)Combining the arguments in [4] and section 2.3 (see above), it is easy to show if( η ( i )) i ≥ are exchangeable random variables, then for all t > 0, ( η t ( i )) i ≥ are ex-changeable. An application of de Finetti’s theorem, yields that X t = lim N →∞ X Nt exists a.s. (5.3)Using the definition of the model, it is easy to see that ( ψ N was defined by (3.1)) X Nt = X N + K Nt + (cid:90) [0 ,t ] × ]0 , ψ N ( X Ns − , u, p, v, w ) ¯ M ( ds, du, dp, dv, dw ) − N (cid:90) [0 ,t ] × [0 , u ≤ X Ns − η s − ( N +1)=0 M N ( ds, du ) , where K Nt is a martingale of jump size ± N . We have emma 5.1. (cid:104)K N (cid:105) t = (cid:90) t ϕ N ( s ) ds where, ϕ N ( s ) = Λ(0) X Ns (1 − X Ns ) . Proof : For each 1 ≤ i < N , let P i be a Poisson process with intensity Λ(0)( N − i ).At time t ∈ P i , we have∆ X Nt = N , if η t − ( i ) = 1 and η t − ( N ) = 0 − N , if η t − ( i ) = 0 and η t − ( N ) = 10 , otherwise.Now, let A i = { η t ( i ) = 1 , η t ( N ) = 0 } ,B i = { η t ( i ) = 0 , η t ( N ) = 1 } . We have P ( A i | X Nt ) = P ( B i | X Nt ) = NN − X Nt (1 − X Nt ) , from which, we deduce that (cid:104)K N (cid:105) t = 1 N Λ(0) N ( N − NN − X Nt (1 − X Nt )= Λ(0) X Nt (1 − X Nt )The result is proved. (cid:3) Now, let Y Nt = X N + (cid:90) [0 ,t ] × ]0 , ψ N ( X Ns − , u, p, v, w ) ¯ M ( ds, du, dp, dv, dw ) − N (cid:90) [0 ,t ] × [0 , u ≤ X Ns − η s − ( N +1)=0 M N ( ds, du ) . We have X Nt = K Nt + Y Nt , ∀ t ≥ ∀ T ≥ ≤ t ≤ T sup N ≥ | ϕ N ( s ) |≤ C a.s. Using the last identity, we deduce by Aldous’ tightness criterion (see Aldous [1]) that {K Nt , t ≥ , N ≥ } is tight in D ([0 , ∞ )) . ince K N is tight, there exists a subsequence of the sequence K N such that K N ⇒ K weakly in D ([0 , ∞ )) , where K is a continuous martingale (since the jumps of K N are of size ± N ) such that < K > t = (cid:90) t cX s (1 − X s ) ds, (5.5)where c = Λ(0). The main result of this section is Theorem 5.2. Suppose that X N → x a.s, as N → ∞ . Then the [0 , − valued process { X t , t ≥ } defined by (5.3) is the (unique in law) solution to the stochastic differentialequation X t = x − α (cid:90) t X s (1 − X s ) ds + (cid:90) t (cid:112) Λ(0) X s (1 − X s ) dB s + (cid:90) [0 ,t ] × ]0 , p ( u ≤ X s − − X s − ) ¯ M ( ds, du, dp ) , (5.6) where ¯ M is the compensated measure M defined in section 3.2, and B is a standardBrownian motion. The identification of the limiting equation is done similarly as in the proof of The-orem 3.5. Strong uniqueness of the solution to (5.6) follows again from Dawson and Li[8], and weak uniqueness could also be proved by a duality argument.Since Kingman’s coalescent comes down from infinity, we have fixation in our newmodel in finite time as soon as Λ(0) > Acknowledgements We thank an anonymous Referee, who drew our attention to the work of H´enard [17],which permitted us to add the subsection 4.4 to our original version. Other Referee’sremarks helped us to improve other points of our first version.This work has been supported by the ANR project MANEGE, and by the Infec-tiopˆole Sud Foundation. References [1] D. Aldous, Stopping times and tightness, Ann. Probab . , 586-595, 1989.[2] D. Aldous, Exchangeability and related topics, in Ecole d’´et´e St Flour 1983 , Lec-ture Notes in Math. , 1–198, 1985.[3] B. Bah, Le mod`ele du look-down avec s´election, Phd Thesis, Univ. Aix–Marseille,2012. 4] B. Bah, E. Pardoux, and A. B. Sow, A look–down model with selection, StochasticAnalysis and Related Topics , L. Decreusefond et J. Najim Ed, Springer Proceed-ings in Mathematics and Statistics Vol , 2012.[5] J. Bertoin and J. F. Le Gall. Stochastic flows associated to coalescent processes II:Stochastic differential equations. Ann. Inst. Henri Poincar´e Probabilit´es et Statis-tiques , 307-333, 2005.[6] J. Bertoin and J. F. Le Gall. Stochastic flows associated to coalescent processesIII: Limit theorems. Illinois J. 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Griffiths, The Lambda-Fleming-Viot process and a connection with Wright-Fisher diffusion, arXiv:1207.1007.[16] T.E. Harris, The theory of branching processes , Springer 1963.[17] O. H´enard. The fixation line, arXiv:1307.0784[18] S. Krone, C. Neuhauser, Ancestral processes with selection, Theor. Popul. Biol, , 210–237, 1997.[19] C. Neuhauser, S. Krone, The genealogy of samples in models with selection, Ge-netics Ann. Probab . , 1870-1902, 1999.[21] S. Sagitov. The general coalescent with asynchronous mergers of ancester lines. J.Appl. Prob . , 1116-1125, 1999.[22] J. Schweinsberg, A necessary and sufficient condition for the Lambda-coalescentto come down from infinity., Electron. Commun. Probab. , , 1–11, 2000. oubacar Bah I2M, Aix–Marseille Universit´e, 39, rue F. Joliot Curie, F 13453Marseille cedex 13. [email protected] Pardoux (corresponding author) Aix-Marseille Universit´e, CNRS, CentraleMarseille, I2M, UMR 7373 13453 Marseille, France. [email protected] Bah I2M, Aix–Marseille Universit´e, 39, rue F. Joliot Curie, F 13453Marseille cedex 13. [email protected] Pardoux (corresponding author) Aix-Marseille Universit´e, CNRS, CentraleMarseille, I2M, UMR 7373 13453 Marseille, France. [email protected]