Journal of Theoretical Probability, Vol. 11, No. 1, 1998
Limit Processes for Age-Dependent
Branching Particle Systems
I. Kaj 1 and S. Sagitov2
Received March 18, 1995; revised November 22, 1996
We consider systems of spatially distributed branching particles in Rd. The
particle lifelengths are of general form, hence the time propagation of the system
is typically not Markov. A natural time-space-mass scaling is applied to a
sequence of particle systems and we derive limit results for the corresponding
sequence of measure-valued processes. The limit is identified as the projection
on Rd of a superprocess in R + x Rd. The additive functional characterizing the
superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.
KEY WORDS: Superprocess; age-dependent branching; Markov renewal
process; Levy process.
1. INTRODUCTION
1.1. (^, K, T )-Superprocesses
Dynkin introduced Markov (£, K, W )-superprocesses t ->Xt with values in
a set of measures on a general space E, in Dynkin.(3) We recall briefly the
following simple but basic case. Take E = Rd and let t ->£t be a timehomogeneous Markov process in Rd. Let Kt = K(0, t; £) denote a continuous additive functional of E and *P(X), A > 0, a nonlinear function of a
certain form to be specified. Let (f, u) = f\ f ( x ) u(dx) denote integration
with respect to a measure ju on Rd. The measure-valued superprocess
1
2
Department of Mathematics, Uppsala University, Box 480, S-751 06, Uppala, Sweden.
Institute of Theoretical and Applied Mathematics, NAS of Kazakhstan, Almaty, Kazakhstan.
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860/11/1-15
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Kaj and Sagitov
t ->X t , t > 0, associated with the triple (£, K, V) can be characterized
through its Laplace functions
for p ranging in a set of positive functions on Rd. The function V t O ( x ) is
the unique solution of the nonlinear integral equation
where expectation refers to £ conditioned on £0 = x. Introducing the random time-change
Eq. (1.1) may be written in the equivalent form
The (£, K, !F)-superprocess could be viewed as a model for a Et-flow
of mass subject to U-fluctuations at dKt-rate, c.f. discussion in Section 1.2
next.
The next remarks reveal some features of the superprocess that we are
interested in. Since J0 is a stopping time we can consider the bi-partitioning
of the state space into
The set I0 can be thought of as the reactive area where transformation of
mass occurs. In its complement of dKt -rate zero, only deterministic massmotion takes place. Put
For a piece of mass at point £t the value 9t estimates the remaining time
of linear motion until the next mass-transformation. The enriched motion
(9 t , £t) is a Markov process in R+x Rd. Hence one can think of the
( ( 9 t , £t,),K, y^-superprocess on R+ x Rd as an "age-structured" (£, K, W)superprocess.
The example of the single point catalytic super-Brownian motion is
particularly illuminating. This model is characterized by ( B t , Lt, W), where
Bt is Brownian motion in R and Lt is its local time at zero. In this model
Age-Dependent Branching Superprocess
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the mass outside zero moves in a linear fashion with I0= {0} being the
source of nonlinear mass transformation. Here Js is the pure jump inverse
Brownian local time at zero and 9t>0 almost surely for any t. See e.g.,
Fleischmann(5) for an introduction to such models.
1.2. Particle Picture
The particle picture of the simplified (£, K, f)-superprocess is the
interpretation of Xt as a limiting mass distribution at time t of a system of
independent particles moving in Rd according to £. Each particle is alive
for a random time which is governed by the accumulated lifelength intensity K(a, a + t), picked up by its path from the time of birth a and onwards.
More exactly, it is implicit in Dynkin's construction that particles are killed
elastically: the lifelengths are of the type r = inf{f > 0: K(a, a + t) > T},
where T is an exponentially distributed random variable independent of
everything else. Equivalently, the conditional probability given a and £ of
surviving time a + t is given by exp — K(a, a + t). Observe that the recursive
relation
where T1 = T and { T k } k > 1 are i.i.d., defines a birth process along the path £.
At time of death each particle independently of path and lifelength
gives birth to a random number of daughters given by a nonnegative
integer-valued random variable v of finite mean. Daughters immediately
start independent E-motions at the place of death of the mother.
The superprocess arises in the limit of many particles of small mass
and high speed branching. It can be assumed that in the scaling limit the
centered offspring variables v — 1 has a limit £. The nonlinear function ¥ is
defined by
By the properties of i.i.d random variables ^(A) has a canonical representation, see Assumption 1(b).
The construction in Dynkin (3) is carried out for a time-inhomogeneous
Markov process £ in a general state space E with the offspring function W
allowed to depend on both time and location of branching.
1.3. Purpose and Plan of the Paper
We extend the scope of Eq. (1.1) to the case when Kt denotes a continuous nondecreasing adapted functional of £, which is not necessarily
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additive, but fulfills a weaker condition put on its inverse Js. Given, for
such K, a triple (£, K, *F) we introduce a unique measure-valued process on
Rd that is characterized by the solution of (1.1). This process, in general
non-Markov, is obtained by projection on Rd of a true superprocess with
respect to a Markov process in R+ xRd, which is subject to a specific
initial age-structure. If Kt is again additive we obtain the (£, K, Y(-superprocess. This construction is the topic of Section 2.
In Section 3, we consider a wider class of branching particle systems
in Rd with general birth process Sk, such that the associated lifelength distributions are of general form and possibly of infinite mean. The process Sk
is the counterpart of the time-change Js. Correspondingly,
the embedded Markov renewal process counting generations along a line
of descent, is the counterpart of Kt. Specializing back to the Markov case,
this point of view offers an interpretation of the additive functional Kt as
the compensator of Nt.
The general branching particle system naturally defines a non-Markov
process t -> Zt with states in the measures on Rd. In Section 4 we prove a
scaling limit theorem for Zt and identify the limit as a projected (£, K, f)superprocess on Rd, which belongs to the class introduced in Section 2. To
prove that the scaled sequence of probability measures is tight, we take
advantage of the residual life process associated with Nt and study Zt by
means of a measure-valued Markov process t —> Zt in the extended state
space of measures on R + x Rd. Whereas Zt describes the particle distribution on Rd at time t, Zt also records the (residual) age-structure among
particles alive at t.
As can be expected the results are most complete in the case when the
motion and the lifelength are independent. For this case we give a simplified statement of the limit theorem in Section 5. In the dependent case
we examine a one-dimensional example closer. It is based on Brownian
particles on the line who die and reproduce only when they pass a barrier
farther out from the origin than the place where their mothers died. The
limiting birth process in this example is the Brownian first passage time
process. Similar examples in higher dimensions will be studied elsewhere.
Several readers of a preprint version of this paper have remarked that
non-Markov processes of the type studied here naturally fall within the
setting of historical superprocesses, since with respect to a sufficiently rich
filtration any process is Markov. It would be interesting to see a treatment
of such examples as those in Section 5 based on the historical process
approach, possibly the present work could inspire a study of that kind.
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2. AGE-STRUCTURED SUPERPROCESS
2.1. Age-Structure and Projection
Assumption 1. Suppose we have
(a) a time-homogeneous Markov process (Q, F, ( F t ) , £ t , T t , Px),
with cadlag paths in Rd, filtration ( F t ) , the shift operator Tt acting on
paths w e Q by T t w ( s ) = w(s + t) and conditional distributions Px such that
P x (E 0 = x) = 1, x e Rd;
(b) constants b, c > 0 and a measure F on R+ with j£° (s A s2) x
F(ds) < oo, such that
(c)
a right-continuous family J= {Js, s > 0} of stopping times with
such that for each x and s > 0 there is a Px-negligible set outside which
and suppose the increments K(s, t] = Kt — Ks of the continuous inverse
Kt = inf{s>0: Js> t} satisfies
(d)
Remark. The condition (d) is taken from the recent monograph
Dynkin, (4) Section 3.3.3, and adapted to our situation; in Dynkin (3)
stronger conditions on exponential moments of the additive functional Kt
are presupposed.
Introduce the remaining time to next point of increase of Kt, defined
by
Note that <9 t _ =0 at each jump of 9t, that KJ1= t, that Js and Kt are
related as in Section 1.1, but that Kt may not be additive.
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230
We consider now the Markov process obtained as the extended
motion
with cadlag paths in R+ x Rd, possibly extended filtration (Ft) such that
( 9 t , £ t ) e F t , shift operator T t and initial distributions
Lemma 1. With respect to the extended motion ( 9 t , £t) the process
Kt is an additive functional.
Proof.
By (2.1),
Due to Assumption 1 ( c ) and again (2.1),
Hence
Since
we obtain
and hence by (2.2)
We can now introduce the ( ( 9 , £ ) , K , SP)-superprocess { X t , t > 0} on
say (Q, F , P u , x ) , with values in the set M ( R + x R d ) of finite Radon
measures on R+x Rd. According to the general results of Dynkin, (4)
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Ch. 3.4, in order to infer the existence of X we must verify that the additive
functional K, of ( 9 t , £ t ) satisfies
where Eu, x is expectation with respect to Pu,x. However, in view of
Assumption 1(d) these conditions are fulfilled. In fact,
Let Eu,
introduce
x
denote the expectation operator for the superprocess and
for f E # + (R+ x Rd), the set of nonnegative continuous bounded functions
on R+ x Rd. The log-Laplace function Vtf is the unique solution of the
nonlinear integral equation
We call Xt the age-structured superprocess (given by the triple (£t, Kt, ¥ ) ) .
Definition 1 (Projected Superprocess). Consider the age-structured
superprocess Xt given by the triple (£, K, V). For any random measure
X 0 (dx) on Rd suppose we start off Xt at time t = 0 with the extended initial
measure
We call the M(R d )-valued process Xt with distribution P defined by
the (£, K, *F )-projected superprocess.
If Kt is an additive functional of Et then X t (dx) itself is the (£, K, *P)superprocess.
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2.2. Generalized Integral Equation
One aim of the construction in the previous subsection is to reveal the
probabilistic meaning of the crucial Eq. (1.1) for some non-additive, continuous functionals Kt. Consider the case X0 = dx, let Px and Ex denote the
corresponding law and expectation operator and define for p e b + (Rd), a
nonnegative bounded continuous function in Rd,
Lemma 2. Under Assumption 1, for any c e&+(Rd) the function
V t p ( x ) defined in (2.4) is the unique solution to the equation
Proof. To prove uniqueness, suppose u t ( x ) and v t ( x ) are two nonnegative and therefore bounded solutions. It is well-known that f is
Lipschitz. Hence for some constant C and t < t0
where || • || denotes the uniform norm in Rd. By Assumption 1(d) we may
choose t0>0 so small that sup x E x K(0, t0] < C-1. Taking supremum of
the left side of the inequality first over x and then over t < t0 shows that
ut and vt coincide on [0, t0]. This is enough since the next part of the proof
shows that the solution is a semigroup.
Fix p e %+ (Rd). By Definition 1,
where q(u, x) = $(x). By (2.3) with f = 9),
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where we changed variables v = Ks. However, by (2.1)
so
Now we claim that, almost surely,
In fact, according to (2.10)
But TJv o J0 = 0 in view of the relation for Js in Assumption 1(c). Hence
which is (2.6). Together with (2.5) we now have
finishing the proof.
3. BRANCHING MODEL
3.1. Age-Dependent Branching Particle Systems
The model we introduce generalizes that described in Section 1.2. Our
emphasis is on point (b) below which expands the range of admissible
lifelength distributions. Suppose we have
(a)
1(a);
the Markov process £ = {£ t , t > 0} as in Section 2, Assumption
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(b) a strictly positive stopping time r;
(c) a distribution {pk} on {0, 1,2,...} such that Pk >0, Ek=0 pk= 1
and Ek=0 kPk < 00 with generating function h(s) = Ek=0 PkSk.
Then we can introduce an age-dependent branching particle system
which is characterized by the properties:
(i)
each particle has random birth and death times;
(ii) given that a particle is born at time r its path is distributed as
Tr°t;
(iii) given the path and the birthtime r of a particle, its lifelength
distribution is that of T r °r;
(iv) the only interaction between the particles is that the birth time
and place of a particle coincides with the death time and place of its
mother;
(v) the random number of offspring at the time of death is independent of both path and lifelength of the particle and has distribution
{Pk}.
We will refer to such a system as a (£, r, h)-particle system. One can follow
the approach in Dynkin,(3) using the lonescu-Tulcea theorem, for the
construction of a probability space (Q, F, P) governing a (£, T, h)-particle
system. Again we write Px and Ex in the case when the initial particle starts
at x.
We construct a general birth process {Sk, k > 0} on the path d by the
recursion
where {rk, k > 1} are the lifelengths of succesive daughters. The birth process Sk provides a time change such that the sequence £ ° Sk, k >1, is an
embedded Markov chain. The counting process
gives the generation number of the particle alive at t. The regenerative
process of residual life time associated to Nt is given by
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3.2. Particle System Integral Equation
It is customary to represent a branching particle system via a measurevalued process. Hence we identify the states of the (£t, T, h)-particle system
with the measures
where the sum is over all particles alive at t and x1, x2,... their positions. In
general, Zt is not a Markov process. In addition we consider the extended
states of the system, which are given by specifying for each particle the pair
(u, x ) e ( 0 , oo )x Rd, where u denotes the remaining time until the particle
dies and x the position at time t of observation. The system is represented
now by the point measure
on (0, co) x Rd. Due to the independence in the model (c.f. 3.1(v)), Zt,
t > 0, is a time-homogeneous Markov process. We always consider rightcontinuous versions of the processes.
We have the interpretation
Zt( [ 0, v] x B) = the number of particles in B c Rd that will die by time t + v
Moreover,
Z t ( d y ) :=Z t (R + x dy) = particle distribution on Rd at time t
Put
and define the function
where it is assumed that the initial particle is born at time 0 at location x.
Introduce the notation
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Lemma 3. The function Q t ( p ( x ) for the (£, T, h)-particle system Zt
satisfies the integral equation
Proof. To emphasize the dependence on initial condition write temporarily Zx for the state at time t under Px. Then
where the j-labeled summands on the right side are independent copies and
the sum vanishes in the case v = 0. Therefore
Rewrite as
Now suppose S1 < t. Replace t by t — S1 and S1 by S2 in relation (3.2). We
obtain in the same manner
Hence, by the strong Markov property of £t
Therefore, inserting (3.4) into (3.3)
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By repeating the steps leading from (3.3) to (3.5) recursively we obtain
4. A LIMIT THEOREM
4.1. Result
Here we present conditions on a sequence Z ( n ) , n>1, of (E ( n ) , r(n),
h (-particle systems which ensure its weak convergence (under a proper
scaling) to a (£, K, !f (-projected superprocess. In particular we restrict to
subcritical and critical branching, see (A2). The supercritical case would
require some further assumptions and additional work in order to obtain
the relevant bounds. The gain, however, would be relatively minor.
We write M = M ( R d ) for the set of finite Radon measures on Rd
equipped with the topology of weak convergence, and we let D ( [ 0 , oo), M]
denote the set of cadlag paths with values in M and furnish them with the
usual Skorokhod topology (J 1 -topology). We also introduce the sets D+ =
D( [ 0, 00 ), R +) and D = D( [ 0, oo), Rd) of cadlag paths x( t) in R + and Rd,
respectively. Moreover, let D+ = D+ ([0, oo),R+) denote the subset of D +
with the properties that x ( t ) is monotone and limt -> x ( t ) = oo and with
the relative topology inherited from D +.
We impose the following assumptions on the sequence Z (n) :
(n)
(A1) the sequence of Markov processes d(n) conforms with Assumption 1 ( a ) and possesses a weak limit process £t in D, where for any t > 0
and q e b + (Rd)
(A2)
that
with Uas in Assumption 1 ( b ) there is a sequence B n -> oo such
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238
(A3)
satisfies
the sequence of processes { J ( n ) , s > 0}, where J(n) : = S
( n )
,
for any continuous bounded function G: D+ x D -> R;
(A4) the following moment condition holds:
(A5)
for some measure X0 e M ( R d ) we have the weak convergence
Theorem 1. Suppose (A1)-(A5) hold. If the weak limit process
{Js, s > 0} in (A3) is a proper time-change for £ in the sense of Assumption
1(c) and we put
then the (£, K, !f)-projected superprocess X exists. If its log-Laplace
function x -> V t O ( x ) defined in (2.4) is continuous in Rd then the weak
convergence of processes,
holds in D ( R + , M).
Remark. If K is an additive functional of I then the (£,, K, 'F)-projected
superprocess X is the (£, K, 'F)-superprocess. Therefore Theorem 1 widens
the class of branching particle systems which in the scaling limit yield
superprocesses.
After a preliminary lemma the proof of the theorem is given in the
next three subsections organized with respect to convergence of one-dimensional distributions, convergence of finite-dimensional distributions and
tightness.
Lemma 4. Under the assumptions of the theorem the (£, K, *P)projected superprocess X exists and the corresponding log-Laplace function
(t, x) |-> V t O ( x ) is jointly continuous in R+ x Rd.
Age-Dependent Branching Superprocess
Proof.
any n
239
We first verify that Assumption 1(d) is fulfilled. Indeed, for
which vanishes as first n -> oo and then t \ s in view of (A3) with G(j, y) =
fcl { s < j v < t }
dv and ( A 4 ) .
It was observed in Section 2 that now, under our assumptions, the
(£, K, U)-projected superprocess X exists and is characterized by the
unique solution Vtp of the equation in Lemma 2.
Since we assume continuity of V t p ( x ) in x, to prove the stated joint
continuity it suffices to show the continuity in t of V t q ( x ) , uniformly in x.
Slightly stronger, for fixed t we prove k0_t (e) -> 0 as e -> 0, where
By Lemma 2,
Using the simple bound V t ( p ( x ) < ||q|| and the uniform Lipschitz property
of U, see e.g., Lemma 4.3.2 in Dawson,(1)
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In particular, for any t 1 >0,
By (4.2) we may choose t1 so small that supxExK(0, t1] < 1/2C. Then the
strong continuity of £t, and (4.2) implies that k0, t 1 (e)->0 as e-> 0. Next,
consider the interval [t1, t2] and choose t2 — t1 sufficiently small that
supx E x K ( t 1 , t2] < 1/2C. Then
concluding
This extends the continuity to [0, t2] and hence to any finite interval
[0, t], by induction.
4.2. Convergence of One-Dimensional Distributions
Lemma 5. Fix t > 0. Let F(u, x) be a bounded continuous function
on R+ x Rd. The map E: D+ x D ->R defined by
is bounded and continuous.
Proof. Recall that lims->00 js = 00 since j e D+. Hence E is bounded.
Pick a sequence ( j n , yn) that converges in the joint Skorokhod topology on D+ x D towards some function (j, y). Let Xn(s) denote a continuous scaling of time which provides the uniform convergence
and
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241
for any t > 0. Since js -> oo and, for each n, jn -> oo as s -> oo we can find
T< oo such that for sufficiently large n
The properties of Xn(s) now imply the claim.
Define
To prove the next lemma we show that V(n)O converges to the limiting logLaplace function V t q. In fact, we will show that the convergence holds
uniformly in both x and t. The uniformity in space is essential in order to
allow for an arbitrary initial distribution and also for our proof of tightness. Uniformity in time, on the other hand, is merely a step of convenience
within our proofs bearing no significance for the weak convergence.
Lemma 6. The one-dimensional distributions of n - 1 Z ( n ) converge to
those of X.
Proof.
where
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Fix t > 0 and p e b + (R d ). Now,
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Kaj and Sagitov
As n -> oc, I1 goes to zero uniformly in t and .v because of Assumption
(A1).
As a preliminary for estimating the remaining terms, recall that we use
the notation J(n)= S (n) [BnS] + 1 and note that with a dummy function F
For I2 we have
By (4.3) with F=1,
so I2 vanishes as n —> oc uniformly in s and t due to Assumptions (A2) and
(A4).
Turning to I3, by the uniform Lipschitz property of V,
To rewrite I4, apply the change-of-variable v = Ks to obtain as in (1.2)
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243
With F(s, x) = V( V t _ s p ( x ) ) we then have by (4.3)
By Lemma 4, V t p ( x ) and hence F(s, x) is jointly continuous. By (A3),
Lemma 5 and the continuous mapping theorem I4 tends to zero uniformly
in x as n tends to infinity. Uniformity in t follows from the well-known fact
that if a sequence of monotone functions converges everywhere to a continuous limit then the convergence is uniform on every compact.
Put
Summing up, we have obtained
where c (n) = 0(1) as n -> oo for any fixed t. Now choose t1 so small that
which is possible according to (A4). Then
whence V ( n ) p ( x ) converges uniformly in (x, t) on R d x [ 0 , t1] towards
Vtp(x).
Fix a time interval [0, T] and consider the partition
For a proof by induction on i, suppose we have shown already that V ( n ) p
converges uniformly on Rdx [0, it1] towards V t p ( x ) . With straightforward
modifications the previous proof applies in order to extend the convergence
to Rd x [0, (i+1)t 1 ], hence to Rd x [0, T].
This concludes the proof that the one-dimensional distributions of
n - 1 Z ( n ) converge to those of X.
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4.3. Convergence of Finite-Dimensional Distributions
We write s = (s 1 ,..., sp) for a sequence of real numbers. The limit process X t (dx) is characterized via its Laplace transforms for p-dimensional
distributions given by
Similarly as for the case p = 1 in (2,5) we also define functions V P (u, x).
The p-dimensional version of (2.3) can then be written in accordance with
Dynkin,(3) Lemma 4.1, as
where t—s = (t1 —s, t2 — s,..., tp — s) and
By the same procedure as in the proof of Lemma 2 we obtain from this
with the same notational conventions as earlier.
Now define
Lemma 7. The function Q p ( x ) satisfies the equation
Proof, This follows from rather tedious but straightforward elaborations of the proof of Lemma 3.1.
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245
Introduce
Lemma 8. We have the convergence
Proof. The proof is by induction on p. Lemma 6 provides the case
p = 1. Suppose the statement in the lemma is true for all values of the
parameter up to p - 1. As in the proof of Lemma 6 we obtain
where the first four terms on the right side are obvious counterparts to the
one-dimensional case. The term I5 is given by
After simple estimates we obtain
uniformly in x and t.
The proof can now be completed exactly as for Lemma 6.
4.4. Tightness
In this section we complete the proof of Theorem 1 by showing that
the sequence n - 1 Z ( n ) is tight in M. A method based on Aldous criterion for
proving the tightness of such measure-valued processes deals with scalar
processes
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Kaj and Sagitov
involving an arbitrary bounded continuous function p in b + (R d ); compare
Dawson, (1) Section 4.6 and Sagitov. (8) According to this approach one
should establish convergence in probability
assuming that £n are nonrandom, and { Tn} is a sequence of stopping times
with respect to the filtration (Ft) such that
The proof of (4.4) is based on the next two lemmas. The first lemma is just
the standard observation that subcritical and critical branching correspond
to the supermartingale property of the total size of the system. We omit the
proof.
Lemma 9. The total number of particles I ( n ) : = Z ( n ) ( R d )
supermartingale property
Lemma 10. Denote
Under the conditions of the theorem
Proof.
where
and
Due to the strong Markov property of the process Zt
has the
Age-Dependent Branching Superprocess
247
Thereby, recalling that I(n) denotes the total mass of the branching particle
system,
for any t1 > 0. This splits our task into proving
and
To check (4.5) observe that
This reduces (4.5) down to
which follows from the uniform convergence
and the continuity of the family of limit operators,
obtained in Lemma 4.
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It remains to prove the convergence (4.6). We use Lemma 9 and a
maximal inequality for positive supermartingales, see e.g., Revuz and
Yor, (6) II (1.15). Take tz e R + . Then
Thus
We conclude that
But the righthand side tends to zero as t2 -> oo because the sequence {I(n)}
is tight.
To finish the proof of the convergence (4.4) observe that due to the
relation (4.6) the sequence {nn(Tn)} is tight. This means that from any subsequence of indices {n} we are able to extract a further subsequence {nk}
such that
This together with Lemma 8 gives the convergence in distribution
Thus (4.4) is fulfilled proving tightness. Hence the proof of Theorem 1 is
complete.
5. SPECIAL CASES AND EXAMPLES
The "nonlinear part" of the conditions used for the passage to the limit
procedure in Theorem 1 are stated in terms of convergence of the scaled
partial sums J(n). In Section 5.1 we show that when T with distribution
function G is independent of £t these conditions could be set directly upon G.
In Section 5.2 we discuss another example of an age-dependent particle
system and establish a limit theorem.
Age-Dependent Branching Superprocess
249
5.1. Lifelength and Motion Independent
If in the situation of Section 2 we assume that Kt and £t, hence 9,
and £t are independent, then Eq. (2.3) takes the form
where the probability measures M t (dv) and the function Ht are deterministic. Corollary 1 (later) of Theorem 1 describes a class of admissible
pairs ( H t , M t (dv)).
An interesting observation is that the projected superprocesses we
obtain in this case in a sense are Markov superprocesses themselves. To see
this, specialize as in (2.10) by chosing f(u, x) = p(x) and obtain
where the alternate notation S t p ( x ) = E x p ( £ t ) is used as well. Of course,
this is also the result of taking expectations in (1.1). However, (5.2) is the
defining relation for the superprocess with time-inhomogeneous deterministic branching rate dH t . Hence the (£, K, V)-projected superprocess is
the same process as an (£, H, *F )-superprocess, where Ht = EK t . We refer
to this process as the time-inhomogeneous Markov (£, H, 'F)-superprocess.
We discuss alternative scaled particle systems leading to this process in a
remark following the next result.
Considering now the scaled particle systems Z(n), suppose that lifelength and motion of particles are independent and let G(n)(t) denote the
distribution function of T ( n ) . For simplicity we also assume in this section
that under scaling the process £(n) = £t is independent of n. The characteristics 0(n) and N (n) of the imbedded renewal process are independent of
£t in this case and the conditions in the theorem simplify. We make the
following assumption:
(A3 t ) there exists a real valued right-continuous function T(t), t > 0,
with T(0 + ) < oo, and a constant a e (0, oo) such that
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(to exclude accumulation of renewal points we assume T ( 0 + ) = co when
a = 0). Let {n(t), t > 0} denote the Levy process with n(0) = 0 and
where 77({0}) = 0 and
Put
Corollary 1. Suppose lifelengths and motion are independent and
that £t is independent of the scaling parameter n. Under conditions (A2),
(A3') and (A5) and if the function (u, x) -> Vt f(u, x) is continuous
the sequence of processes { n - 1 Z ( n ) , t > 0} converges weakly in D(R+,
M ( R + x Rd)) as n -> oo to the ( ( 9 i , £), K1, !F)-superprocess.
The corresponding weak limit of the scaled branching particle system
{ Z ( n ) , t > 0}, given by the (E, K1, !f)-projected superprocess, can be identified with the time-inhomogeneous Markov (£, H, !F)-superprocess Yt,
where Ht is continuous on R +, satisfies H0 = 0 and is implicitly determined
in terms of the function
by the equality
The measures M,(dv) appearing in (5.1) are given by
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251
Remarks. (A) The limiting age-structure process Xt :=X t (dv x Rd),
obtained by projection on the age variable only, is the ( 3 i , K1, !P)-superprocess. Consider Eq. (5.1) applied with f(u, x) = O ( u ) . The resulting function V t O ( u ) := V t f(u, x) is independent of x and solves
whereas VtO(u) = O(u — t) if u > t. For this case, Corollary 1 proves the
limit theorem for age-dependent branching processes stated by Sagitov.(7)
(B)
When a = 0 and
(B denotes beta-function) we get Ht = ty. The particle lifelengths in this case
have stable distributions with index y. According to (5.2) the projected
superprocess is characterized by
The same process can be obtained as the scaling limit of a branching particle system with deterministic lifelength intensity subject to a power-y
decrease relative to macroscopic time. We are lead to the following interpretation: starting from newborn particles only, the succesive appearence of
long-living particles in the (£,, T, h)-particle system will lead to a slow-down
effect in branching rate, which in the scaling limit manifests itself as an
overall power decrease relative to an outside clock.
Eq. (5.6) with y = 1 sets off the class of Dawson-Watanabe superprocesses.
Proof. We consider the ( ( 9 ( n ) , E), T(n), h(n)) branching particle system,
where 9t was introduced in (3.1), and like to deduce the weak convergence
of this extended system. However, to avoid having to reformulate the
theorem for that purpose we simply apply our result directly in the sense
that the process £(n) in the theorem is taken to be £(n) = (d(n), £) for this
application. It is certainly a serious abuse of notation in this proof to
let n - 1 Z ( n ) of the the Corollary correspond to n - 1 Z ( n ) of the Theorem,
hopefully excused by the resulting relativley short argument. In fact, we
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claim that under the assumptions of the Corollary all assumptions for
Theorem 1 hold.
Condition (A3'), which arises in the theory of infinitely divisible distributions, implies the weak convergence
towards the Levy process 77, introduced in (5.3), which is a proper timechange in the sense of Assumption 1(c). Furthermore, due to the generalized
version of the Dynkin-Lamperti renewal theorem, see Sagitov,(8) we have
uniformly in t on any compact set. Thus, because of the independence in
the model, Assumptions (A1) and (A3) are verified. To check A4 we
assume for the moment that H0 = 0 and that Ht is a continuous function,
these properties of Ht will be established in the course of the proof. Then
(independently of x), so (A4) holds. Therefore, by Theorem 1, the branching particle system converges weakly to the (($ 1 , £), K1, !F)-projected
superprocess. However, in this independent case this limit process coincides
with the ( ( 9 1 , £,), K1, !F)-superprocess. In particular, the sequence of
marginal distributions obtained by restriction to the spatial motion £,, i.e.,
the sequence of particle systems Z(n), converges to the corresponding projection of the limit process.
It remains to verify the formulae (5.4) and (5.5), which we now do in
terms of Laplace transforms. Since n(0) = 0,
Therefore,
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253
Now it is easy to see that
which is the Laplace transform version of (5.4).
Turning to the continuity of the function Ht observe first that when
a > 0 the function Ht, in fact, is absolutely continuous since
The last relation follows from
which in turn is a consequence of (5.4) and
When a = 0 Eq. (5.8) yields
for almost all t > 0. Given t > 0 take a sequence {Sk} such that Sk -> 0+ as
k -> oo and
The estimates
and the condition T(0+) = oo imply the continuity of the function Ht.
It remains to prove (5.5). The decomposition
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Kaj and Sagitov
holds, almost surely, with K°s distributed as K1s. This implies
By Laplace transformation,
Together with (5.7) this yields
Thus we have
On the other hand, due to (5.4),
Combining the last two equations we deduce
The asserted relation (5.5) follows after differentiating with respect to s.
5.2. An Example Based on the Maximum of Brownian Motion
For this final example we let £t denote standard Brownian motion on
the real line and, for simplicity, take !f(A) = A2. Consider a symmetric
binary branching particle system where the branching epochs are given in
units of the current running maximum of the Brownian particles. Introducing an ordering of particles called for convenience left/right, we mean
by this that with equal probabilities a particle is either killed or splits into
two daughter particles at that moment when it reaches for the first time a
point which is at a unit distance to the right of where it was born. More
Age-Dependent Branching Superprocess
255
exactly, we suppose that the lifelength of a particle born at time r at the
point y is the functional of its own path ( £ t ) t > r with £r = y given by
This yields a particle system Zt with birth process (Sk)k>0 such that
Sk = Jk, where we let Js, s > 0, with J0 = 0 denote the right continuous first
passage time process associated to £t.
Attempting now to obtain a limit process Xt for a scaled system
n - 1 Z ( n ) , consider a system with the order of magnitude of n Brownian particles each of mass 1/n and define T(n) as T earlier but modified so that particles branch after travelling the distance 1/n towards the right during their
life. In the notation of Theorem 1 we have £(n) = £t, Bn = n, ¥ ( n ) ( A ) = A2,
A(n) = 1 so that Assumptions (A1) and (A2) are trivially fulfilled. Moreover,
Hence £J(n)s = ([ns] + 1)/n ->s = £Js so (A3) follows immediately. Also the
moment condition (A4) is readily verified.
Putting
it is clear what we should expect to obtain in the limit n —> oo. In fact, in
this example we even have n - 1 N ( t n ) =n-1N[nt]-> Kt.
According to our results the (£, K, !P)-projected superprocess exists
and by Lemma 2 its log-Laplace function V t p ( x ) is the unique solution of
the cumulant Eq. (1.1), that is
Rewriting as in (1.2), using KJv = v,
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This can be written
where
is the Brownian first passage time density of Jv, v > 0.
In order to conclude the convergence of n - 1 Z ( n ) to Xt using Theorem 1,
it remains to show the continuity of x —> V t p ( x ) . To this end, suppose
x < y and observe that
Hence
The desired continuity now follows by an application of dominated
convergence.
ACKNOWLEDGMENTS
This research was partly funded by a travel grant from the Royal
Swedish Academy of Sciences for the support of joint research projects
between Sweden and the former Soviet Union. The authors are most grateful for a number of corrections and inspiring comments provided by an
anonymous referee.
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