On exceptional times for generalized Fleming Viot processes with mutations

Transcription

1 Stoch PD: Anal Comp 214 2:84 12 DOI 1.17/s On exceptional times for generalized Fleming Viot processes with mutations J. Berestyci L. Döring L. Mytni L. Zambotti Received: 1 April 213 / Accepted: 14 February 214 / Published online: 6 March 214 Springer Science+Business Media New Yor 214 Abstract If Y is a standard Fleming Viot process with constant mutation rate in the infinitely many sites model then it is well nown that for each t > the measure Y t is purely atomic with infinitely many atoms. However, Schmuland proved that there is a critical value for the mutation rate under which almost surely there are exceptional times at which the stationary version of Y is a finite sum of weighted Dirac masses. In the present wor we discuss the existence of such exceptional times for the generalized Fleming Viot processes. In the case of Beta-Fleming Viot processes with index α ]1, 2[ we show that irrespectively of the mutation rate and α the number of atoms is almost surely always infinite. The proof combines a Pitman Yor type representation with a disintegration formula, Lamperti s transformation for self-similar processes and covering results for Poisson point processes. Keywords Fleming Viot processes Mutations xceptional times xcursion theory Jump-type SD Self-similarity J.Berestyci L. Döring L. Zambotti B Laboratoire de Probabilités et Modéles Aléatoires, Université Paris 6, 4 Place Jussieu, Paris Cedex 5, France lorenzo.zambotti@upmc.fr J. Berestyci julien.berestyci@upmc.fr L. Döring leif.doering@upmc.fr L. Mytni Faculty of Industrial ngineering and Management, Technion Israel Institute of Technology, 32 Haifa, Israel leonid@ie.technion.ac.il

2 Stoch PD: Anal Comp 214 2: Mathematics Subject Classification 2 Primary 6J8 Secondary 6G18 1 Main result The measure-valued Fleming Viot diffusion processes were first introduced by Fleming and Viot [2] and have become a cornerstone of mathematical population genetics in the last decades. It is a model which describes the evolution forward in time of the genetic composition of a large population. ach individual is characterized by a genetic type which is a point in a type-space. The Fleming Viot process is a Marov process Y t t on M 1 = { ν : ν is a probability measure on } for which we interpret Y t B as the proportion of the population at time t which carries a genetic type belonging to a Borel set B of types. In particular, the number of different types at time t is equal to the number of atoms of Y t with the convention that the number of types is infinite if Y t has absolutely continuous part. Fleming Viot superprocesses can be defined through their infinitesimal generators Lφμ= δ 2 φμ μdvδ v dy μdy δμvδμy + μdva δφμ v, δμ 1.1 acting on smooth test-functions where δφμ/δμv = lim ɛ + ɛ 1 {φμ + ɛδ v φμand A is the generator for a Marov process in which represents the effect of mutations. Here δ v is the Dirac measure at v. It is well nown that the Fleming Viot superprocess arises as the scaling limit of a Moran-type model for the evolution of a finite discrete population of fixed size if the reproduction mechanism is such that no individual gives birth to a positive proportion of the population in a small number of generations. For a detailed description of Fleming Viot processes and discussions of variations we refer to the overview article of thier and Kurtz [19] and to theridge s lecture notes [17]. The first summand of the generator reflects the genetic resampling mechanism whereas the second summand represents the effect of mutations. Several choices for A have appeared in the literature. In the present wor we shall wor in the setting of the infinitely-many-alleles model where each mutation creates a new type never seen before. Without loss of generality let the type space be =[, 1]. Then the following choice of A gives an example of an infinite site model with mutations: Afv = θ f y f vdy, 1.2 for some θ>. The choice of the uniform measure dyis arbitrary we could choose the new type according to any distribution that has a density with respect to the Lebesgue

3 86 Stoch PD: Anal Comp 214 2:84 12 measure, all that matters is that the newly created type y is different from all other types. With A as in 1.2, mutations arrive at rate θ and create a new type piced at random from according to the uniform measure, therefore the corresponding process is sometimes called the Fleming Viot process with neutral mutations. Let us briefly recall two classical facts concerning the infinite types Fleming Viot process described above. For any initial condition Y : i If there is no mutation, then, for all t > fixed, the number of types is almost surely finite. ii If the mutation parameter θ is strictly positive, then, for all t > fixed, the number of types is infinite almost surely. This can be deduced e.g. from the explicit representation of the transition function given in thier and Griffiths in [18]. A beautiful complement to i and ii was found by Schmuland for exceptional times that are not fixed in advance: Theorem 1.1 Schmuland [35] For the stationary infinitely-many-alleles model P t > : #{types at time t} < = { 1 if θ<1, if θ 1. Schmuland s proof of the dichotomy is based on analytic arguments involving the capacity of finite dimensional subspaces of the infinite dimensional state-space. In Sect. 6 we reprove Schmuland s theorem with a simple proof via excursion theory, that yields the result for arbitrary initial conditions. In the series of articles [5 7], Bertoin and Le Gall introduced and started the study of -Fleming Viot processes, a class of stochastic processes which naturally extends the class of standard Fleming Viot processes. These processes are completely characterized by a finite measure on ], 1] and a generator A. Similarly to the standard Fleming Viot process, these processes can be defined through their infinitesimal generator Lφμ = 1 y 2 dy μdaφ1 yμ + yδ a φμ + μdva δφμ v, 1.3 δμ and the sites of atoms are again called types. For A =, the generator formulation only appeared implicitly in [6] and is explained in more details in Birner et al. [1] and for A as in 1.2 it can be found in Birner et al. [9]. The dynamics of a generalized Fleming Viot process Y t t are as follows: at rate y 2 dy a point a is sampled at time t > according to the probability measure Y t da and a point-mass y is added at position a while scaling the rest of the measure by 1 y to eep the total mass at 1. The second term of 1.3 is the same mutation operator as in 1.1. For a detailed description of -Fleming Viot processes and discussions of variations we refer to the overview article of Birner and Blath [8].

4 Stoch PD: Anal Comp 214 2: In the following we are going to focus only on the choice = Beta2 α, α,the Beta distribution with density f u = C α u 1 α 1 u α 1 du, C α = 1 Ɣ2 αɣα, for α ]1, 2[, and mutation operator A as in 1.2. The corresponding -Fleming Viot process Y t t is called Beta-Fleming Viot process or α, θ-fleming Viot process and several results have been established in recent years. The α, θ-fleming Viot processes converge wealy to the standard Fleming Viot process as α tends to 2. It was shown in [1] that a -Fleming Viot process with A = is related to measurevalued branching processes in the spirit of Perin s disintegration theorem precisely if is a Beta distribution this relation is recalled and extended in Sect. 2.3 below. If we chose α ]1, 2[ and Y uniform on [, 1], then we find the same properties i and ii for the one-dimensional marginals Y t unchanged with respect to the classical case 1.1, 1.2. In fact, for a general -Fleming Viot process, i is equivalent to the requirement that the associated -coalescent comes down from infinity see for instance [2]. Here is our main result: contrary to Schmuland s result, α, θ-fleming Viot processes with α ]1, 2[ and θ> never have exceptional times: Theorem 1.2 Let Y t t be an α, θ-fleming Viot superprocess with mutation rate θ> and parameter α ]1, 2[. Then for any starting configuration Y for any θ>. P t > : #{types at time t} < = One way one can get a first rough understanding of why this should be true is by using a heuristic based on the duality between -Fleming Viot processes and -coalescents. If -Fleming Viot processes describe how the composition of a population evolve forward in time, -coalescents describe how the ancestral lineages of individuals sampled in the population merge as one goes bac in time. The fact that coalescents describe the genealogies of -Fleming Viot processes can be seen from Donelly and Kurtz [14] so-called loodown construction of Fleming Viot processes and was also established through a functional duality relation by Bertoin and Le Gall in [5]. The coalescent which corresponds to the classical Fleming Viot process is the celebrated Kingman s coalescent. Kingman s coalescent comes down from infinity at speed 2/t, i.e. if one initially samples infinitely many individuals in the populations, then the number of active lineages at time t in the past is N t and N t 2/t almost surely when t. It is nown see [6] or more recently [28] that the process N t, t has the same law as the process of the number of atoms of the Fleming Viot process. For a Beta-coalescent that is a -coalescent where the measure is the density of abetaα, 2 α variable with parameter α 1, 2 we have N t c α t 1/α 1 almost surely as t see[3, Theorem 4]. Therefore Kingman s coalescents comes down from infinity much quicer than Beta-coalescents. Since the speed at which the generalized Fleming Viot processes looses types roughly corresponds to the speed at

5 88 Stoch PD: Anal Comp 214 2:84 12 which the dual coalescent comes down from infinity, it is possible that α, θ-fleming Viot processes do not lose types fast enough, and hence there are no exceptional times at which the number of types is finite. 2 Auxiliary constructions To prove Theorem 1.2 we construct two auxiliary objects: a particular measure-valued branching process and a corresponding Pitman Yor type representation. Those will be used in Sect. 5 to relate the question of exceptional times to covering results for point processes. In this section we give the definitions and state their relations to the Beta-Fleming Viot processes with mutations. All appearing stochastic processes and random variables will be defined on a common stochastic basis, G, G t, P that is rich enough to carry all Poisson point processes PPP in short that appear in the sequel. 2.1 Measure-valued branching processes with immigration We recall that a continuous state branching process CSBP in short with α-stable branching mechanism, α ]1, 2], is a Marov family P v v of probability measures on càdlàg trajectories with values in R +, such that v e λx t = e v u t λ, v,λ, 2.1 where for ψ : R + R +, ψu := u α, we have the evolution equation u t λ = ψu tλ, u λ = λ. See e.g. [27] for a good introduction to CSBP. For α = 2, ψu = u 2 is the branching mechanism for Feller s branching diffusion, where P v is the law of the unique solution to the SD t X t = v + 2Xs db s, t, 2.2 driven by a Brownian motion B t t. On the other hand, for α ]1, 2[, ψu = u α gives the so-called α-stable branching processes which can be defined as the unique strong solution of the SD t X t = v + X 1/α s dl s, t, 2.3 driven by a spectrally positive α-stable Lévy process L t t, with Lévy measure given by

6 Stoch PD: Anal Comp 214 2: x> c α x 1 α dx, c α := αα 1 Ɣ2 α. Note that strong existence and uniqueness for 2.3 follows from the fact that the function x x 1/α is Lipschitz outside zero, and hence strong existence and uniqueness holds for 2.3 until X hits zero. Moreover X, being a non-negative martingale, stays at zero forever after hitting it. For a more extensive discussion on strong solutions for jumps SDs see [22] and [32]. The main tool that we introduce is a particular measure-valued branching process with interactive immigration MBI in short. For a textboo treatment of this subject we refer to Li [31]. Following Dawson and Li [12], we are not going to introduce the MBIs via their infinitesimal generators but as strong solutions of a system of stochastic differential equations instead. On, G, G, P, let us consider a Poisson point process N = r i, x i, y i i I on,,, adapted to G and with intensity measure νdr, dx, dy := 1 r> dr c α 1 x> x 1 α dx 1 y> dy. 2.4 Throughout the paper we adopt the notation Ñ := N ν, i.e. Ñ is the compensated version of N.Itwasshownin[12] that the solution to 2.3 has the same law as the unique strong solution to the SD X t = X + ],t] R + R + 1 y<xr x Ñ dr, dx, dy 2.5 with X = v. Now we are going to switch to the measure-valued setting. The real-valued process X in 2.3, 2.5 describes the evolution of the total mass of the CSBP starting at time zero at the mass X = v. We are going to consider all initial masses v [, 1] simultaneously, constructing a process X t t taing values in the space M F [,1] of finite measures on [, 1], endowed with the narrow topology, i.e. the trace of the wea- topology of C[, 1]. Assume that at time t =, X is a finite measure on [, 1] with cumulative distribution function Fv, v [, 1], and denote X t v := X t [,v], t,v [, 1]. Then the measure-valued branching process X t t can be constructed in such a way that for each v, X t v t solves 2.5 with X = Fv, and with the same driving noise for all v [, 1]. In what follows, we deal with a version of 2.5 including an immigration term only depending on the total-mass X t 1:

7 9 Stoch PD: Anal Comp 214 2:84 12 X t v = Fv + 1 y<xr v x Ñ dr, dx, dy + I v ],t] R + R + v [, 1], t, t gx s 1 ds, 2.6 where I v, v [, 1] is the cumulative distribution function of a finite measure on [, 1] and we assume G g : R + R + is monotone non-decreasing, continuous and locally Lipschitz continuous away from zero. Definition 2.1 An M F [,1] -valued process X t t on, G, G t, P is called a solution to 2.6if it is càdlàg P-a.s., for all v [, 1], setting X t v := X t [,v], X t v v [,1],t satisfies P-a.s Moreover, a solution X t t is strong if it is adapted to the natural filtration F t generated by N. Finally, we say that pathwise uniqueness holds if P X 1 t = X 2 t, t = 1, for any two solutions X 1 and X 2 on, G, G t, P driven by the same Poisson point process. Here is a well-posedness result for 2.6: Theorem 2.2 Let F and I be as above. For any immigration mechanism g satisfying Assumption G, there is a strong solution X t t to 2.6 and pathwise uniqueness holds until T := inf{t : X t [, 1] = }. The proof of Theorem 2.2 relies on ideas from recent articles on pathwise uniqueness for jump-type SDs such as Fu and Li [22]orDawsonandLi[12]. Our equation 2.6 is more delicate since all coordinate processes depend on the total-mass X t 1. The uniqueness statement is first deduced for the total-mass X t 1 t and then for the other coordinates interpreting the total-mass as random environment. To construct a wea solution we use a pathwise Pitman Yor type representation as explained in the next section. 2.2 A Pitman Yor type representation for interactive MBIs Let us denote by the set of càdlàg trajectories w : R + R + such that w =, w is positive on a bounded interval ],ζw[ and w on[ζw,+ [. We recall the construction of the excursion measure of the α-stable CSBP P v v, also called the Kuznetsov measure, see[3, Section4]or[31, Chapter 8]: For all t, let K t dx be the unique σ -finite measure on R + such that

8 Stoch PD: Anal Comp 214 2: e λ x K t dx = u t λ = λ 1 α 1 + α 1t 1 α, λ, R + where we recall that the function u t λ t is the unique solution to the equation u t λ + t u s λ α ds = λ, t, λ. We also denote by Q t x, dy the Marov transition semigroup of P v v. Then there exists a unique Marovian σ -finite measure Q on with entrance law K t t and transition semigroup Q t t, i.e. such that for all < t 1 < < t n, n N, Qw t1 dy 1,...,w tn dy n, t n < ζw = K t1 dy 1 Q t2 t 1 y 1, dy 2 Q tn t n 1 y n 1, dy n. 2.7 By construction 1 e λw s Qdw = us λ = λ 1 α 1 + α 1s 1 α, s, λ, 2.8 and under Q, for all s >, conditionally on σw r, r s, w t+s t has law P ws. The σ -finite measure Q is called the excursion measure of the CSBP 2.3. By 2.8, it is easy to chec that for any s > w s Qdw = λ u sλ = lim λ= λ 1 + λ α 1 α α 1s 1 α = In Duquesne Le Gall s setting [15], under the σ -finite measure Q with infinite total mass, w has the distribution of l a e a under nde, where nde is the excursion measure of the height process H and l a is the local time at level a. For the more general superprocess setting see for instance Dynin and Kuznetsov [16]. We need now to extend the space of excursions as follows: D := {w : R + R + : s, w on[, s], w s }, i.e. D is the set of càdlàg trajectories w : R + R + such that w is equal to on [, sw], w is positive on a bounded interval ]sw, sw + ζw[ and w on [sw + ζw,+ [. Fors, we denote by Q s dw the σ -finite measure on D given by w Q s dw := 1 s w s Qdw, 2.1 D

9 92 Stoch PD: Anal Comp 214 2:84 12 i.e. Q s is the image measure of Q under the map w γ t := 1 t s w t s, t Let us consider a Poisson point process s i, u i, a i,w i i I on R + R + D with intensity measure Ɣds, du, da, dw := δ ds δ du Fda + ds du I da Q s dw 2.12 where F and I are the cumulative distribution functions appearing in 2.6. An atom s i, u i, a i,w i is a population that has immigrated at time s i whose size evolution is given by w i and whose genetic type is given by a i. The coordinate u i is used for thinning purposes, to decide whether or not this particular immigration really happened or not. Theorem 2.3 Suppose g : R + R + satisfies Assumption G. Then, for all v [, 1], there is a unique càdlàg process Z t v, t on, G, G t, P satisfying P-a.s. { Z t v = s i = wi t 1 a i v + s i > wi t 1 a i v1u i gz si 1, t >, Z v = Fv Moreover, we can construct on, G, G t, P a PPP N with intensity ν given by 2.4 such that Z solves 2.6 with respect to N. If I 1 = 1, then in the special case of branching mechanism ψλ = λ 2 and constant immigration rate g θ, the total-mass process X t = X t 1 for 2.6 also solves { dx t = 2X t db t + θ dt, t, X = F1. for which Pitman and Yor obtained the excursion representation in their seminal paper [34]. Remar 2.4 The recent monograph [31] by Zenghu Li contains a full theory of this ind of Pitman Yor type representations for measure-valued branching processes, see in particular Chapter 1. We present a different approach below which shows directly how the different Poisson point processes in 2.6 and in 2.13 are related to each other. The most important feature of our construction is that it relates the excursion construction and the SD construction on a pathwise level. Observe that an immediate and interesting corollary of Theorem 2.3 is the following: Corollary 2.5 Let g be an immigration mechanism satisfying assumption G and let X t t be a solution to 2.6. Then almost surely, X t is purely atomic for all t.

10 Stoch PD: Anal Comp 214 2: In the proof of our Theorem 1.2 we mae use of the fact that the Pitman Yor type representation is well suited for comparison arguments. If g can be bounded from above or below by a constant, then the righthand side of 2.6 can be compared to an explicit PPP for which general theory can be applied. 2.3 From MBI to Beta-Fleming Viot processes with mutations Let us first recall an important characterization started in [6] and completed in [12] which relates Fleming Viot processes, defined as measure-valued Marov processes by the generator 1.3, and strong solutions to stochastic equations. Theorem 2.6 Dawson and Li [12] Let be the Beta distribution with parameters 2 α, α. Suppose θ and M is a non-compensated Poisson point process on, [, 1] [, 1] with intensity ds y 2 dy du. Then there is a unique strong solution Y t v t,v [,1] to Y t v = v + y [ 1 u Ys v Y s v ] t Mds, dy, du + θ [v Y s v]ds, ],t] [,1] [,1] v [, 1], t, 2.14 and the measure-valued process Y t [,v] := Y t v is an α, θ-fleming Viot process started at uniformly distributed initial condition. xistence and uniqueness of solutions for this equation was proved in Theorem 4.4 of [12] while the characterization of the generator of the measure-valued process Y is the content of their Theorem 4.9. We next extend a classical relation between Fleming Viot processes and measurevalued branching processes which is typically nown as disintegration formula. Without mutations, for the standard Fleming Viot process this goes bac to Konno and Shiga [26] and it was shown in Birner et al. [1] that the relation extends to the generalized -Fleming Viot processes without immigration if and only if is a Betameasure. Our extension relates α, θ-fleming Viot processes to 2.6 with immigration mechanism gx = αα 1Ɣαθ x 2 α and for θ = gives an SD formulation of the main result of [1]. Theorem 2.7 Let Fv = I v = v and let g : R + R + be defined by gx = αα 1Ɣαθ x 2 α for some α 1, 2. Let then X t t be the unique solution of to 2.6 in the sense of Definition 2.1 such that and Define X t 1 =, t T := inf{s > : X s 1 = }. t St = αα 1Ɣα X s 1 1 α ds

11 94 Stoch PD: Anal Comp 214 2:84 12 Y t dv = X S 1 tdv, t X S 1 t1 Then Y t t is well-defined, i.e. S 1 t <T for all t, and is an α, θ-fleming Viot process, i.e. a strong solution to 2.14 with = Beta2 α, α. The proof of the theorem is different from the nown result for θ =. To prove that X S 1 t1 > for all t, Lamperti s representation for CSBPs was crucially used in [1]. This idea breas down in our generalized setting since the total-mass process X t 1 is not a CSBP. Our proof uses instead the fact that for all θ the total-mass process is self-similar and an interesting cancellation effect of Lamperti s transformation for self-similar Marov processes and the time-change S. In [1] we study a generalized version of the total mass process X t 1, t and we show that the extinction time T = inf{t : X t 1 = } is finite almost surely if and only if θ<ɣα. Otherwise T = almost surely. We will see in the proof of Theorem 2.7 that in both cases lim t S 1 t = T a.s. Theorem 2.7 thus gives some partial information on the behavior of X t near the t extinction time T : XS 1 Corollary 2.8 As t the probability-valued process t dv X S 1 t 1 converges t wealy to the unique invariant measure of Y t, t. As t T, almost surely, there exists a random sequence of times t 1 < t 2 <...< T tending to T such that the sets A i = support of X ti are pairwise disjoints. This corollary is a direct consequence of the result, due to Donnelly and Kurtz [13,14], that the α, θ-fleming Viot process as well as its loodown particle system is strongly ergodic and of Theorem 2.7. For the sae of self-containdeness, a setch of the proof is given in Sect. 7 which specialize and explicits the arguments of Donelly and Kurtz to our case. 3 Proof of Theorems 2.2 and 2.3 Recall that s i, u i, a i,w i i I is a Poisson point process on R 3 + D with intensity measure Ɣ given as in 2.12, and that we use the notation We are going to show that for all v [, 1] there exists a unique càdlàg process Z t v, t solving { Z t v = s i = wi t 1 a i v + s i > wi t 1 a i v1 ui gz si 1, t >, Z = Fv. 3.1

12 Stoch PD: Anal Comp 214 2: Fig. 1 Definition of N.Ontheleft-hand side we represent the point process s i,w i, a i. Observe that s 4 = while s 1, s 2, s 3 >. On the right-hand side we show how the w i are combined to construct the noise N Then we are going to construct a PPP N with intensity dr c α 1x > x 1 α dx dy such that, for all v [, 1], Z is solution of The Pitman Yor type representation with predictable random immigration We start by replacing the immigration rate gz s 1 s> in the right-hand side of 3.1 with a generic F t -predictable process V s s, that we assume to satisfy V t and t V s ds < + t ; 3.2 this will be useful when we perform a Picard iteration in the proof of existence of solutions to 2.6 and 3.1. Then we consider { Z t v := s i = wi t 1 a i v + s i > wi t 1 a i v1 ui V si, t >, v [, 1], Z v := Fv, v [, 1]. 3.3 Then we want to show that there is a noise N on, G, G, P such that Z is a solution of an equation of the type Definition of N Let us consider a family of independent random variables U ij i, j N such that U ij is uniform on [, 1] for all i, j N. We also assume that U ij i, j N is independent

13 96 Stoch PD: Anal Comp 214 2:84 12 of the PPP s i, u i, a i,w i. Then, for all atoms s i, u i, a i,w i in the above PPP, we define the following point process N i := r i j, xi j, yi j j J i : 1 r i j j J i is the family of jump times of r wi r ; 2 for each r i j we set x i j := wi r i j w i r i j, yi j := wi r i j U ij. 3.4 We note that N i is not expected to be a Poisson point process. For each N we set L := Fa and L t := wt i, t >, a i <a,u i V si L t := sup L t, t. 3.5 We consider a PPP N = r j, x j, y j j with intensity measure ν given by 2.4 and independent of s i, u i, a i,w i i,u ij i, j N,V t t. We set for any non-negative measurable f = f r, x, y fdn := 1 u V s f r, x, y + Lr N dr, dx, dy + f r, x, y + L r N dr, dx, dy. 3.6 The filtration we are going to wor with is F t := σ s i, u i, a i,wr i, U ij, V r, ri, x i, y i i : r t, s i t, ri t, i, j N, t. We are going to prove the following Proposition 3.1 N is a PPP with intensity νdr, dx, dy = dr c α x 1 α dx dy. Proof For f = f r, x, y wenowset I t := 1 u V s f r, x, y + Lr N dr, dx, dy. ],t] R + R + Since wt i = ifs i t, V is predictable and we can write L t := 1 ai <a wt i + 1 ai <a wt i, s i = s i >,u i V si then we obtain that L is predictable. Hence, I t is F t -measurable and for t < T

14 Stoch PD: Anal Comp 214 2: I T I t F t = 1 u V s f r, x, y + Lr N dr, dx, dy F t ]t,t ] R + R + = 1 u V s 1 s <t f r, x, y + Lr N dr, dx, dy F t ]t,t ] R + R u V s 1 s t f r, x, y + Lr N dr, dx, dy F t ]t,t ] R + R + We will need the following two facts: 1 Conditionally on wt and s < t the process w +t has law P wt this follows for instance from Let w t, t be a CSBP started from w with law P w.letm = r i, x i, y i be a point process which is defined from w and a sequence of i.i.d. uniform variables on [, 1] as N is constructed from w and U ij i, j N. Then for any positive function f = f r, x, y f r, x, ymdr, dx, dy [,T ] R + R + = w f r, x, y1 y wr νdr, dx, dy. [,T ] R + R + Let us start with the case s < t. Using the above facts we see that 1 u V s 1 s <t ]t,t ] R + R + = 1 u V s 1 s <t = 1 u V s 1 s <t f r, x, y + Lr N dr, dx, dy F t f r, x, y + Lr N dr, dx, dy wt, Lḳ c α 1 [L r,lr +w r [y f r, x, y dr dx dy x1+α F t ]t,t ] R + R + ]t,t ] R + R + F t

15 98 Stoch PD: Anal Comp 214 2:84 12 Let us now consider the case s > t. 1 u V s 1 s t f r, x, y + Lr N dr, dx, dy F t ]t,t ] R + R + = lim 1 u V s 1s t f r, x, y + Lr 1 s +ɛ<r N dr, dx, dy F ɛ t ]t,t ] R + R + = lim 1 u V s 1s t f r, x, y + Lr 1 s +ɛ<r N dr, dx, dy ws ɛ +ɛ, Lḳ F t ]t,t ] R + R + = lim 1 u V s 1 s t 1 ɛ [L r,lr +w r [ y f r, x, y1 c α s +ɛ<r dr dx dy x1+α F t ]t,t ] R + R + c α = 1 u V s 1 s t 1 [L r,lr +w r [y f r, x, y dr dx dy x1+α F t ]t,t ] R + R + where we need to introduce the indicator that r > s + ɛ to get a sum of CSBP started from a positive initial mass and thus be in a position to apply the above fact. We conclude that I T I t F t = ]t,t ] R + R + 1 ],sup L r [y f r, x, y dr c α dx dy x1+α F t, Therefore by the Definition 3.6 of N ]t,t ] R + R + = = ]t,t ] R + R + ]t,t ] R + R + f r, x, y N dr, dx, dy F t 1],L r [y + 1 ]L r, [y f r, x, y dr f r, x, y dr c α dx dy. x1+α c α dx dy x1+α By [23, Theorem II.6.2], a point process with deterministic compensator is necessarily a Poisson point process, and therefore the proof is complete. Proposition 3.1 tells us how to construct a Poisson noise N from the s i, u i, a i,w i. Let us now show that Z solves 2.6 with this particular noise. F t.

16 Stoch PD: Anal Comp 214 2: Proposition 3.2 Let Z satisfy 3.3. Then for all v, Zv, N satisfies P-a.s. Z t v = Fv + ],t] R + R + 1 y<zr v x Ñ dr, dx, dy + I v Proof Using an idea introduced by Dawson and Li [11], we set for n N Z n t t V s ds, t. v := wt i 1 s i + n 1 t. 3.7 a i v,u i V si Note that Q{w 1/n > } <+ for all n 1, so that Zt n is P-a.s. given by a finite sum of terms. Moreover, by the properties of PPPs, s i, u i, a i,w i : w1/n i > is a PPP with intensity δ ds δ du Fda+ds du I da 1 w1/n > Qdw. Moreover Zt n Z t as n + for all t. Now we can write Z n t v = Mn t v + J n t v, with M n t v := wt i wi s i si + 1 t, n n a i v,u i V si Jt n v := a i v,u i V si w i s i + 1 n 1 si + n 1 t. 3.8 Let us concentrate on M n first. We can write, for s i + 1 n t, w i t wi s i + 1 n = r x Ñ i dr, dx, dy [s i + 1 n,t] R+ R + 1 y<wi where N i is defined in 3.4 and Ñ i dr, dx, dy is the compensated version of N i : Ñ i dr, dx, dy := N i dr, dx, dy 1 y<w i r νdr, dx, dy, with ν defined in 2.4. We set A i,n := {y, r : L i r y < Li r + wi r 1 si + 1n r }, B v n := a i v,u i V si A i,n. Since Q{w 1/n > } <+, only finitely many {A i,n } i such that u i V si are non-empty P-a.s and, moreover, the {A i,n } i are disjoint. Then by 3.6

17 1 Stoch PD: Anal Comp 214 2: Ai,n y, r x Ñ dr, dx, dy ],t] R + R + = 1 si + n 1 t = w i t wi s i + 1 n 1 y<wi r x Ñ i dr, dx, dy [s i + n 1,t] R+ R + 1 si + 1 n t so that 1 Bv y, r x Ñ dr, dx, dy = n ],t] R + R + a i v,u i V si = M n t v. wt i wi s i + n 1 1 si + n 1 r We need first the two following technical lemmas. Lemma 3.3 For a F t -predictable bounded process f t : R + R we set Then we have M t := ],t] R + R + f r y x Ñ dr, dx, dy, t. sup M t C f 2 r y dr dy + f r y dr dy. t [,T ] [,T ] R + [,T ] R + Proof Recall that ν α dx = c α x 1 α dx.weset J 1,t := f r y 1 x 1 x Ñ dr, dx, dy, t, ],t] R + R + J 2,t := f r y 1 x>1 x Ñ dr, dx, dy, t. ],t] R + R + Then, by Doob s inequality, 2 sup J 1,t sup J 1,t 2 t [,T ] t [,T ] 4 c α x 1 α dx ],1] [,T ] R + f 2 r y dr dy

18 Stoch PD: Anal Comp 214 2: while sup t [,T ] J 2,t 2 ]1, [ c α x α dx f r y dr dy. [,T ] R + Lemma lim n z z 1n 1 Qdz =. n 2 lim n z 1 1 z 1n 1 Qdz =. n Proof First recall from 2.9 that z 1 n Qdz = 1 for all n. The proof of 1 is based on the estimate 1 e x 1 e x for x [, 1] which follows from differentiating both sides. Of course, the inequality also implies that x 2 1 x 1 ex1 e x, x. We apply this estimate to the excursion measure: z z 1n 1 Qdz e z 1 1 e z 1 n Qdz n n = e z 1 Qdz z 1 e z 1 n Qdz. 3.9 n n Next, by 2.8, z 1 e z 1 n Qdz = d n dλ u 1 n 1/nλ = λ=1 1 + α 1 1 α/α 1 1, n so that 3.9 combined with z 1 Qdz = 1 proves 1. For 2 we use that x1 x>1 n e e 1 x1 e x to get z 1 n 1 z 1n >1 Qdz e e 1 z 1 n 1 e z 1 n Qdz which goes to zero as argued above. Lemma 3.5 For all v and T we have lim n sup Z t v Zt n v =, t [,T ]

19 12 Stoch PD: Anal Comp 214 2:84 12 Z t v t is P-a.s. càdlàg and P-a.s. Z t v = Fv + ],t] R + R + 1 y<zr v x Ñ dr, dx, dy + I v Proof We have obtained above the representation First, let us note that B v n Bv := t V s ds. Zt n v = n 1 Bvy, r x Ñ dr, dx, dy + J n t v. 3.1 ],t] R + R + a i v,u i V si A i, A i := { } y, r : Lr i y < Li r + wi r, and moreover B v \ Bn v = } {y, r : Lr i y < Li r + wi r 1 si + 1n >r a i v and the latter union is disjoint. If we set then and by Lemma 3.3 M t v := M t v M n t v = ],t] R + R + 1 Bvy, r x Ñ dr, dx, dy, ],t] R + R + 1 B v \B v n y, r x Ñ dr, dx, dy 1 C sup M t Mt n T T 1 B v \B v y, r dr dy + n t [,T ] s i + = n 1 T w i r dr + a i v,u i V si s i T a i v,u i V si s i T 1 B v \Bn v y, r dr dy s i + 1 n T wr i dr. 3.11

20 Stoch PD: Anal Comp 214 2: Then we get = = T n 1 = Fv a i v,u i V si 1 si = a i v Fv s i + 1 n T s i T s i + 1 n T s i T wr i dr T w r Qdwdr + T 1 T + n wr i dr + V s I v V s I v a i v,u i V si 1 si > s+ 1 n T s T s + 1 s ds, n s i + 1 n T s i T wr i dr w r s Qdwdrds where the last equality follows by 2.9. By our assumptions on V the right hand side in the above display converges to, as n. Hence 3.11 also converges to, as n. Let us now deal with J n t. Note that we can write where J n t+ n 1 v = a i v,u i V si w i s i + 1 n A n t := <s i t 1 si t = A n t + t w i 1 1 si = + I v n a i v w i s i + 1 n t 1 ai v,u i V si I v V s ds, V s ds, and A n t t is a martingale such that A n =. We have by an analog of Lemma 3.3 and its proof sup t [,T ] A n t 2 K V z z 1n 1 Qdz + 2K V z 1 1 z 1n >1 Qdz, n n T where K V := I v V s ds. The righthand side tends to zero as n by Lemma 3.4. Analogously w i 1/n 1 s i = Fv a i v

21 14 Stoch PD: Anal Comp 214 2: F1 z z 1n 1 Qdz + 2F1 n z 1 n 1 z 1n >1 Qdz, which again tends to as n by Lemma 3.4. Therefore [ sup Z t v Zt n v ]. t [,T ] and, passing to a subsequence, we see that a.s. sup Z t v Z n t v 3.12 t [,T ] observe that in fact we don t need to tae a subsequence since Z n t is monotone nondecreasing in n. In particular, a.s. Z t v, t is càdlàg and we obtain Z t v = Fv + ],t] R + R + 1 Bvy, r x Ñ dr, dx, dy + I v It remains to prove that a.s. B v ={y, r : y < Z r v}. By definition a.s. Z r v = a i v,u i V si w i r, r. t V s ds. If a i v and u i V si, then L i r +wi r Z r v, so that B v {y, r : y < Z r v}. On the other hand, if y < Y r v, then there is one j such that a i <a j,u i V si w i r = L j r y < L j r + w j r. Therefore we have obtained the desired results. The proof of Proposition 3.2 is complete. 3.2 Proof of Theorem 2.3 With a localisation argument we can suppose that g is globally Lipschitz. Let us first show uniqueness of solutions to 3.1. Let v = 1. If Zt i, t for i = 1, 2 is a càdlàg process satisfying 3.1 with v = 1, then taing the difference we obtain

22 Stoch PD: Anal Comp 214 2: t Zt 1 Zt 2 I 1 = I 1 t ds z t s Qdz gz 1 s gz 2 s ds gz 1 s gz 2 s, where the second equality follows by 2.9. By the Lipschitz-continuity of g and the Gronwall Lemma we obtain Z 1 = Z 2 a.s., i.e. uniqueness of solutions to 3.1. The next step is to use an iterative Picard scheme in order to construct a solution of 3.1 and thus of 2.6. Let v := 1, and let us set Z t := and for all n Zt n+1 := wt i 1 u i 1 + wt i 1 u i gzs n i, t. s i = s i > By recurrence and monotonicity of g, Z n+1 t Z n t and therefore a.s. there exists the limit Z t := lim n Z n t. To show that Z is actually the solution of 3.1 we show first that it is càdlàg by proving that the convergence holds in a norm that maes the space of càdlàg processes on [, T ] complete and then by proving that 3.1 holds almost surely for each fixed t. Let us first show that Z n is a Cauchy sequence for the norm Z =sup t [,T ] Z t for which first we set Z n, t := Z n++1 t Z n+1 t = s i > w i t 1 gz n s i <u i gz n+ s i. By an analog of Proposition 3.2 we can construct a PPP N n, with the intensity measure 1 r> dr c α 1 x> x 1 α dx 1 y> dy such that for all t t Zt n, = t 1 y<z n, r x Ñ n, dr, dx, dy + I 1 [ gz n+ s gz n s ] ds. Then by the Lipschitz-continuity of g with the Lipschitz constant L, and by Lemma 3.3 sup Zt n, t [,T ] T T Zs n, ds + T Zs n, ds + I 1L Zs n 1, ds. We show now that the right hand side in the latter formula vanishes as n + uniformly in. Indeed

23 16 Stoch PD: Anal Comp 214 2:84 12 t t Zt n+1 =, Zt = F1 + gzs n ds C + L Zs n ds. Then by recurrence Zt n+1 Ce tl and by monotone convergence we obtain that Z t Ce tl. By dominated convergence it follows that Z n+1 t i.e. the sequence T T Zs n ds Z s ds, T Zs n ds is Cauchy and we conclude that Z n Z in the sense of the above norm and therefore Z is almost surely càdlàg. The above argument also show that Z t = wt i 1 u i 1 + wt i 1 u i gz si, s i = s i > holds almost surely for each fixed t and therefore for all t, i.e. Z is a solution of 3.1 forv = 1. Setting V s := gz s 1 and applying Proposition 3.2, we obtain 3.1 and the proof of Theorem 2.3 is complete. 3.3 Proof of Theorem 2.2 Let us start from existence of a wea solution to 2.6; by Theorem 2.3 we can build a process Z t v, t,v [, 1] and a Poisson point process N dr, dx, dy such that 3.1 and 2.6 hold. Now, we set X t := wt i δ u i + wt i 1 gz si 1> δ ui /gz si 1, s i = s i > where δ a denote the Dirac mass at a; by construction it is clear that X t v := X t [,v], for all v [, 1], is a solution to 2.6. It remains to prove that X t t is càdlàg in the space of finite measures on the space [, 1]. By Lemma 3.5, for all v [, 1], X t v t is càdlàg; by countable additivity, a.s. X t v t is càdlàg for all v Q [, 1]; then, by the compactness of [, 1], it is easy to see that X t t is càdlàg: for instance, a.s. any limit point of X tn,fort n t and t n t, is equal on each interval ]a, b], a, b Q [, 1], tox t b X t a = X t ]a, b]. Therefore, we have proved that X t t isasolutionto2.6 in the sense of Definition 2.1. It remains to prove pathwise uniqueness. Let X i t t, i = 1, 2, be two solutions to 2.6 driven by the same Poisson noise N and let us set Xt iv := Xi t [,v], v [, 1]. Let us first consider the case v = 1: then Xt i 1, t, i = 1, 2, solves a particular case of the equation considered by Dawson and Li [12, 2.1]; therefore, by [12, Theorem 2.5], PXt 11 = X t 2 1, t = 1.

24 Stoch PD: Anal Comp 214 2: Let us now consider v<1; in this case the equation satisfied by Xt i v, t depends on Xt i 1, t and therefore the uniqueness result by Dawson and Li does not apply directly. Instead, we consider the difference D t := Xt 1v X t 2v so that the drift terms cancel since X 1 1 = X 2 1. Hence, D t, t can be treated as if g were identically equal to. The same proof as in [12] shows that PXt 1v = X t 2v, t = 1. Finally, since a.s. the two finite measures X1 t and X 2 t are equal on each interval ]a, b], a, b Q [, 1], they coincide. Therefore, pathwise uniqueness holds for 2.6. Finally, in order to obtain existence of a strong solution, we apply the classical Yamada-Watanabe argument, for instance in the general form proved by Kurtz [25, Theorem 3.14]. 4 Proof of Theorem 2.7 We consider the immigration rate function gx = αα 1Ɣαθ x 2 α, x. Now g is not Lipschitz-continuous, so that Theorem 2.3 does not apply directly. However, by considering g n x = αα 1Ɣαθx n 1 2 α, we obtain a monotone nondecreasing and Lipschitz continuous function for which Theorem 2.3 yields existence and uniqueness of a solution X n t v, t,v to 2.6. We now define T :=, T n := inf{t > : X n t 1 = n 1 } and X t v := n 1 X n t v 1 T n 1 t<t n. Since X t 1 t has no downward jumps, it follows that T := sup n T n is equal to inf{s > : X s 1 = }, and moreover X t 1 = for all t T. By pathwise uniqueness, if n m then X n t v = X m t v on {t T m }, and therefore X t v, t,v isasolutionto2.6 forgx = αα 1Ɣαθ x 2 α with the desired properties. Pathwise uniqueness follows from the same localisation argument. To prove that the right-hand side of 2.15 is well-defined, i.e. the denominator is always strictly positive, we are going to apply Lamperti s representation for selfsimilar Marov process. A positive self-similar Marov process of index w is a strong Marov family P x x> with coordinate process denoted by U t t in the Sorohod space of càdlàg functions with values in [, + [, satisfying the law of cu c 1/w t t under P x is given by P cx 4.1 for all c >. Lamperti has shown in [29] that this property is equivalent to the existence of a Lévy process ξ such that, under P x, the process U t T t has the same law as x exp ξa 1 tx 1/w t, where A 1 t := inf{s : A s > t} and At := t 1 exp w ξ s ds.

25 18 Stoch PD: Anal Comp 214 2:84 12 We now use Lamperti s representation to find a surprisingly simple argument for the well-posedness of Lemma 4.1 The right-hand side of 2.15 is well-defined for all v [, 1] and t. Proof In Lemma 1 of [1] it was shown that, if L is a spectrally positive α-stable Lévy process as in 2.3, solutions to the SD t X t = X + t X 1/α s dl s + αα 1Ɣαθ X 2 α s ds 4.2 trapped at zero induce a positive self-similar Marov process of index 1/α 1.The corresponding Lévy process ξ has been calculated explicitly in [1, Lemma 2.2], but for the proof here we only need that ξ has infinite lifetime and additionally a remarable cancellation effect between the time-changes. Since, by Lemma 1 of Fournier [21], the unique solution to the SD 4.2 forx = 1 coincides in law with the unique solution to X t = 1 + ],t] R + R + 1 y<xs x Ñ ds, dx, dy + αα 1Ɣαθ t Xs 2 α ds, we see that the total-mass process X t 1 t and exp ξ A 1 t are equal in law t up to first hitting. Applying the Lamperti transformation for t < T yields St := = = t t X s 1 1 α ds exp1 αξ A 1 s ds A 1 t = A 1 t exp1 αξ s expα 1ξ s ds so that S and A are reciprocal for t < T. Plugging this identity into the Lamperti transformation yields = X T 1 = lim X t 1 = lim expξ A 1 t T t T t = lim expξ St. 4.3 t T For the second equality we used left-continuity of X 1 at T which is due to Sect. 3 of [29] because the Lévy process ξ does not jump to. Using that ξ t > for

26 Stoch PD: Anal Comp 214 2: any t [,, from4.3 we see that S explodes at T, that is ST =. Since S and S only differ by the factor αα 1Ɣα, it also holds that ST = so that X S 1 t1 > for all t. We can now show how to construct on a pathwise level the Beta-Fleming Viot processes with mutations the measure-valued branching process. Proof of Theorem 2.7 Suppose N is the PPP with compensator measure ν that drives the strong solution of 2.6 with atoms r i, x i, y i i I,,,. Then we define a new point process on,,, by Mds, dz, du := δ { Sr i, i I x i Xr i 1+x i 1, y i yi Xr i 1 Xr i 1 } ds, dz, du. If we can show that the restriction M of M to,, 1, 1 is a PPP with intensity measure M ds, dz, du = ds C αz 2 z 1 α 1 z α 1 dz du and furthermore that R t v := X S 1 tv X S 1 t1 t,v [,1] is a solution to 2.14 with respect to M, then the claim follows from the pathwise uniqueness of Step 1: We have R t v = X S 1 t v X S 1 t 1 = = X v X 1 + S 1 t [ Xr v + x1 y Xr v X r 1 + x1 y Xr 1 X ] r v N νdr, dx, dy X r 1 S 1 t [ Xr v + x1 y Xr v + X r v X r 1 + x1 y Xr 1 X r 1 x1 y X r v X r 1 + x1 ] y X r 1 X r v X r 1 2 νdr, dx, dy + αα 1Ɣαθv S 1 t 1 X r 1 X r 1 2 α dr θ S 1 t [ Xr v + x1 y Xr v = v + X r 1 + x1 y Xr 1 S 1 t + αα 1Ɣαθv X r 1 1 α dr θ S 1 t X r v X r 1 2 X r 1 2 α dr X ] r v N dr, dx, dy X r 1 S 1 t X r v X r 1 X r 1 1 α dr.

27 11 Stoch PD: Anal Comp 214 2:84 12 To verify the third equality, first note that due to Lemma II.2.18 of [24] the compensation can be split from the martingale part and then can be canceled by the compensator integral since integrating-out the y-variable yields S 1 t [ x1 y X r v X r 1 + x1 ] y X r 1 X r v X r 1 2 c α x 1 α dr dx dy =. To replace the jumps governed by the PPP N by jumps governed by M note that by the definition of M we find, for measurable non-negative test-functions h for which the first integral is defined, the almost sure transfer identity S 1 t = h t 1 Sr, x, X r 1 + x1 y Xr 1 y N dr, dx, dy X r 1 hs, z, u Mds, dz, du 4.4 or in an equivalent but more suitable form S 1 t h r, = x X S 1 r 1+x1 y XS 1 r 1, t y X S 1 r 1 N dr, dx, dy 1 h S 1 s, z, u Mds, dz, du. 4.5 Since the integrals are non-compensated we actually defined M in such a way that the integrals produce exactly the same jumps. Let us now rewrite the equation found for R in such a way that 4.5 can be applied: R t v = v + S 1 t [ x1y Xr v X r 1 X r vx1 y Xr 1 X r 1 + x1 y Xr 1X r 1 N dr, dx, dy + αα 1Ɣαθ [ vx r 1 1 α X r v X r 1 X r 1 1 α] dr = v + S 1 t x S 1 t X r 1 + x1 y Xr 1 ]

28 Stoch PD: Anal Comp 214 2: [ 1 y Xr v X ] r v X r 1 1 y X r 1 N dr, dx, dy + αα 1Ɣαθ S 1 t [ vx r 1 1 α X r v X r 1 X r 1 1 α] dr. The stochastic integral driven by N can now be replaced by a stochastic integral driven by M via 4.5: t R t v = v + 1 [ z 1 uxs 1 s 1 X S 1 s v ] R S 1 s v1 uxs 1 s 1 X S 1 s 1 Mds, dz, du + θ = v + + θ t [ v Rs v ] ds t 1 t ] z [1 u Rs v R s v1 u 1 Mds, dz, du [ v Rs v ] ds. By monotonicity in v, R t v 1 so that the du-integral in fact only runs up to 1 and the second indicator can be sipped: t R t v = v + +θ t 1 1 [ ] z 1 u Rs v R s v M ds, dz, du [ v Rs v ] ds. This is precisely the equation we wanted to derive. Step 2: The proof is complete if we can show that the restriction M of M to, [, 1] [, 1] is a PPP with intensity M ds, dz, du = ds C α z 1 α 1 z α 1 dz du. For this sae, we choose a non-negative measurable predictable function W :,, 1, 1 R bounded in the second and third variable and compactly supported in the first, plug-in the definition of M and use the compensator measure ν of N to obtain via 4.4

29 112 Stoch PD: Anal Comp 214 2:84 12 t 1 1 = = t 1 W s, z, um ds, dz, du S 1 t W 1 u 1 W s, z, umds, dz, du Sr, 1 y Xr 1 1 x, X r 1 + x1 y Xr 1 y N dr, dx, dy X r 1 which, by predictable projection and change of variables, equals S 1 t W Sr, 1 y Xr 1 1 x X r 1 + x1, y Xr 1 1 y c α x 1 α dr dx dy. X r 1 Now we substitute the three variables r, x, y in this order, using C α = for the substitution of r and the identity 1 x g x 1 α dx = a α gzz 1 α 1 z α 1 dz a + x for the substitution of x to obtain t 1 1 W s, z, um ds, dz, du t 1 1 = W s, z, uc α z 1 α 1 z α 1 ds dz du. 1 αα 1Ɣα c α It now follows from Theorems II.4.8 of [24] and the definitions of c α, C α that M is a PPP with intensity ds C α z 2 z 1 α 1 z α 1 dz du. 5 Proof of Theorem 1.2 Let us briefly outline the strategy for the proof: In order to show that the measurevalued process Y, P-a.s., does not possess times t for which Y t has finitely many atoms,

30 Stoch PD: Anal Comp 214 2: by Theorem 2.7 it suffices to show that P-a.s. the same is true for the measure-valued branching process X. In order to achieve this, it suffices to deduce the same property for the Pitman-Yor type representation up to extinction, i.e. we need to show that P #{v ], 1] :Z t v Z t v >} =, t ], T [ = The upshot of woring with Z instead of Y is that things are easier due to a comparison property that is not available for Y. More precisely, we are going to prove that with probability 1, the number of immigrated types alive is infinite at all times, therefore proving that the result in Theorem 1.2 is indeed independent of the starting configuration Y. We start the proof with a technical result on the covering of a half line by the shadows of a Poisson point process defined on some probability space, G, G t, P. Suppose s i, h i i I are the points of a Poisson point process on,, with intensity dt dh. For a point s i, h i we define the shadow on the half line R + by s i, s i + h i which is precisely the line segment covered by the shadow of the line segment connecting s i, and s i, h i with light shining in a 45 degrees angle from the above left-hand side. Shepp proved that the half line R + is almost surely fully covered by the shadows induced by the points s i, h i i I if and only if 1 exp 1 t h t dh dt =. 5.2 The reader is referred to the last remar of [36]. For our purposes we need the following variant: Lemma 5.1 Suppose is a PPP with intensity dt dh and Shepp s condition 5.2 holds, then P # { s i t : s i, h i and s i + h i > t } =, t > = 1, i.e. almost surely every point of R + is covered by the shadows of infinitely many line segments. Proof The proof is an iterated use of Shepp s result for the sequence of restricted Poisson point processes obtained by removing all the atoms s i, h i with h i > 1 from, i.e. restricting the intensity measure to [, 1 ]. Since Shepp s criterion 5.2 only involves the intensity measure around zero, the shadows of all point processes cover the half line. Consequently, if there is some t > such that t is only covered by the shadows of finitely many points s i, h i, then t is not covered by the shadows generated by for some large enough. But this is a contradiction to Shepp s result applied to. Now we want to apply Shepp s result to the Pitman-Yor type representation. We want to prove that 5.1 holds for any θ>.letussetforallɛ> T ɛ := inf{t > : Z t 1 ɛ}.

31 114 Stoch PD: Anal Comp 214 2:84 12 Then it is clearly enough to prove that for all ɛ> P #{v ], 1] :Z t v Z t v >} =, t ], T ɛ [ = 1. In order to connect the covering lemma with the question of exceptional times, we use the comparison property of the Pitman-Yor representation to reduce the problem to the process Z ɛ explicitly defined by Z ɛ t v = s i > w i t 1 u i vαα 1Ɣαθɛ 2 α, v [, 1], t. 5.3 Setting N t := # { v ], 1] :Z t v Z t v > }, Nt ɛ := # { v ], 1] :Zt ɛ v Z t ɛ v >}, it is obvious by the definition of Z and Z ɛ that We are now prepared to prove our main result. PN t N ɛ t, t ], T ɛ[} = Proof of Theorem 1.2 Due to 5.4 we only need to show that almost surely v Zt ɛ v has infinitely many jumps for all t > and arbitrary ɛ>. To verify the latter, Lemma 5.1 will be applied to a PPP defined in the sequel. If denotes the Poisson point process with atoms s i,w i, u i i I from which Zt ɛ v is defined, then we define a new Poisson point process l via the atoms s i, h i, u i i I := s i,lw i, u i i I, where lw := inf{t > : w t = } denotes the length of the trajectory w. In order to apply Lemma 5.1 we need the intensity of l. Using the definition of Q and the Laplace transform duality 2.8 with the explicit form 1 e λw t Qdw = λ 1 α 1 + α 1t 1 α, we find the distribution Qlw > h = Qw h > = lim Q1 λ + e λw h = lim u hλ = α 1h 1 α λ +

32 Stoch PD: Anal Comp 214 2: Differentiating in h shows that l is a Poisson point process on R + R + R + with intensity measure l α dt, dh, du = dt α 1h 1 α dh du. Plugging-in the new definitions leads to N ɛ t = number of non-zero summands of Z ɛ t 1 t = # { s i t : s i,w i, u i and w i t s i 1 ui αα 1Ɣαθɛ 2 α > } t = # { s i t :s i,w i, u i and lw i >t s i, u i αα 1Ɣαθɛ 2 α} t = # { s i t :s i, h i, u i l and s i +h i >t, u i αα 1Ɣαθɛ 2 α} t. 5.6 There is one more simplification that we can do. Let us define l,ɛ as a Poisson point process on,, with intensity measure l,ɛ dt, dh = αα 1Ɣαθɛ2 α dt α 1 α/1 α h 1 α dh, 5.7 then by the properties of Poisson point processes we have the equality in law {s i, h i : s i, h i, u i l and u i αα 1Ɣαθɛ 2 α } d = l,ɛ. Then 5.6 yields N ɛ t t d = # { s i t : s i, h i l,ɛ and s i + h i > t } t. Now we are precisely in the setting of Shepp s covering results and the theorem follows from Lemma 5.1 if 5.2 holds. Shepp s condition can be checed easily for l,ɛ for 5.7 independently of θ and ɛ. α 6 A Proof of Schmuland s Theorem In this section we setch how our lines of arguments can be adopted for the continuous case corresponding to α = 2. The proofs go along the same lines reduction to a measure-valued branching process and then to an excursion representation for which the covering result can be applied but are much simpler due to a constant immigration structure. The crucial difference, leading to the possibility of exceptional times, occurs in the final step via Shepp s covering results. Proof of Schmuland s Theorem 1.1 We start with the continuous analogue to Theorem 2.2. Suppose W is a white-noise on,,, then one can show via the standard Yamada-Watanabe argument that there is a unique strong solution to