THE HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY LUKE MILLER AND HELMUT H. PITTERS
Abstract. We quantify the manner in which the beta coalescent Π = {Π(t), t ≥ 0}, with parameters a ∈ (0, 1), b > 0, comes down from infinity. Approximating Π by its restriction Π n to [n] := {1, . . . , n}, the suitably rescaled block counting process n−1 #Π n (tna−1 ) has a deterministic limit, c(t), as n → ∞. An explicit formula for c(t) is provided in Theorem 1. The block size spectrum (c1 Π n (t), . . . , cn Π n (t)), where ci Π n (t) counts the number of blocks of size i in Π n (t), captures more refined information about the coalescent tree corresponding to Π. Using the corresponding rescaling, the block size spectrum also converges to a deterministic limit as n → ∞. This limit is characterized by a system of ordinary differential equations whose ith solution is a complete Bell polynomial, depending only on c(t) and a, that we work out explicitly, see Corollary 1.
1. Introduction and summary of results 1.1. Multiple merger coalescent processes. For any finite measure Λ on the unit interval there exists a (unique in law) Markov process Π with state space PN , the set of all set partitions of the positive integers N, such that for each n ∈ N the restriction Π n of Π to [n] := {1, . . . , n} n is a continuous-time Markov chain with the following dynamics: when any R 1 Πk−2has m blocks, 2 ≤ k ≤ m specific blocks merge into a single block at rate λm,k := 0 x (1 − x)m−k Λ(dx). The process Π is called a multiple merger coalescent process or Λ-coalescent, cf. [4]. To each Λ coalescent Π one can assign a corresponding tree, the coalescent tree, in an obvious fashion. This is made precise in Section 2. 1.2. Statement of main results. Consider the beta(a, b) coalescent (see definition in Section 2) Π with parameters a ∈ (0, 1), b ≥ 0. In the first part of our note we study the behaviour of the frequency of the total number of blocks of Π for small times. To this end, we show in Theorem 1 that with a suitable rescaling of time the frequency of the total number of blocks of Π has a scaling limit, more precisely, as n → ∞ we have convergence to a deterministic limit ( ) 1 a−1 Γ (a + b) {n−1 #Π n (tna−1 ), t ≥ 0} → 1+ t ,t ≥ 0 (2 − a)Γ (b) in the Skorokhod topology. We obtain the limit as the solution c(t) of the ordinary differential equation d Γ (a + b) c(t) = − c(t)2−a , dt (1 − a)(2 − a)Γ (b)
(t ≥ 0),
c(0) = 1,
(1)
of Bernoulli type. In the second part we work out in Proposition 1 a scaling limit (after a suitable rescaling) for the so-called block size spectrum {cΠ n (t), t ≥ 0},
(2) 1
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LUKE MILLER AND HELMUT H. PITTERS
of Π n , where PA denotes the set of all partitions of a non-empty set A and for π ∈ P[n] cπ := (c1 π, . . . , cn π) denotes the so-called type of π defined by ci π := #{B ∈ π : #B = i} for any i ∈ N. The block size spectrum can be thought of as a summary statistics that encodes “almost all” information on a subtree of the coalescent tree spanned by n leaves. The tree spanned by the leaves l1 , . . . , ln , also called the spanning tree, is the smallest subtree in the coalescent tree with leaves l1 , . . . , ln . To be more specific, given {cΠ n (t), t ≥ 0} one can recover the corresponding subtree in the coalescent tree up to the choice of branches that merge at each branch point and up to the labelling of the leaves. In fact, in order to reconstruct this subtree on n leaves, due to the exchangeability of Π all one needs to do is choose at each branch point k branches among the existing branches uniformly at random (if k branches are to merge), and randomly label its leaves by 1, . . . , n (i.e. according to a permutation of {1, . . . , n} picked uniformly at random). We focus on the behaviour of the average number of blocks of a given size, and therefore rescale the state space of the block size spectrum by n−1 . Fix d ∈ N arbitrarily. We show that the evolution of the frequency of blocks of size ≤ d in Π n with the same time rescaling as before converges to a deterministic limit, namely {n−1 (c1 Π n (tna−1 ), . . . , cd Π n (tna−1 )), t ≥ 0} → {(c1 (t), . . . , cd (t)), t ≥ 0}, as n → ∞ in [0, 1]d in the Skorokhod topology, where by Corollary 1 ! • 1 c(t)2−a 1−a •−1 • (−c(t) ) , (1 − a) Bi (i ∈ N). ci (t) = i! 1−a
(3)
Here, for any two sequences v• = (vk )k∈N and w• = (wk )k∈N Bi (v• , w• ) :=
i X
vl Bi,l (w• )
l=1
denotes the ith complete Bell polynomial (associated with (v• , w• )), where X Y Bi,l (w• ) := w#B π∈P[i],l B∈π
denotes the (i, l)th partial Bell polynomial (associated with w• ) and P[i],l denotes the set of all partitions of [i] that contain l blocks. Moreover, for x ∈ R, k ∈ N let xk := x(x − 1) · · · (x − k + 1) denote the falling factorial power, xk := x(x + 1) · · · (x + k − 1) the rising factorial power and we agree on x0 := x0 := 1, and for any function f : N → R we write f (•) as a shorthand for the sequence (f (k))k≥1 . We find the functions ci (t) by computing their generating function X G (t, x) := ci (t)xi (t ≥ 0, x ∈ [0, 1]). i≥1
Theorem 3 states that G is given by 1 a−1 Γ (a + b) G (t, x) = c(t) − (1 − x)a−1 + t (2 − a)Γ (b)
(x ∈ (−1, 1), t ≥ 0).
It is remarkable to see the similarity between the subtrahend in formula (4) for G , namely 1 a−1 Γ (a + b) a−1 g(t, x) := c(t) − G (t, x) = (1 − x) + t (2 − a)Γ (b)
(4)
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
3
and c(t). In fact, g(t, x) solves the partial differential equation ∂t g(t, x) = −
Γ (a + b) g(t, x)2−a , (1 − a)(2 − a)Γ (b)
(5)
with boundary condition g(0, x) = 1 − x. This partial differential equation should be compared to the ordinary differential equation for c(t), equation (1). We interpret this as a form of selfsimilarity of the limiting block frequency spectrum of the beta coalescents expressed in terms of generating functions. 2. Preliminaries A partition of a set A is a set, π say, of nonempty pairwise disjoint subsets of A whose union is A. The members of π are called the blocks of π. Let #A denote the cardinality of A and let PA denote the set of all partitions of A. A Λ coalescent Π is said to come down from infinity if P {#Π(t) < ∞} = 1 for all t > 0 and is said to stay infinite if P {#Π(t) = ∞} = 1 for all t > 0. In Proposition 23 of [4] Pitman showed the dichotomy that if Λ does not charge 1, i.e. Λ({1}) = 0, then the corresponding coalescent either comes down from infinity or stays infinite almost surely. Schweinsberg [5] showed that a Λ coalescent that does not charge 1 comes down from infinity if and only if X (γn(1) )−1 < ∞, (6) n≥2 (1) γn
P := nl=2 nl λn,l (l − 1) is the rate at which the number of blocks decreases. where Let (Ω, F, P) denote the probability space underlying Π. If Π comes down from infinity, we will (as is often done implicitly) identify for any ω ∈ Ω the path t 7→ Π(t)(ω) with a rooted tree whose leaves are labelled by N. More formally, the set of nodes of the tree corresponding to t 7→ Π(t)(ω) is T (ω) := {(t, B) : t ≥ 0, B ∈ Π(t)(ω)}. If we interpret T (ω) as a genealogical tree, (s, B) ∈ T (ω) means that individual B is alive at time s, and if for two points (s, B), (t, C) ∈ T (ω) we have that s ≤ t and B ( C, then C is interpreted as an ancestor of B alive at time t. For any two points (s, B), (t, C) ∈ T (ω) let m((s, B), (t, C))(ω) := inf{u > s ∨ t : both A and B are subsets of a common block in Π(u)(ω)} denote the time back to the most recent common ancestor of (s, B) and (t, C). It can be shown that T (ω) together with the metric d(ω) defined by d((s, B), (t, C))(ω) := (m((s, B), (t, C))(ω) − s) + (m((s, B), (t, C))(ω) − t) is an R-tree. This is done formally in Example 3.41 of [2]. Informally, d(ω) yields the genealogical distance between any two points in T (ω). One may wonder whether the (random) tree T can be described more explicitly. One way to study T is by “exploring” it via subtrees, namely, if we consider any n of its leaves labelled l1 , . . . , ln ∈ N, their spanning tree will correspond to a Λ n-coalescent with leaves labelled l1 , . . . , ln , as is apparent from the consistency of Λ coalescents. As we increase the sample size n, we explore larger and larger subtrees of T . However, the topology of a subtree spanned by n leaves is rather involved. In order to work out explicitly the asymptotic behaviour of a subtree when the number n of leaves grows without bound in what follows, we restrict ourselves to beta coalescents that come down from infinity, i.e. we take Λ to be the beta(a, b) distribution with density Λ(dx) = B(a, b)−1 xa−1 (1 − x)b−1 dx (x ∈ (0, 1)),
(7)
4
LUKE MILLER AND HELMUT H. PITTERS
R1 where for a, b > 0 the beta function with parameters a, b is defined as B(a, b) := 0 xa−1 (1 − x)b−1 dx. Notice that according to Schweinsberg’s characterization of coalescents that come down from infinity, equation (6), the beta(a, b) coalescent comes down from infinity if and only if a ∈ (0, 1), cf. Example 15 in [5]. Without further mention, we assume hereafter that Λ is the beta(a, b) distribution for some a ∈ (0, 1), b > 0. We use the convention that empty sums equal 0 and empty products equal 1 throughout. 3. Block counting process Before we turn to the scaling limit of the block size spectrum we study the simpler block counting process N n := {N n (t), t ≥ 0}. Recall that N n (t) = #Π n (t) counts the number of blocks R1 in Π n (t). Moreover, recall that the Gamma function is defined as Γ (a) := 0 e−x xa−1 dx for any positive real number a ∈ (0, ∞) and may be meromorphically continued to the entire complex plane. Also recall the identity B(a, b) = Γ (a)Γ (b)/Γ (a + b) for a, b > 0 that we will use repeatedly. In order to find the correct time scaling for N n , notice that the rate at which the number of blocks decreases has asymptotics n X n Γ (a + b) λn,l (l − 1) ∼ n2−a , l (1 − a)(2 − a)Γ (b) l=2
as n → ∞, which is proved in Lemma 3. Consequently, in order to see a nontrivial limit of N n when its state space is rescaled by n−1 , we should rescale time by a factor on the order of na−1 . Let therefore C n := {C n (t), t ≥ 0} be defined by C n (t) := n−1 N n (tτn )
(n ≥ 2, t ≥ 0),
(8)
where τn ∼ na−1 as n → ∞. The process C n is a continuous-time Markov chain with state space En := n−1 [n], initial state n C (0) = 1, absorbing state n−1 and evolves according to the following dynamics: l−1 nc a transition c 7→ c − occurs at rate λnc,l (2 ≤ l ≤ nc). (9) n l For 2 ≤ k ≤ m if there are currently m blocks in Π, we will see any k specific blocks merge at rate Z 1 B(k − 2 + a, m − k + b) ak−2 bm−k λm,k := xk−2 (1 − x)m−k Λ(dx) = = . (10) B(a, b) (a + b)m−2 0 Remark 1. Equation (10) yields the recursive formula λm,k+1 =
a+k−2 λm,k b+m−k−1
(2 ≤ k ≤ m − 1).
(11)
This should be compared to the recursive formula λm,k = λm+1,k + λm+1,k+1
(2 ≤ k ≤ m)
for arbitray Λ given by Pitman in [4], Lemma 18. Combined, these formulae yield λm+1,k =
b+m−k λm,k a+b+m−2
(2 ≤ k ≤ m),
and can be used to efficiently compute the rates of the beta(a, b) coalescent by an algorithm.
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
5
We prepare the proof of the scaling limit for cn by establishing some Lemmata. For k ∈ N define n n X X n n γn(k) := λn,l (l − 1)k , and γn(k) := (12) λn,l lk . l l l=2
l=2
(1)
(2)
The asymptotic behaviour of γn , γn as n → ∞ will play a crucial rôle in the proof of our first result, Theorem 1. Lemma 1. For a, b > 0, a natural number n ∈ N and an integer z ∈ Z we have an bn+z
∼
Γ (b) a−b−z n , Γ (a)
(13)
as n → ∞. Proof. We calculate an bn+z
=
Γ (a + n) Γ (b) Γ (b) a−b−z ∼ n , Γ (a) Γ (b + n + z) Γ (a)
as n → ∞.
Moreover, for d ∈ N, x ∈ Rd and k ∈ Nd0 let xk :=
Qd
i=1
xki i and |x| =
Pd
i=1 |xi |.
Lemma 2. Fix d ∈ N and k, n ∈ Nd0 . Then for a, b ∈ R X
k |l| n−|l|
l a b
d Y ni
li
i=1
l∈Nd 0
= a|k| n|k| (a + b + |k|)|n|−|k| .
(14)
Proof. We first prove the statement for k = 0 by an induction on d. Hence, for d = 1 the statement reads n X n l n−l ab = (a + b)n , l l=0
and this is true, since the sequence (ak )k≥1 of rising factorial powers is a sequence of polynomials of binomial type, as is well known. Suppose now that (14) holds for some d ∈ N. Then X
|l| |n|−|l|
a b
i=1
l∈Nd+1 0
=
X
d+1 Y
ni li
|l| n1 +...+nd −|l|
a b
l∈Nd 0
= (a + b)
d nX d+1 Y ni nd+1 (a + |l|)ld+1 (b + n1 + · · · + nd − |l|)nd+1 −ld+1 l l i d+1 i=1 ld+1 =0
n1 +···+nd
(a + b + n1 + . . . + nd )nd+1 = (a + b)|n| ,
where we used the induction hypothesis in the second equality. Now suppose |k| > 0. Performing the index shift m = l − k in the first step and applying the statement for k = 0 in the last step,
6
LUKE MILLER AND HELMUT H. PITTERS
we obtain X
lk a|l| b|n|−|l|
l∈Nd 0
d Y ni i=1
li
X
=
(m + k)k a|m|+|k| b|n|−|k|−|m|
i=1
m∈Nd 0
=a
|k|
d Y
X
|m| |n|−|k|−|m|
(a + |k|)
b
d Y ki (mi + ki ) i=1
m∈Nd 0
X
= a|k| nk
ni mi + ki
(a + |k|)|m| b|n|−|k|−|m|
d Y ni − ki i=1
m∈Nd 0
ni mi + ki
mi
= a|k| nk (a + b + |k|)|n|−|k| ,
which completes the proof. Lemma 3. For a ∈ (0, 1) as n → ∞ we have that γn(1) ∼
Γ (a + b) n2−a . (1 − a)(2 − a)Γ (b) (1)
(15) (1)
(0)
Proof. Firstly, note the relation γn = γn − γn . Using Lemma 2 with d = k = 1, we find ! n (a + b − 1)n−1 bn−1 (1) − . γn = − 1−a (a + b)n−2 (a + b)n−2 The first summand in above paranthesis vanishes, if a + b = 1. If a + b 6= 1 we have (a + b − 1)n−1 Γ (a + b) ∼ n−2 Γ (a + b − 1) (a + b) as n → ∞ by Lemma 1. On the other hand, Lemma 1 yields bn−1 Γ (a + b) 1−a ∼ n , Γ (b) (a + b)n−2 as n → ∞, and since a < 1, we obtain Γ (a + b) 2−a n , γn(1) ∼ (1 − a)Γ (b) as n → ∞. (0) Now focus on γn : n n X X n 1 n l−2 n−l γn(0) = λn,l = a b n−2 l l (a + b) l=2 l=2 =
(16)
n X n 1 (a − 2)l bn−l (1 − a)(2 − a)(a + b)n−2 l=2 l
1 = (1 − a)(2 − a)
(a + b − 2)n (2 − a)nbn−1 + bn + (a + b)n−2 (a + b)n−2
!
Γ (a + b) 2−a n , (2 − a)Γ (b) where we applied Lemma 1 once more and distinguished the cases a + b = 2 and a + b 6= 2. Putting everything together, the claim follows. ∼
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
7
(2)
Lemma 4. We have γn ∼ n2 as n → ∞. (2)
(2)
(1)
Proof. We can write γn = γn − γn , since (l − 1)2 = l2 − (l − 1). Applying Lemma 2 with d = 1 and k = 2 we find that n n X X 1 n al−2 bn−l 2 n γn(2) = l = (a − 2)l bn−l l2 = n(n − 1). l (a + b)n−2 l (a − 2)2 (a + b)n−2 l=2
l=2
From this and Lemma 3 we conclude
(2) γn
=
(2) γn
−
(1) γn
∼ n2 as n → ∞.
For a metric space (E, r) we denote by DE ([0, ∞)) the space of right-continuous functions from [0, ∞) into E having left limits. Moreover, by C(E), respectively C ∞ (E), we denote the continuous, respectively smooth functions (that is functions that have derivatives of arbitrary order) from E to R. Theorem 1. Let τn be of order na−1 . Then as n → ∞ we have convergence {C n (t), t ≥ 0} → {c(t), t ≥ 0}
(17)
in D[0,1] ([0, ∞)) in the Skorokhod topology, where c(t) solves the ordinary differential equation of Bernoulli type d Γ (a + b) c(t) = − c(t)2−a (t ≥ 0), (18) dt (1 − a)(2 − a)Γ (b) with boundary condition c(0) = 1. The solution to (18) is given by 1 1 a−1 1−a (2 − a)Γ (b) Γ (a + b) t = . c(t) = 1 + (2 − a)Γ (b) (2 − a)Γ (b) + Γ (a + b)t
(19)
Proof. The jump chain (Jkn )k≥0 of C n (t) = n−1 N n (tτn ) has transition probabilities nc λ nc,l −1 l λnc if c > n , 2 ≤ l ≤ nc, l − 1 n l−1 n ) := P J1 = c − J =c = 1 µn (c, c − if c = n−1 , l = 1, n n 0 0 otherwise. Denoting by λn (c) the total rate of C n in state c ∈ En for any f ∈ C ∞ ([0, 1]) the generator of cn is given by Z Gn f (c) = λn (c) (f (c0 ) − f (c))µn (c, dc0 ) En
nc X l−1 nc λnc,l = τn λnc f (c − ) − f (c) n l λnc l=2 nc X l−1 0 nc = τn − f (c) + R2 (ϑn,l ) λnc,l , n l
(20)
l=2
where we used Taylor’s approximation in the third equality. Taylor’s approximation ensures the existence of a value ϑn,l ∈ (c − (l − 1)/n, c) such that the remainder term R2 (ϑn,l ) is given, for instance, by its Lagrange form 2 1 l−1 R2 (ϑn,l ) = f 00 (ϑn,l ). 2 n First notice that by Lemma 3 nc τn (1) Γ (a + b) τn X nc (l − 1)λnc,l = − γnc →− c2−a − n l n (1 − a)(2 − a)Γ (b) l=2
8
LUKE MILLER AND HELMUT H. PITTERS
as n → ∞, since τn is chosen to be of order na−1 , and a < 1 by assumption. Since f has derivatives of arbitrarily high order on [0, 1], f 00 attains its supremum kf 00 k∞ := supx∈[0,1] |f 00 (x)|. Consequently, as n → ∞ we obtain nc nc X X nc nc τn τn λnc,l R2 (xn,l ) ≤ 2 kf 00 k∞ λnc,l (l − 1)2 l 2n l l=2
l=2
1 τn (2) ∼ c2 τn kf 00 k∞ → 0, = 2 kf 00 k∞ γnc 2n 2 where we used Lemma 4 in the third step. This shows the convergence lim sup |Gn f (c) − Gf (c)| → 0,
n→∞ c∈En
(21)
where the operator G is defined by Gf (c) := −
Γ (a + b) c2−a f 0 (c). (1 − a)(2 − a)Γ (b)
(22)
Since [0, 1] 3 c 7→ −c2−a Γ (a+b)/(1−a)(2−a)Γ (b) is Lipschitz continuous, Theorem 2.1 in Chapter 8 of [1] yields that the set C ∞ ([0, 1]) is a core for G, and the closure of {(f, Gf ) : f ∈ C ∞ ([0, 1])} is single-valued and generates a Feller semigroup {T (t)} on C([0, 1]). By Theorem 2.7 in Chapter 4 of [1] there exists a process c corresponding to {T (t)}. To prove that C n converges in D[0,1] ([0, ∞)) in the Skorokhod topology to c as n → ∞, it suffices by Corollary 8.7 of Chapter 4 to show that (21) holds for all f in a core for the generator G, which we have just done. Remark 2. Kingman’s coalescent can be thought of as the beta(a, b) coalescent obtained in the limit as a → 0. In this case we recover for the rescaled block counting process the hydrodynamic limit 2 c(t) = (t ≥ 0), 2+t which is well known, cf. [7, Equation (2.15)]. Instead of the restriction Π n of the beta coalescent Π, we now rescale the latter process, namely n n n for each n ∈ N let C∞ = {C∞ (t), t ≥ 0} be defined by C∞ (t) := n−1 #Π(tτn ). In particular, n notice that the initial state is C∞ (0) = ∞. Theorem 2. Let (τn ) be of order na−1 . Then as n → ∞ we have for a < 1 convergence n {C∞ (t), t ≥ 0} → {c∞ (t), t ≥ 0}
in DR ([0, ∞)) in the Skorokhod topology and the deterministic limit is given by 1 a−1 Γ (a + b) c∞ (t) := t . (2 − a)Γ (b)
(23)
(24)
Proof. For the most part the calculations are identical to the ones in the proof of Theorem 1. n Notice that the process C∞ has state space E∞,n := n−1 N ∪ {∞} and initial state C n (0) = ∞. Because of the consistency of the n-Λ-coalescents, i.e. Π n is equal in distribution to the restriction n of Π to [n], the generators of C n and C∞ are of precisely the same form, except that the n n generator of C operates on functions from f : n−1 [n] → R, whereas C∞ operates on functions −1 n f : n N ∪ {∞} → R. For this reason the generator calculations for C∞ are identical to the ones n given in the proof of Theorem 1. In particular, the limit c∞ (t) of C∞ (t) as n → ∞ satisfies the ordinary differential equation (18) with boundary condition c∞ (0) = ∞.
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
9
We solve (18) with this new boundary condition as follows. For any M > 0 let cM denote 1 a−1 Γ (a+b) the solution of (18) with initial condition cM (0) = M, i.e. cM (t) = M a−1 + (2−a)Γ t . (b) By definition limM →∞ cM (0) = c∞ (0), and, trivially, the c0M converge to c0∞ uniformly as M → ∞, since they are all identical. This implies the existence of a function c∞ such that Γ (a+b) 2−a the cM converge uniformly to c∞ and c0∞ (t) = limM →∞ c0M (t) = − (1−a)(2−a)Γ . Hence (b) c(t) 1 a−1 Γ (a+b) c∞ (t) = (2−a)Γ solves the ordinary differential equation as required. (b) t
4. Block size spectrum d+1 n For d ∈ N let the rescaled block size spectrum (Cin )d+1 i=1 := (Ci (t), t ≥ 0)i=1 be defined by
Cin (t) := n−1 ci Π n (tτn ), i ∈ [d],
n X
cnd+1 (t) := n−1
ci Π n (tτn ).
(25)
i=d+1
For l ∈ Nd+1 with |l| > 1 we say that an l-merger occurs in Π n if among the merging blocks there 0 are l1 singletons, l2 blocks of size 2, ..., ld blocks of size d and ld+1 blocks of size at least d + 1. The d −1 process (Cin )d+1 {0, . . . , n}d+1 \ {0}, initial state (1, 0, . . . , 0), absorbing i=1 has state space En := n −1 state (0, 0, . . . , n ) and evolves according to the following dynamics: a transition
c 7→ c −
l − eklk∧(d+1) n
occurs at rate
λn|c|,|l|
d+1 Y i=1
if c ∈ End and li ≤ ci for all i ∈ [d + 1], where klk := unit vector in Rd+1 . ∂ Let ∂i = ∂x denote the ith partial derivative. i
Pd+1 i=1
nci , li
(26)
ili and ei = (δij )d+1 j=1 denotes the ith
Proposition 1. Fix d ∈ N. For a sequence (τn ) of order na−1 and a < 1 we have convergence n (C1n (t), . . . , Cd+1 (t)) → (c1 (t), . . . , cd+1 (t)),
in D[0,1]d+1 ([0, ∞)) in the Skorokhod topology, where the latter process is deterministic with initial state (c1 (0), . . . , cd+1 (0)) = (1, 0, . . . , 0) and generator Gf (c) :=
d i X Γ (a + b) X ci |c|1−a + am−2 |c|2−a−m − Γ (b) i=1 1 − a m=2
X l∈Nd 0
d Y clkk ∂i f (c) lk !
(27)
k=1
|l|=m,klk=i
+
d+1 r Γ (a + b) |c|2−a X X m−2 + a − Γ (b) 2 − a r=1 m=2
X
d+1 Y
k=1 l∈Nd+1 0 |l|=m,klk=r
clkk ∂d+1 f (c). lk !
κd+1 Proof. For a function f : Rd+1 → R and a vector κ ∈ Nd+1 let Dκ := ∂1κ1 · · · ∂d+1 and 0 Qd+1 κd+1 f (κ) (x) := ∂1κ1 · · · ∂d+1 f (x). Moreover, let κ! := i=1 κi ! and for any vector x ∈ Rd+1 let Qd+1 xκ := i=1 xκi i . Letting λn (c) denote the total rate of (cni )ni=1 in state c ∈ End . Using a Taylor
10
LUKE MILLER AND HELMUT H. PITTERS
expansion we obtain for the generator Gn of (cni )d+1 i=1 Z Gn f (c) := λn (c)
(f (c0 ) − f (c))µ(c, dc0 ) Y nci λn|c|,|l| d+1 f (c − (l − eklk∧(d+1) )/n) − f (c) λn|c| i=1 li
X
= τn λn|c|
l∈Nd+1 0 |l|>1,l≤nc
X
= τn
l∈Nd+1 ,|l|>1 0
κ∈Nd+1 ,|κ|=1 0
X
+
X
−
κ∈Nd+1 ,|κ|=2 0
(l − eklk∧(d+1) )κ κ D f (c) n
(l − eklk∧(d+1) )κ κ D f n2 κ!
c − ϑn,l
(28)
l − eklk∧(d+1) n
d+1 Y nci λn|c|,|l| , li i=1
for any f ∈ C ∞ ([0, 1]d+1 ), c ∈ End and for some ϑn,l ∈ [0, 1]d+1 . Let kf k∞ := supx∈[0,1]d+1 |f (x)|. Part 1. Let us first consider summands corresponding to |κ| = 2 by focusing on T
(2)
l − eklk∧(d+1) τn X X (l − eklk∧(d+1) )κ κ D f c − ϑn,l (n) := 2 n κ! n d+1 d+1
(29)
κ∈N0 l∈N0 |l|>1 |κ|=2
× λn|c|,|l|
d+1 Y
nci , li
i=1
Evidently, in this case there exist (possibly equal) i, j ∈ [d + 1] with κ = ei + ej . Notice that ( κ
κ
(l − eklk∧(d+1) ) ≤ l ≤
li2 li lj
if κ = 2ei for some i ∈ [d + 1], if κ = ei + ej for some i, j ∈ [d + 1], i 6= j.
Hence, for fixed i ∈ [d + 1] and κ = 2ei T 2 (n) = τn
X l∈Nd+1 0 |l|>1
1 (l − eklk∧(d+1) )2ei D2ei f 2n2
≤ kf (2ei ) k∞
c − ϑn,l
l − eklk∧(d+1) n
λn|c|,|l|
d+1 Y k=1
nck lk
d+1 Y nck τn X 2 l λ i n|c|,|l| n2 lk d+1 l∈N0 |l|>1
= kf (2ei ) k∞
τn (a + b)n|c|−2 n2
k=1
X
li2 a|l|−2 bn|c|−|l|
l∈Nd+1 0 |l|>1
d+1 Y k=1
nck lk
d+1 Y nck X kf (2ei ) k∞ τn 2 |l| n|c|−|l| . = li (a − 2) b lk (1 − a)(2 − a) (a + b)n|c|−2 n2 d+1 l∈N0 |l|>1
k=1
(30)
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
11
2
Writing li2 = li + li , we find τn
X
(a + b)n|c|−2 n2 =
2 li (a
|l| n|c|−|l|
− 2) b
l∈Nd+1 0 |l|>1
τn (a + b)n|c|−2 n2
d+1 Y k=1
nck lk
(1 − a)(2 − a)nci (nci − 1)(a + b)n|c|−2
∼ (a − 2)(a − 1)c2i na−1 → 0, as n → ∞, and τn
X
(a + b)n|c|−2 n2
li (a − 2)|l| bn|c|−|l|
l∈Nd+1 0 |l|>1
k=1
τn
= (a − 2)
d+1 Y
nck lk
nci (a + b − 1)n|c|−1 − bn|c|−1
(31)
(a + b)n|c|−2 n2 ci na−2 |c|1−a n−2 ∼ (a − 2)Γ (a + b) − → 0, Γ (a + b − 1) Γ (b)
as n → ∞, where we applied Lemma 2 (with k = 2ei in the first case and k = ei in the second) and Lemma 1. For fixed i, j ∈ [d + 1] such that i 6= j and κ = ei + ej we obtain τn n2
X
li lj D
ei +ej
f
,|l|>1 l∈Nd+1 0
l − eklk∧(d+1) c − ϑn,l n
λn|c|,|l|
d+1 Y k=1
nck lk
d+1 X Y nck kf (ei +ej ) k∞ τn |l| n|c|−|l| ≤ li lj (a − 2) b (1 − a)(2 − a) n2 (a + b)n|c|−2 lk d+1 k=1
l∈N0 |l|>1
=
τn kf (ei +ej ) k∞ (a − 2)(a − 1)ci cj n2 (a + b)n|c|−2 (1 − a)(2 − a) n2 (a + b)n|c|−2
= kf (ei +ej ) k∞ ci cj na−1 → 0, as n → ∞, where we applied Lemmata 1 and 2 (with k = ei + ej ). Summarizing, we showed that T 2 (n) vanishes as n → ∞. Part 2. We now focus on |κ| = 1, i.e. we consider T
(1)
(n) := −
d+1 Y nck τn X X κ κ (l − eklk∧(d+1) ) D f (c)λn|c|,|l| n lk d+1 d+1 k=1
l∈N0 κ∈N0 |l|>1 |κ|=1
=−
τn
d+1 X
n(a + b)n|c|−2
i=1
∂i f (c)
X l∈Nd+1 0 |l|>1
(li − 1{i=klk∧(d+1)} )a|l|−2 bn|c|−|l|
d+1 Y k=1
nck . lk
Now consider in T (1) (n) the summands corresponding to a fixed i ∈ [d + 1]. We partition these summands and analyse their asymptotics seperately as follows. Firstly, by Lemma 2 (with k = ei )
12
LUKE MILLER AND HELMUT H. PITTERS
we have that
Oi (n) :=
=
τn n(a + b)n|c|−2
X
li a|l|−2 bn|c|−|l|
l∈Nd+1 0 |l|>1
d+1 Y k=1
nck lk
d+1 X Y nck 1 τn li (a − 2)|l| bn|c|−|l| (a − 1)(a − 2) n(a + b)n|c|−2 lk d+1 k=1
l∈N0 |l|>1
1 τn (a − 2)nci (a + b − 1)n|c|−1 − (a − 2)bn|c|−1 nci (a − 1)(a − 2) n(a + b)n|c|−2 (32) ci τn = (a + b − 1)n|c|−1 − bn|c|−1 a − 1 (a + b)n|c|−2 Γ (a + b) a−1 1 (|c|n)1−a Γ (a + b) 1−a ∼ ci n − ∼− ci |c| , a−1 Γ (a + b − 1) Γ (b) (a − 1)Γ (b) =
as n → ∞. Secondly, for any fixed i ∈ [d + 1] the summand corresponding to the indicator 1{i=klk∧(d+1)} has asymptotic behaviour
Ii (n) := −
=−
=−
τn n(a + b)n|c|−2 τn
n(a + b)n|c|−2 τn n(a + b)n|c|−2
X
1{i=klk∧(d+1)} a
|l|−2 n|c|−|l|
b
l∈Nd+1 0 |l|>1
k=1
X
|l|−2 n|c|−|l|
a
b
l∈Nd+1 0 |l|>1,klk=i
d+1 Y k=1
n|c|∧i
X
X
m−2 n|c|−m
a
b
m=2
nck lk
d+1 Y
k=1 l∈Nd+1 0 |l|=m,klk=i
i Γ (a + b) X m−2 2−a−m a |c| Γ (b) m=2
X
d+1 Y
k=1 l∈Nd+1 0 |l|=m,klk=i
nck lk
i Γ (a + b) a−2 X m−2 2−a−m −a−(m−2) m ∼− n a |c| n n Γ (b) m=2
=−
d+1 Y
nck , lk
X
d+1 Y
k=1 l∈Nd+1 0 |l|=m,klk=i
clkk lk !
clkk , lk !
as n → ∞. However, Id+1 (n) must be treated as a special case. Using the set equality {l ∈ Nd+1 : |l| > 1, klk ≥ d + 1} = Nd+1 \ 0 d {l ∈ Nd+1 : |l| ≥ 2, klk ∈ [d]} ∪ {l ∈ Nd+1 : 0 ≤ |l| ≤ 1} , 0 0
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
13
it follows that Id+1 (n) :=
−
τn
X
(1 − a)(2 − a)n(a + b)n|c|−2
d X
r X
r=1 m=2
−
X
(a − 2) b
l∈Nd+1 0
k=1 l∈Nd+1 0 |l|=m,klk=r l n|c|−l
(a − 2) b
l∈Nd+1 0
d+1 Y
nck lk
k=1
d+1 Y k=1
d+1 Y
X
(a − 2)m bn|c|−m
|l| n|c|−l
nck lk
nck lk
!
n|c|
−b
.
|l|=1
We now study each of the summands in Id+1 (n). Using Lemma 2 (with k = 0) we find that the first summand d+1 X Y nck τn (a − 2)|l| bn|c|−l lk (1 − a)(2 − a)n(a + b)n|c|−2 d+1 k=1
l∈N0
τn (a + b − 2)n|c| (1 − a)(2 − a)n(a + b)n|c|−2 Γ (a + b) na−2 ∼ (1 − a)(2 − a)Γ (a + b − 2)
=
vanishes as n → ∞. For the second summand, we find −
τn
d X r X
(1 − a)(2 − a)n(a + b)n|c|−2
r=1 m=2
∼−
r d Γ (a + b) X X m−2 a Γ (b) r=1 m=2
d+1 Y
X
X
(a − 2)m bn|c|−m
k=1 l∈Nd+1 0 |l|=m,klk=r
d+1 Y
k=1 l∈Nd+1 0 |l|=m,klk=r
nck lk
clkk , lk !
as n → ∞. Using X
|l| n|c|−l
(a − 2) b
l∈Nd+1 0
d+1 Y k=1
nck lk
d+1 X
=
(a − 2)bn|c|−l ncm = (a − 2)bn|c|−l n|c|,
m=1
|l|=1
as n → ∞ we find for the final summand −
τn (1 − a)(2 − a)n(a +
X b)n|c|−2
l∈Nd+1 0 |l|=1
l n|c|−l
(a − 2) b
d+1 Y k=1
nck lk
− bn|c| ∼
Γ (a + b) |c|2−a (2 − a)Γ (b)
as n → ∞. Pd Since, by definition, T (1) (n) = − i=1 (Oi (n) + Ii (n))∂i f (c), we obtain the convergence lim sup |Gn f (c) − Gf (c)| = 0,
n→∞ c∈E d
n
for all f ∈ Cc∞ ([0, 1]d+1 ).
(33)
14
LUKE MILLER AND HELMUT H. PITTERS
Since G operates on real functions defined on the bounded domain E d := [0, 1]d+1 whose boundary is not smooth, a direct analysis of the corresponding semigroup, respectively process, as done in the one-dimensional case in the proof of Theorem 1, is nontrivial, cf. [6]. Instead, we proceed via the theory of martingale problems. We say that a process is a martingale, if it is a martingale with respect to its natural filtration. Since Gn is the generator of a Markov jump process, Z t Mn (t) := f ((cni (t))d+1 ) − Gn f ((cni (s))d+1 i=1 i=1 )ds 0
is a martingale for each f ∈ B(E d ) with compact support. Hence, if some subsequence of d+1 2 d {(cni )d+1 i=1 , m ≥ 2} converges in distribution to (ci )i=1 , then for each f ∈ Cc (E ) Z t f ((ci (t))d+1 Gn f ((ci (s))d+1 i=1 ) − i=1 )ds 0
is a martingale by the continuous mapping theorem (cf. Corollary 1.9 of Chapter 3 in [1]) and Problem 7 of Chapter 7 in [1], since (cni )d+1 i=1 is bounded by 1 and Mn (t) is uniformly integrable, 2 and so (ci )d+1 i=1 is a solution of the martingale problem for {(f, Gf ) : f ∈ Cc (E)}. Once we show d+1 d+1 d+1 that the function b = (bi )i=1 from [0, ∞) × R to R , defined for t ≥ 0, c ∈ Rd+1 by Pi P Qd clkk i |c| − c1−a + m=2 am−2 |c|2−a−m if c ∈ [0, 1]d+1 , i ∈ [d], k=1 lk ! l∈Nd |l|=m,klk=i Γ (a + b) P Qd+1 clkk Pd+1 Pr |c|2−a bi (t, c) := bi (c) := − 2−a + r=1 m=2 am−2 if c ∈ [0, 1]d+1 , i = d + d+1 Γ (b) k=1 lk ! l∈N 0 |l|=m,klk=r 0 otherwise satisfies the conditions of Theorem 3.10 of Chapter 5 in [1], then Theorem 2.6 of Chapter 8 implies that the martingale problem for {(f, Gf ) : f ∈ Cc2 (E)} is well-posed. This is indeed the case, since cb(c) =
d+1 X
ci bi (c),
i=1
and moreover, using ci ≤ |c| ≤ 1 d X
ci bi (c) ≤ K
i=1
d X
˜ d |c|3−a ≤ K ˆ d |c|2 ci |c|2−a Kd,i ≤ K
i=1
and cd+1 bd+1 (c) ≤ |c|
d+1 X r X
|c|m Kd,r,m ≤ Kd |c|3 ≤ Kd |c|2 ,
r=1 m=2
˜ d, K ˆ d , Kd,r,m denote suitable constants. It is now straightforward to where the K, Kd,i , Kd , K verify the conditions of Corollary 8.16 of Chapter 4 in [1] which implies the convergence in the statement. Proposition 1 implies that for each i ∈ N ci (t) is a solution of the ODE c0i (s) −
i Γ (a + b) Γ (a + b) X m−2 ci (s)c(s)1−a = a c(s)2−a−m (a − 1)Γ (b) Γ (b) m=2
X
d Y clkk (s) , lk !
k=1 l∈Nd 0 |l|=m,klk=i
(34)
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
15
or c0i (s) =
=
Γ (a + b) (a − 1)(a − 2)Γ (b)
X
(a − 2)|l| c(s)2−a−|l|
l∈Ni0 ,klk=i
i G X (a − 2)m c(t)2−a−m i! m=1
i Y ck (s)lk lk !
(35)
k=1
i!
X Qi l∈Nd 0
lk k=1 lk !(k!)
i Y
(k!ck (t))lk
k=1
|l|=m,klk=i
=
i G X (a − 2)m c(t)2−a−m Bi,m (w• ) i! m=1
=
G c(t)2−a Bi (v• , w• ), i!
where G := Γ (a + b)/(1 − a)(2 − a)Γ (b)
(36)
and v• = (vk ), w• = (wk ) are sequences defined by vk := (a − 2)k c(t)−k and wk := k!ck (t). Consider now the generating function X ci (t)xi (x ∈ [−1, 1], t ≥ 0). (37) G (t, x) := i≥1
We write ∂t for the partial derivative
∂ ∂t
with respect to t.
Lemma 5. The generating function G solves the partial differential equation Γ (a + b) ((c(t) − G (t, x))2−a − c(t)2−a ) (x ∈ (−1, 1), t ≥ 0) ∂t G (t, x) = − (1 − a)(2 − a)Γ (b) (38) with boundary condition G (0, x) = x for x ∈ [−1, 1]. Proof. We use the well known fact, cf. [4, Equation (1.11)], that for any two sequences (vk ), (wk ), the exponential generating function of the associated complete Bell polynomials (Bk (v• , w• )) is given by X xk Bk (v• , w• ) = v(w(x)), k! k≥1
where these quantities are defined, and v, respectively w, P denotes the exponential P generating function of v• = (vk ), respectively w• = (wk ), i.e. v(θ) := k≥1 vk θk /k!, w(x) := k≥1 wk xk /k!. For our particular choice of (vk ) and (wk ), we find X θj X θj θ 2−a v(θ) := vj = (a − 2)j = (1 − ) − 1, j! j!c(t)j c(t) j≥1
w(x) :=
X
j≥1
k
wk
k≥1
X x = ck (t)xk = G (t, x), k! k≥1
for |θ| < c(t), x ∈ [−1, 1]. Noticing that |G (t, x)| < c(t) for |x| < 1 we obtain X X xi ∂t G (t, x) = c0i (t)xi = Gc(t)2−a Bi ((a − 2)• c(t)−• , •!c• (t)) i! i≥1
i≥1
2−a
= Gc(t)
v(G (t, x)) = G((c(t) − G (t, x))2−a − c(t)2−a ).
16
LUKE MILLER AND HELMUT H. PITTERS
Theorem 3. The generating function G is given by 1 a−1 Γ (a + b) a−1 t G (t, x) = c(t) − (1 − x) + (2 − a)Γ (b)
(x ∈ (−1, 1), t ≥ 0).
(39)
Proof. First, for fixed x ∈ (0, 1) consider the transformation g(t, x) := c(t) − G (t, x)
(t ≥ 0).
It is straightforward to verify that g solves the Bernoulli differential equation ∂t g(t, x) = −
Γ (a + b) g(t)2−a , (1 − a)(2 − a)Γ (b)
(40)
with boundary condition g(0, x) = 1 − x. Notice the remarkable similarity between this partial differential equation and the ordinary differential equation in (18) for the total number of blocks. We interpret this as a form of self-similarity in terms of generating functions. It is straightforward to solve (40) and obtain 1 a−1 Γ (a + b) a−1 . (41) g(t, x) = (1 − x) + t (2 − a)Γ (b) Corollary 1. For the deterministic limit {(c1 (t), . . . , cd (t)), t ≥ 0} we have ! • 1 c(t)2−a • 1−a •−1 Bi (i ∈ [d], t ≥ 0). ci (t) = (−c(t) ) , (1 − a) i! 1−a
(42)
Proof. From the definition (37) of G it is clear that we can compute its coefficient ci (t) for instance by evaluating its ith partial derivative with respect to x at x = 0. To this end, it will prove useful to write G as a composition, namely G (t, x) = c(t) − (f ◦ g)(x), where f (x) :=
Γ (a + b) x+ t (2 − a)Γ (b)
1 a−1
and
g(x) := (1 − x)a−1 .
We can now find a formula for the ith partial derivative of G by an application of Faà di Bruno’s formula, cf. [3], which states that X Y di (f ◦ g)(x) = f (#π) (g(x)) g (#B) (x), i dx π∈P[i]
(43)
B∈π
for any two real functions f, g that are at least i times differentiable, where f (j) denotes the jth derivative of f . In our case, for j ∈ N the jth derivatives are 1 j a−1 −j 1 Γ (a + b) (j) f (x) = x+ t , (44) a−1 (2 − a)Γ (b) and g (j) (x) = (−1)j (a − 1)j (1 − x)a−1−j .
(45)
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
17
Plugging this into (43) yields 1 i G (t, x) ci (t) = − ∂x i! x=0 1 #π a−1 −#π Y 1 X 1 Γ (a + b) =− 1+ t (−1)#B (a − 1)#B i! a−1 (2 − a)Γ (b) π∈P[i] B∈π #π Y c(t) X 1 =− (−1)#π c(t)#π(1−a) (1 − a)#B i! a−1 B∈π
π∈P[i]
=
i 2−a X
c(t) i!
1 1−a
m
1 1−a
•
m=1
2−a
c(t) = i!
Bi
(−c(t)1−a )m−1 Bi,m ((1 − a)• ) ! 1−a •−1
(−c(t)
)
, (1 − a)
•
.
In complete analogy to our discussion of the block counting process of Π n , define the pron n n n cess (C∞,1 (t), . . . , C∞,d+1 (t), t ≥ 0) via C∞,i (t) := n−1 ci Π(tτn ) for i ∈ [d] and C∞,d+1 (t) = P −1 n n n i≥d+1 ci Π(tτn ). In particular, (C∞,1 , . . . , C∞,d+1 ) has initial state (∞, 0, . . . , 0). Theorem 4. Fix d ∈ N. For any sequence (τn ) such that τn ∼ na−1 as n → ∞ and a < 1 we have convergence n n (C∞,1 (t), . . . , C∞,d+1 (t), t ≥ 0) → (c∞,1 (t), . . . , c∞,d+1 (t), t ≥ 0),
as n → ∞ in D[0,1]d+1 ([0, ∞)) in the Skorokhod topology, where the latter process is deterministic with initial state (∞, 0, . . . , 0) and given by ! • 1 c∞ (t)2−a • Bi (i ∈ [d], t ≥ 0). (46) c∞,i (t) = , (1 − a) i! 1−a n n d Proof. The process (C∞,1 (t), . . . , C∞,d+1 (t), t ≥ 0) has state space E∞,n = (n−1 N ∪ {∞})d+1 and initial state (∞, 0, . . . , 0). The limiting process (c∞,1 (t), . . . , c∞,d+1 (t), t ≥ 0) solves the system of ordinary differential equations (34) with initial conditions c∞,1 (0) = ∞ and c∞,i (0) = 0 for i ≥ 2. We find the solution of this system of ordinary differential equations as follows. First, for some M ∈ N let (cM,1 (t), . . . , cM,d+1 (t), t ≥ 0) denote the solution of the system of ordinary differential equations (34) but with initial conditions cM,1 (0) = M and cM,i (0) = 0 for i ≥ 2. The corresponding generating function X GM (t, x) := cM,i (t)xi (47) i≥1
solves the partial differential equation (38) with boundary condition GM (0, x) = M x. Letting 1 a−1 P Γ (a+b) cM (t) := i≥1 cM,i (t), hence cM (t) = M a−1 + (2−a)Γ t as in the proof of Theorem 2, the (b) function gM (t, x) := cM (t) − GM (t, x)
(48)
18
LUKE MILLER AND HELMUT H. PITTERS
solves the partial differential equation (40) with initial condition gM (0, x) = M (1 − x) and therefore 1 a−1 Γ (a + b) gM (t, x) = (M (1 − x))a−1 + t . (49) (2 − a)Γ (b) In complete analogy to Corollary 1 we let 1 a−1 Γ (a + b) f (x) := x + t and (2 − a)Γ (b)
gM (x) := (M (1 − x))a−1 ,
(j)
so gM (t, x) = (f ◦ gM )(x). Since gM (x) = M a−1 (−1)j (a − 1)j (1 − x)a−1−j , we obtain 1 i ci (t) = − ∂x GM (t, x) i! x=0 1 #π a−1 −#π Y 1 1 X Γ (a + b) a−1 =− M t + (−1)#B (a − 1)#B M a−1 i! a−1 (2 − a)Γ (b) π∈P[i] B∈π #π X Y 1 cM (t) (−1)#π cM (t)#π(1−a) (1 − a)#B M a−1 =− i! a−1 π∈P[i] i X
B∈π
m
cM (t) 1 (−1)m cM (t)m(1−a) Bi,m ((1 − a)• )M m(a−1) i! m=1 1−a ! • 1 c∞ (t)2−a Bi → , (1 − a)• i! 1−a
=−
as M → ∞, since (cM (t)/M )m(1−a) = (1 +
Γ (a + b)t M 1−a )−m → 1 (2 − a)Γ (b)
as M → ∞.
Acknowledgement. This is part of the authors’ PhD theses. They would like to thank their supervisor Alison Etheridge for her advice and helpful discussions. H. H. P. also thanks Julien Berestycki and Matthias Birkner as well as Martin Möhle and Elmar Teufl for stimulating discussions, and thankfully acknowledges financial support by the foundation “Private Stiftung Ewald Marquardt für Wissenschaft und Technik, Kunst und Kultur”. References [1] S. N. Ethier and T. G. Kurtz. Markov processes – characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. [2] S. N. Evans. Probability and real trees, volume 1920 of Lecture Notes in Mathematics. Springer, Berlin, 2008. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005. [3] W. P. Johnson. The curious history of Faà di Bruno’s formula. Amer. Math. Monthly, 109(3):217–234, 2002. [4] J. Pitman. Coalescents with multiple collisions. Ann. Probab., 27(4):1870–1902, 1999. [5] J. Schweinsberg. A necessary and sufficient condition for the Lambda-coalescent to come down from infinity. Electronic Communications in Probability [electronic only], 5:1–11, 2000. [6] M. G. Ulmet. Properties of semigroups generated by first order differential operators. Results Math., 22(34):821–832, 1992. [7] J. A. Wattis. An introduction to mathematical models of coagulation–fragmentation processes: A discrete deterministic mean-field approach. Physica D: Nonlinear Phenomena, 222(1-2):1–20, Oct. 2006.
HYDRODYNAMIC LIMIT OF BETA COALESCENTS THAT COME DOWN FROM INFINITY
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Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK E-mail address:
[email protected] Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK E-mail address:
[email protected]
