Branching Particle Systems in Spectrally One-sided Levy Processes

0.42 MB·English

by  Hui He

1
(Version: Jan/04/2012)
Branching Particle Systems in Spectrally
One-sided L´evy Processes 1
Hui He, Zenghu Li and Xiaowen Zhou
Beijing Normal University and Concordia University
2
1
0 Abstract
2

n We investigate the branching structure coded by the excursion above
a
zero of a spectrally positive L´evy process. The main idea is to identify the
J
4 level of the L´evy excursion as the time and count the number of jumps
upcrossing the level. By regarding the size of a jump as the birth site of
]
R a particle, we construct a branching particle system in which the particles
P undergo nonlocal branchings and deterministic spatial motions to the left
.
h on the positive half line. A particle is removed from the system as soon as it
t
a reaches the origin. Then a measure-valued Borel right Markov process can
m
be defined as the counting measures of the particle system. Its total mass
[ evolves according to a Crump-Mode-Jagers branching process and its sup-

1 port represents the residual life times of those existing particles. A similar
v
result for spectrally negative L´evy process is established by a time reversal
0
9 approach. Properties of the measure-valued processes can be studied via
8
the excursions for the corresponding L´evy processes.
0
.
1
0
AMS 2010 subject classifications: Primary 60J80, 60G51; Secondary 60J68.
2
1
: Keywords: L´evy process, spectrally one-sided, subordinator, branching par-
v
i ticle system, non-local branching, Crump-Mode-Jagers branching process.
X
r
a
1 Introduction
Branching processes embedded in processes with independent increments have been stud-
ied by many authors. The study yields detailed information and understandings in the
two classes of processes. In particular, Dwass [7] constructed branching processes from
simple random walks. To study random walks in random environment Kesten et al [11]
constructed a Galton-Watson process with geometric offspring law from a simple random
walk. Multitype branching processes have also been introduced in the study of random
walks in random environment; see [8, 10, 12] and the references therein. Since continuous
state branching processes and Brownian motions arise as the scaling limits of Galton-
Watson processes and simple random walks, respectively, we may naturally expect some
1Supported by NSFC (11071021, 11131003, 11126037), 985 Project and NSERC grants.
2
branching structures embedded in a Brownian motion. The well-known Knight-Ray the-
orem brings an answer to this question; see also [16, 21].
Le Gall and Le Jan [18, 19] recovered a deep connection between general continuous
state branching processes and spectrally positive L´evy processes. Furthermore, Duquesne
and Le Gall [4, 5] showed that the branching points of a L´evy tree constructed in [18] are
of two types: binary nodes (i.e. vertex of degree three), which are given by the Brownian
part of the L´evy process, and infinite nodes (i.e. vertex of infinite degrees), which are
given by the jumps of the L´evy process. The size of the jump is also called the size of the
corresponding infinite node (or the mass of the forest attached to the node).
In the interesting recent work [14], Lambert used spectrally positive L´evy processes
for the first time to code random splitting trees. In the population dynamics represented
by the splitting tree, the number of individuals evolves according to a binary Crump-
Mode-Jagers process. It was proved in [14] that the contour process of the splitting
tree truncated up to a certain level is a spectrally positive L´evy process reflected below
this level and killed upon hitting zero. From this result Lambert derived a number of
properties of the splitting tree and the Crump-Mode-Jagers process.
The purpose of this paper is to give a formulation of the branching structures of
spectrally one-sided L´evy processes in terms of measure-valued processes, which we call
single-birth branching particle systems. Those structures are undoubtedly conveyed by
the random splitting trees, so we could have derived the results from those of Lambert
[14]. However, we think a simple construction of the branching particle systems directly
from the L´evy process is of interest. In addition, we show that the branching systems are
Borel right Markov processes in a suitable state space and characterize their transition
semigroups using some simple quasi-linear integral equations. Those properties make
the branching systems easier to handle than the Crump-Mode-Jagers processes. A more
precise description of the branching structures is given in the next paragraph.
Let us consider a typical trajectory of the spectrally positive L´evy process with nega-
tive drift {S : t ≥ 0} started from a > 0 and killed upon hitting zero; see Figure 1(cid:48). Let
t
{y : i = 1,2,3} denote the sizes of jumps. Then the sample path of a branching particle
i
system can be obtained in the following way: At time zero, an ancestor starts off from
a > 0 and moves toward the left at the unit speed. At times z and z , it gives birth to
1 3
two children at positions y and y , respectively. At time z , the first child of the ancestor
1 3 2
gives birth to a child at position y . Once an individual hits zero, it is removed from the
2
system. So the ancestor dies at time a and its two children die at times z +y and z +y ,
1 1 3 3
respectively.
From the structures described above, we use a time reversal to derive a similar result
for spectrally negative L´evy processes with positive drift. We will see that the branching
systems we encounter here are actually very special cases of the models studied in [3, 20].
Unfortunately, bynowwecanonlytreatL´evyprocesseswithboundedvariationsasin[14].
An interesting open question is to give a description of the branching structures of general
spectrally one-sided L´evy processes in terms of measure-valued branching processes. We
hope to see the precise formulation of such structures in the future.
3
Therestofthispaperisarrangedasfollows. InSection2, weintroducesomebranching
particle systems on the positive half line involving nonlocal branching structures. In
Section 3, we extend the model to the case with infinite branching rates. In Section 4 the
result on the branching structures in spectrally positive L´evy processes with negative drift
isestablished. InSection5,wederivethebranchingstructuresforspectrallynegativeL´evy
processeswithpositivedriftbyatimereversalapproach. Somepropertiesofourbranching
systems are studied in Section 6. In Section 7, we discusses briefly the connection of the
branching systems with the Crump-Mode-Jagers models.
Notations. Write R = [0,∞). Given a metric space E, we denote by B(E) the
+
Banach space of bounded Borel functions on E endowed with the supremum/uniform
norm “(cid:107) · (cid:107)”. Let C(E) be the subspace of B(E) consisting of bounded continuous
functions on E. We use the superscript “+” to denote the subset of positive elements of
the function spaces, e.g., B+(R ) and C+(0,∞). Let M(E) denote the space of finite
+
Borel measures on E endowed with the topology of weak convergence. Let N(E) be the
set of integer-valued measures in M(E). For a measure µ and a function f on E write
(cid:82)
(cid:104)µ,f(cid:105) = fdµ if the integral exists. Other notations will be explained when they first
appear.
2 Branching systems on the positive half line
We begin with the description of a branching system of particles on R . Suppose that
+
α > 0 is a constant, η = η(dx) is a probability measure on (0,∞) and g = g(z) is a
probability generating function with g(cid:48)(1) < ∞. Let {ξ : t ≥ 0} be the R -valued
t +
Markov process defined by ξ := (ξ −t)∨0. Let F(0,·) be the unit mass at δ ∈ N(R ).
t 0 0 +
For x > 0, let F(x,·) be the distribution on N(R ) of the random measure
+
Z
(cid:88)
δ + δ ,
x Yi
i=1
where Z is an integer-valued random variable with distribution determined by g = g(z)
and {Y ,Y ,···} are i.i.d random variables on (0,∞) with distribution η(dx). Here we
1 2
4
assumed that Z and {Y ,Y ,···} are independent.
1 2
Suppose that we have a set of particles on R moving independently according to the
+
law of {ξ : t ≥ 0}. A particle is frozen as soon as it reaches zero. Before that at each
t
α-exponentially distributed random time, the particle gives birth to a random number of
offspring according to the law specified by the generating function g = g(z), and those
offspring are scattered over R independently according to the distribution η(dx). It
+
is assumed as usual that the reproduction of different particles are independent of each
other. Let X¯ (B) denote the number of particles in the set B ∈ B(R ) at time t ≥ 0. By
t +
Dawson et al [3, p.103] one can see that {X¯ : t ≥ 0} is a Markov process on N(R ) with
t +
¯
transition semigroup (Q ) defined by
t t≥0
(cid:90)
e−(cid:104)ν,f(cid:105)Q¯ (µ,dν) = exp{−(cid:104)µ,U¯ f(cid:105)}, f ∈ B+(R ), (2.1)
t t +
N(0,∞)
¯
where (t,x) (cid:55)→ U f(x) is the unique positive solution of
t
(cid:90) t (cid:90) t
e−U¯tf(x) = e−f((x−t)∨0) −α e−U¯t−sf((x−s)∨0)ds+α e−U¯t−sf(0)1 ds
{x≤s}
0 0
(cid:90) t
+α e−U¯t−sf(x−s)1 g((cid:104)η,e−U¯t−sf(cid:105))ds;
{x>s}
0
see also Dawson et al [3, pp.95-96] and Li [20, p.98]. By Proposition 2.9 of [20], the above
equation can be rewritten as
(cid:90) t
e−U¯tf(x) = e−f((x−t)∨0) −α e−U¯t−sf(x−s)1 (cid:2)1−g((cid:104)η,e−U¯t−sf(cid:105))(cid:3)ds. (2.2)
{x>s}
0
By Proposition A.49 of [20], for f ∈ B(R ) there is a unique locally bounded solution
+
(t,x) (cid:55)→ π¯ f(x) to the equation
t
(cid:90) t
π¯ f(x) = f((x−t)∨0)+αg(cid:48)(1) 1 (cid:104)η,π¯ f(cid:105)ds. (2.3)
t {x>s} t−s
0
Moreover, the linear operators (π¯ ) on B(R ) form a semigroup and
t t≥0 +
(cid:107)π¯ f(cid:107) ≤ (cid:107)f(cid:107)eαg(cid:48)(1)t, t ≥ 0. (2.4)
t
Proposition 2.1 For t ≥ 0 and f ∈ B+(R ) we have U¯ f ≤ π¯ f and for t ≥ 0 and
+ t t
f ∈ B+(R ) we have
+
(cid:90)
¯
(cid:104)ν,f(cid:105)Q (µ,dν) = (cid:104)µ,π¯ f(cid:105). (2.5)
t t
N(R+)
Proof. For t ≥ 0 and f ∈ B+(R ) one can use (2.2) and (2.3) to see
+
∂ (cid:12)
π¯ f(x) = U¯ (θf)(x)(cid:12) .
t ∂θ t (cid:12)
θ=0
Then (2.5) follows by differentiating both sides of (2.2). By (2.1), (2.5) and Jensen’s
¯
inequality it is clear that U f(x) ≤ π¯ f(x) for x ≥ 0. By linearity we also have (2.3) and
t t
(2.5) for f ∈ B(R ). (cid:3)
+
5
Proposition 2.2 Foranyf ∈ B+(R )themappingt (cid:55)→ U¯ f(·+t)from[0,∞)toB+(R )
+ t +
is increasing and locally Lipschitz in the supremum norm. Moreover, for any t ≥ r ≥ 0
we have
0 ≤ e−U¯rf(x+r) −e−U¯tf(x+t) ≤ α(t−r). (2.6)
Proof. For any t,x ≥ 0 one can use (2.2) to see
(cid:90) t
e−U¯tf(x+t) = e−f(x∨0) −α e−U¯sf(x+s)1 (cid:2)1−g((cid:104)η,e−U¯sf(cid:105))(cid:3)ds.
{x+s>0}
0
¯
Then we have (2.6). Since t (cid:55)→ U f is locally bounded by Proposition 2.1, we see t (cid:55)→
t
U¯ f(·+t) is increasing and locally Lipschitz in the supremum norm. (cid:3)
t
Proposition 2.3 For any f ∈ B+(R ) the function (t,x) (cid:55)→ U¯ f(x) is the unique locally
+ t
bounded positive solution of
(cid:90) t
U¯ f(x) = f((x−t)∨0)+α 1 (cid:2)1−g((cid:104)η,e−U¯t−sf(cid:105))(cid:3)ds. (2.7)
t {x>s}
0
¯ ¯
Proof. For notational convenience, in this proof we set f(x) = f(0) and U f(x) = U f(0)
t t
for all x ≤ 0 and t ≥ 0. Let 0 = t < t < ··· < t = t be a partition of [0,t]. For x ∈ R ,
0 1 n +
we can write
n
(cid:88)(cid:104) (cid:105)
¯ ¯ ¯
U f(x) = f(x−t)+ U f(x−t )−U f(x−t ) . (2.8)
t t−ti−1 i−1 t−ti i
i=1
¯ ¯
Note that Proposition 2.3 implies U f(x−t )−U f(x−t ) ≥ 0. By (2.2), (2.6)
t−ti−1 i−1 t−ti i
and Taylor’s formula, as t −t → 0,
i i−1
(cid:104) (cid:105)
¯ ¯
U f(x−t )−U f(x−t )
t−ti−1 i−1 t−ti i
(cid:104) (cid:105)
= eU¯t−ti−1f(x−ti−1) e−U¯t−tif(x−ti) −e−U¯t−ti−1f(x−ti−1) +o(ti −ti−1)
= (cid:90) ti−ti−1(cid:2)1+εi(s,x)(cid:3)1{x−ti−1>s}(cid:2)1−g((cid:104)η,e−U¯t−ti−1−sf(cid:105))(cid:3)ds+o(ti −ti−1),
0
where
(cid:104) (cid:105)
εi(s,x) = eU¯t−ti−1f(x−ti−1) e−U¯t−ti−1−sf(x−ti−1−s) −e−U¯t−ti−1f(x−ti−1) .
By Propositions 2.1 and 2.2 one can see that
0 ≤ ε (s,x) ≤ α(t −t )exp(cid:8)(cid:107)f(cid:107)eαg(cid:48)(1)t(cid:9), 0 ≤ s ≤ t −t .
i i i−1 i i−1
It then follows that
(cid:104) (cid:105)
¯ ¯
U f(x−t )−U f(x−t )
t−ti−1 i−1 t−ti i
6
= (cid:90) ti−ti−11{x−ti−1>s}(cid:2)1−g((cid:104)η, e−U¯t−ti−1−sf(cid:105))(cid:3)ds+o(ti −ti−1)
0
= (cid:90) ti 1 (cid:2)1−g((cid:104)η, e−U¯t−sf(cid:105))(cid:3)ds+o(t −t ).
{x>s} i i−1
ti−1
Substituting this into (2.8) and letting max (t − t ) → 0 we obtain (2.7). The
1≤i≤n i i−1
uniqueness of the solution of the equation follows from Proposition 2.18 in [20]. (cid:3)
Theorem 2.4 There is a Borel right transition semigroup (Q ) on N(0,∞) defined by
t t≥0
(cid:90)
e−(cid:104)ν,f(cid:105)Q (µ,dν) = e−(cid:104)µ,Utf(cid:105), f ∈ B+(0,∞), (2.9)
t
N(0,∞)
where (t,x) (cid:55)→ U f(x) is the unique locally bounded positive solution of
t
(cid:90) t
U f(x) = f(x−t)1 +α 1 [1−g((cid:104)η,e−Usf(cid:105))]ds, t ≥ 0,x > 0. (2.10)
t {x>t} {x>t−s}
0
Proof. It is not hard to see that (2.10) is a special cases of (2.21) in [20, p.39]. By (2.2)
¯ ¯
we have U f(0) = f(0) for all t ≥ 0. Consequently, if {X : t ≥ 0} is a Markov process
t t
¯ ¯
with transition semigroup (Q ) defined by (2.1) and (2.7), then {X | : t ≥ 0} is a
t t≥0 t (0,∞)
Markov process in N(0,∞) with transition semigroup (Q ) defined by (2.9) and (2.10).
t t≥0
By Theorem 5.12 of [20], we can extend (Q ) to a Borel right semigroup on the space
t t≥0
of finite measures on (0,∞). Then (Q ) itself is a Borel right semigroup. (cid:3)
t t≥0
By Proposition 2.1 we have the following:
Proposition 2.5 For every f ∈ B(0,∞) there is a unique locally bounded solution
(t,x) (cid:55)→ π f(x) of
t
(cid:90) t
π f(x) = f((x−t)∨0)+αg(cid:48)(1) 1 (cid:104)η,π f(cid:105)ds. (2.11)
t {x>s} t−s
0
Moreover, the linear operators (π ) on B(0,∞) form a semigroup and
t t≥0
(cid:90)
(cid:104)ν,f(cid:105)Q (µ,dν) = (cid:104)µ,π f(cid:105), t ≥ 0,f ∈ B(0,∞). (2.12)
t t
N(0,∞)
Proposition 2.6 We have U f(x) ≤ π f(x) ≤ (cid:107)f(cid:107)eαg(cid:48)(1)t for t ≥ 0,x > 0 and f ∈
t t
B(0,∞).
A Markov process in N(0,∞) with transition semigroup (Q ) defined by (2.9) and
t t≥0
(2.10) will be referred to as a branching system of particles with parameters (g,α,η),
where g is the generating function, α is the branching rate and η is the offspring position
law.
7
3 The system with infinite branching rate
In this section, we consider a system of particles, which can be thought of as a branching
system with infinite branching rate. Let ρ(x) = x for x ∈ (0,∞). Let B (0,∞) be the set
ρ
of Borel functions on (0,∞) bounded by ρ·const. Let C (0,∞) be the subset of B (0,∞)
ρ ρ
consisting of continuous functions. Let M (0,∞) be the set of Borel measures µ on (0,∞)
ρ
satisfying (cid:104)µ,ρ(cid:105) < ∞. Let N (0,∞) be the set of integer-valued measures in M (0,∞).
ρ ρ
We endow M (0,∞) and N (0,∞) with the topologies defined by the convention that
ρ ρ
µ → µ if and only if (cid:104)µ ,f(cid:105) → (cid:104)µ,f(cid:105) for all f ∈ C (0,∞). (3.1)
n n ρ
We say a function (t,x) (cid:55)→ u (x) on [0,∞)×(0,∞) is locally ρ-bounded if
t
sup sup |ρ(x)−1u (x)| < ∞, t ≥ 0.
s
0≤s≤tx∈(0,∞)
Let c > 0 be a constant and let Π(dz) be a σ-finite measure on (0,∞) such that
(cid:104)Π,ρ(cid:105) < c. Given f ∈ B+(0,∞), we consider the following evolution equation:
ρ
(cid:90) t
U f(x) = f(x−t)1 +c−1 1 (cid:104)Π,1−e−Ut−sf(cid:105)ds. (3.2)
t {x>t} {x>s}
0
Lemma 3.1 For each f ∈ B+(0,∞) there is at most one locally ρ-bounded positive
ρ
solution of (3.2).
Proof. Suppose that (t,x) (cid:55)→ U f(x) and (t,x) (cid:55)→ V f(x) are two locally ρ-bounded
t t
solutions of (3.2). Let
l (x) = sup |ρ(x)−1(U f(x)−V f(x))|.
T t t
0≤t≤T
Then for any 0 ≤ t ≤ T we have
(cid:90) t
|U f(x)−V f(x)| ≤ c−1 1 (cid:104)Π,|e−Ut−sf −e−Vt−sf|(cid:105)ds
t t {x>s}
0
(cid:90) t
≤ c−1 1 (cid:104)Π,|U f −V f|(cid:105)ds
{x>s} t−s t−s
0
(cid:90) t
≤ c−1 1 ds(cid:107)l (cid:107)(cid:104)Π,ρ(cid:105) ≤ c−1ρ(x)(cid:107)l (cid:107)(cid:104)Π,ρ(cid:105),
{x>s} T T
0
which implies (cid:107)l (cid:107) ≤ c−1(cid:107)l (cid:107)(cid:104)Π,ρ(cid:105). Then we have (cid:107)l(T)(cid:107) = 0 as (cid:104)Π,ρ(cid:105) < c. (cid:3)
T T
Proposition 3.2 For each f ∈ B+(0,∞), there is a unique locally ρ-bounded positive
ρ
solution (t,x) (cid:55)→ U f(x) of (3.2) and the solution is increasing in (Π,f) ∈ M (0,∞) ×
t ρ
B+(0,∞). Furthermore, the operators (U ) on B+(0,∞) form a semigroup and
ρ t t≥0 ρ
(cid:107)ρ−1U f(cid:107) ≤ (c−(cid:104)Π,ρ(cid:105))−1(cid:107)ρ−1f(cid:107), t ≥ 0. (3.3)
t
8
Proof. Step 1) We first assume that Π ∈ M(0,∞) and f ∈ B+(0,∞). By Theorem 4.1
there is a unique locally bounded positive solution (t,x) (cid:55)→ U f(x) of (3.2). This solution
t
can also be constructed by a simple iteration procedure. In fact, if we let u (t,x) = 0 and
0
define u (t,x) = u (t,x,f) inductively by
n n
(cid:90) t (cid:90) ∞
u (t,x) := f(x−t)1 +c−1 1 ds [1−e−un−1(t−s,z)]Π(dz), (3.4)
n {x>t} {x>s}
0 0
then u (t,x) → U f(x) increasingly as n → ∞; see Proposition 2.18 of [20]. Using this
n t
construction one can see that the solution of (3.2) is increasing in (Π,f) ∈ M(0,∞) ×
B+(0,∞).
Step 2) Next, we assume that Π ∈ M(0,∞) and f ∈ B+(0,∞). Let f = f ∧ k for
ρ k
k ≥ 1. Let (t,x) (cid:55)→ U f (x) be the unique locally bounded positive solution of (3.2) with
t k
f replaced by f . According to the argument above the sequence {U f } is increasing in
k t k
k ≥ 1. By (3.2) and Proposition 2.6 we have
(cid:90) t (cid:90)
U f (x) ≤ (cid:107)ρ−1f (cid:107)ρ(x)+c−1 1 ds U f (z)Π(dz)
t k k {x>s} t−s k
0 R+
(cid:104) (cid:105)
≤ (cid:107)ρ−1f (cid:107)+c−1(cid:107)f (cid:107)(cid:104)Π,1(cid:105)exp{c−1(cid:104)Π,1(cid:105)t} ρ(x).
k k
Thus (t,x) (cid:55)→ U f (x) is locally ρ-bounded. On the other hand, if we set
t k
l (t,x) := sup U f (x),
k s k
0≤s≤t
then
(cid:90)
l (t,x) ≤ (cid:107)ρ−1f (cid:107)ρ(x)+c−1ρ(x) sup U f (z)Π(dz)
k k s k
0≤s≤t (0,∞)
(cid:104) (cid:105)
≤ (cid:107)ρ−1f(cid:107)+c−1(cid:107)ρ−1l (t)(cid:107)(cid:104)Π,ρ(cid:105) ρ(x).
k
It follows that
ρ(x)−1l (t,x) ≤ (cid:107)ρ−1f(cid:107)+c−1(cid:107)ρ−1l (t)(cid:107)(cid:104)Π,ρ(cid:105),
k k
which implies
(cid:107)ρ−1f(cid:107) c(cid:107)ρ−1f(cid:107)
(cid:107)ρ−1l (t)(cid:107) ≤ = . (3.5)
k 1−c−1(cid:104)Π,ρ(cid:105) c−(cid:104)Π,ρ(cid:105)
In particular, we have
(cid:107)ρ−1U f (cid:107) ≤ c(cid:107)ρ−1f(cid:107)(c−(cid:104)Π,ρ(cid:105))−1, t ≥ 0.
t k
Then the limit U f(x) := lim U f (x) exists. It is easy to see that (t,x) (cid:55)→ U f(x) is
t k→∞ t k t
a locally ρ-bounded positive solution of (3.2) satisfying (3.3).
Step 3) In the general case, let Π (dz) = 1 Π(dz) for k ≥ 1. For f ∈ B+(0,∞)
k {z≥1/k}
let (t,x) (cid:55)→ U(k)f(x) be the unique locally ρ-bounded positive solution of (3.2) with
t
9
Π replaced by Π . By the second step, we can define U(k)f by the equation for any
k t
f ∈ B+(0,∞). The sequence {U(k)f} is increasing by the first and the second steps.
ρ t
As in the second step one can see the limit U f(x) := lim U(k)f(x) exists and is a
t k→∞ t
locallyρ-boundedpositivesolutionof(3.2)satisfying(3.3). Theuniquenessofthesolution
follows from Lemma 3.1, which yields the semigroup property of (U ) . (cid:3)
t t≥0
Proposition 3.3 For each f ∈ B (0,∞), there is a unique locally ρ-bounded solution
ρ
(t,x) (cid:55)→ π f(x) of
t
(cid:90) t
π f(x) = f(x−t)1 +c−1 1 (cid:104)Π,π f(cid:105)ds. (3.6)
t {x>t} {x>t−s} s
0
Furthermore, the solution is increasing in (Π,f) ∈ M (0,∞)×B (0,∞) and (π ) is a
ρ ρ t t≥0
semigroup of linear operators on B (0,∞) such that
ρ
(cid:107)ρ−1π f(cid:107) ≤ (c−(cid:104)Π,ρ(cid:105))−1(cid:107)ρ−1f(cid:107), t ≥ 0. (3.7)
t
Proof. For f ∈ B+(0,∞) one can obtain (3.6) by differentiating both sides of (3.2), and
ρ
(3.7) follows by (3.3). By the linearity, the equation has a solution for any f ∈ B (0,∞)
ρ
and (3.7) remains true. By Proposition 3.2 one can see the solution is increasing in
(Π,f) ∈ M (0,∞)×B (0,∞). The uniqueness of the solution follows by a modification
ρ ρ
of the proof of Lemma 3.1. (cid:3)
Theorem 3.4 There is a Borel right semigroup (Q ) on N (0,∞) defined by
t t≥0 ρ
(cid:90)
e−(cid:104)ν,f(cid:105)Q (µ,dν) = e−(cid:104)µ,Utf(cid:105), f ∈ B+(0,∞), (3.8)
t ρ
Nρ(0,∞)
where (t,x) (cid:55)→ U f(x) is the unique locally ρ-bounded positive solution of (3.2). Further-
t
more, we have
(cid:90)
(cid:104)ν,f(cid:105)Q (µ,dν) = (cid:104)µ,π f(cid:105), f ∈ B (0,∞), (3.9)
t t ρ
Nρ(0,∞)
where (t,x) (cid:55)→ π f(x) is the unique locally ρ-bounded solution of
t
(cid:90) t
π f(x) = f(x−t)1 +c−1 1 (cid:104)Π,π f(cid:105)ds. (3.10)
t {x>t} {x>t−s} s
0
Proof. Let (U(k)) be defined as in the last step of the proof of Proposition 3.2. By
t t≥0
Theorem 2.1, we can define a Borel right semigroup (Q(k)) on N(0,∞) by
t t≥0
(cid:90)
e−(cid:104)ν,f(cid:105)Q(k)(µ,dν) = e−(cid:104)µ,Ut(k)f(cid:105), f ∈ B+(0,∞). (3.11)
t
N(0,∞)
In view of (2.12) and (3.7), if µ ∈ N (0,∞) is a finite measure, we can regard Q(k)(µ,·)
ρ t
as a probability measure on N (0,∞). Clearly, N (0,∞) is a closed subset of M (0,∞)
ρ ρ ρ
10
and the latter is an isomorphism of M(0,∞) under the mapping ν(dx) (cid:55)→ xν(dx). By
Theorem1.20of[20]andthelaststepoftheproofofProposition3.2onecansee(3.8)really
defines a probability measure Q (µ,·) on N (0,∞) for any finite measure µ ∈ N (0,∞).
t ρ ρ
By approximating µ ∈ N (0,∞) with an increasing sequence of finite measures, we infer
ρ
the formula defines a probability kernel on N (0,∞). Here (3.2) can be regarded as a
ρ
special form of (6.11) in [20]. By Theorem 6.3 in [20], we can extend (Q ) to a Borel
t t≥0
right semigroup on M (0,∞). Then we infer that (Q ) itself is a Borel right semigroup.
ρ t t≥0
The moment formula (3.9) can be obtained as in the proof of Proposition 2.1. (cid:3)
A Markov process in N (0,∞) with transition semigroup (Q ) defined by (3.2) and
ρ t t≥0
(3.8) will be referred to as a single-birth branching system of particles with offspring
position law Π. Clearly, when Π is a finite measure on (0,∞), this reduces to a special
case of the model introduced in the last section.
4 Subordinators with negative drift
In this section, we give a description of the branching structures in subordinators with
negative drift. Set
C1(R) = {f ∈ C(R) : f is differentiable and has bounded derivative.}
Let c > 0 be a constant and let Π be a σ-finite measure on (0,∞) satisfying (cid:104)Π,ρ(cid:105) < c.
Suppose that {S : t ≥ 0} is a subordinator with negative drift generated by the operator
t
A given by
(cid:90) ∞
Af(x) = [f(x+z)−f(x)]Π(dz)−cf(cid:48)(x), f ∈ C1(R). (4.1)
0
We assume S = a > 0. Our assumption implies that S → −∞ as t → ∞, so the hitting
0 t
time
τ− := inf{t > 0 : S ≤ 0}
0 t
is a.s. finite. For t ≥ 0 set
J(t) := {u ∈ [0,τ−] : S ≤ t < S } (4.2)
0 u− u
with the convention that S = 0. Then we define the measure-valued process
0−
(cid:88)
X = δ , t ≥ 0. (4.3)
t Su−t
u∈J(t)
It is easy to see that X = δ .
0 a
Theorem 4.1 The process {X : t ≥ 0} is a single-birth branching system in N (0,∞)
t ρ
with transition semigroup (Q ) defined by (3.2) and (3.8).
t t≥0