Stochastic Analysis and Related Topics

Series Editors Thomas Liggett Charles Newman Loren Pitt
Stochastic Analysis and Related Topics
H. Körezliowu A. S. Ü stünel Editors
Springer Science+Business Media, LLC
H. Körezlioglu ENST
Library of Congress Cataloging-in-Publication Data
Stochastic analysis and related topics / edited by H. Körezlioglu, A. S. Üstünel.
p. cm. -- (Progress in probability : 31) Includes two main lectures given at the 3rd Silivri meeting in
1990, and other research papers. Includes bibliographical referencess. ISBN 978-1-4612-6731-7 ISBN 978-1-4612-0373-5 (eBook) DOI 10.1007/978-1-4612-0373-5 I. Stochastic analysis. I. Korezlioglu, H. (Hayri)
A. S. (Ali Süleyman) IIl. Series. QA274.2.S7714 519.2--dc20
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Table of Contents
Infinitely Divisible Random Measures and Superprocesses D.A. Dawson. . . . . . . . . . . . . . . . . . . . 1
Dirichlet forms on Infinite Dimensional State Space and Applications M. Rockner . . . . . . . . . . . . . . . . . . . . . . .. 131
Law of Large Numbers and the Central Limit Theorem for Distributions on Wiener Space S. Amine .................... 187
Une Formule d'Itc3 dans des Espaces de Banach et Applications D. Fellah and E. Parooux . . . . . . . . . . . . . . . . 197
Un Calcul Anticipatif sur une Variete Riemannienne Compacte A. Grorud and M. Pontier . . . . . . . . . . . . . . . . 211
Distributions, Feynman Integrals and Measures on Abstract Wiener Spaces G. Kallianpur and A. S. Ustund ..
Small Stochastic Perturbation of a One Dimensional Wave Equation
237
R. Leandre and F. Russo ............... 285
An Ergodic Result for Critical Spatial Branching Processes S. Melearo and S. Roelly. . . . . . . . . . . . . 333
Some Remarks on the Conditional Independence and the Markov Property D. Nualart and A. Alabert . . . . . . . . . . . .
The Wiener Chaos Expansion of certain Radon-Nikodym Derivatives
343
v
Foreword
This volume contains a large spectrum of work: super processes, Dirichlet forms, anticipative stochastic calculus, random fields and Wiener space analysis.
The first part of the volume consists of two main lectures given at the third Silivri meeting in 1990:
1. "Infinitely divisible random measures and superprocesses" by D.A. Dawson,
2. "Dirichlet forms on infinite dimensional spaces and appli cations" by M. Rockner.
The second part consists of recent research papers all related to Stochastic Analysis, motivated by stochastic partial differ ential equations, Markov fields, the Malliavin calculus and the Feynman path integrals.
We would herewith like to thank the ENST for its material support for the above mentioned meeting as well as for the ini tial preparation of this volume and to our friend and colleague Erhan Qmlar whose help and encouragement for the realization of this volume have been essential.
H. Korezlioglu A. S. Ustiinel
INFINITELY DIVISIBLE RANDOM MEASURES
The objective of these lectures is to serve as an
introduction to the theory of measure-valued branching
processes or superprocesses. This class of processes
first arose from the study of continuous state branch
ing in the work of Jirina (1958, 1964) and Watanabe
(1968). It was also linked to the study of stochastic
evolution equations in Dawson (1975). In this intro-
duction we look at two roots of this subject, namely,
spatially distributed birth and death particle systems
and stochastic partial differential equations with
non-negative solutions. In Section 2 we carry out some
exploratory calculations concerning the continuous
limit of branching particle systems and their relation
to stochastic partial differential equations. In ad
dition, we introduce the ideas of local spatial clum
ping with a set of informal calculations that lead to
the prediction that the continuum limit of branching
particle systems in dimensions d?::3 will lead to infi
nitely divisible random measures which are almost
2 D.A. Dawson
divisible random measures culminating in a proof of
the canonical representation theorem in a general set
ting. In Section 4 we introduce some of the general
setting and some basic tools of measure-valued
processes. In Section 5 we consider the class of in
finitely divisible Markov processes known as measure
valued branching processes or superprocesses. In
particular, we give a detailed construction of this
class of processes in a general setting and also give
a brief introduction to the historical process which
describes the genealogy of the superprocess. In Sec
tion 6 we derive analytical and probabilistic repre
sentations of the Palm measures associated with these
processes. The Palm measure is a useful tool in the
study of the fine structure of the fixed time random
measures. In Section 7 we apply the representation of
the Palm measures to obtain results on the carrying
dimension for (cx,d,t3)-super-processes.
It is hoped that these notes will serve two pur-
poses. The first is to provide an introduction to
measure-valued branching processes. The second is to
present in more detail those aspects most closely re
lated to the theory of infinitely divisible random
measures. For this reason, this side of the subject
is developed in considerable generality.
Random Measures and Superprocesses 3
1.1 Stochastic Population Models
the problem of identifying and classifying measure
valued stochastic processes in IRd which arise as
limits in distribution of spatially distributed popu
lation systems.
are particle systems having independent particle mo
tions, and in which particles undergo birth, death and
transformation of type. Such a particle system is des
cribed by a pure-atomic-measure-valued process
N (t)
1 =1
where N(t) is the number of particles and x.(t) de- l
notes the location of the ith particle, at time t. It
is also of interest to study systems with several
types of particles but to keep things simple we will
restrict ourselves to systems consisting of identical
particles.
models. Such a system can be described by a particle
system on cZd in which birth and death rates of a d
particle at location Yk = ck, keZ, depend on the po-
pulation size (local density) at site and are
denoted by A(Yk)' Il(Yk). In addition particles can
migrate according to a random walk on cZd.
Example 1.1. 2. Branching Markov processes.
4 D.A. Dawson
offspring at their current location at the time of
branching; between branching times the x.L) are in- 1
dependent Brownian motions, symmetric stable processes
or other Feller processes in IRd. In the multitype case
particles can also produce offspring particles of dif -
ferent types.
equations
equations. In particular ~(t) = {x.(t): ie eZd} where 1
x.(t) ~ 0 denotes the mass at the lattice site i at 1
time t, and ~ satisfies the system of stochastic dif-
ferential equations
dx.(t) = 1
+ (Ax(t».dt - 1
the discrete Laplacian, that is,
(A~)(i) = e-2 [( L x j ) - xi]
Ij-i I =e
A is
to be discussed in these lectures were motivated by
the following general questions concerning the behav
ior of the population systems introduced above.
(i) What are the possible continuous state measure
valued processes on (Rd which arise in the limit under
various space-time-mass rescalings of the above dis
cretized systems?
(ii) What is the spatial structure of the mass dis-
tribution at time t?
tical physics an approach to the first question would
be to classify the resulting measure-valued processes
into certain natural "universality classes". The most
obvious such class is the collection of deterministic
limit processes described by reaction diffusion equa-
tions. The deterministic limit corresponds to a law
of large numbers limit and has been studied by many
authors (cf. DeMasi and Presutti (1989)). By analogy
with the finite dimensional case (cf. Feller (1951)),
a second possibility is a diffusion process limit and
at a formal level this would correspond to a non
negative solution of a stochastic partial differential
equation. One important example of such a diffusion
limit is given by the class of continuous measure-
valued branching processes. In fact this class of pro
cesses has been intensively studied in recent years
because it is both natural and tractable and hopefully
will serve as a starting point for a more general the
ory of measure-valued processes.
6 D. A. Dawson
We will explore this phenomenon for branching systems
by looking at the phenomenon of clumping at small spa
tial scales.
As mentioned above, deterministic limit processes
are frequently described by reaction diffusion par
tial differential equations of the form
U.3.1) aX(t,X) = ~X(t,X) + R(t,X(t,X»
at where X(t,x) =: 0 denotes the density at location xeD
C IRd at time t, the term involving the Laplacian ~ corresponds to spatial diffusion, and R describes the
density dependent local chemical reaction.
It is then natural to explore the possibility
that the limiting dynamics is random and corresponds
to the solution of a stochastic partial differential
equation:
at • + Q(t,X(t,x»W(t,x)
form this becomes
U.3.3) J(X(t,X)-X(O,X»c/>(X)dX
sure (cf. Walsh (1986), El Karoui and Meleard (1990))
and At generates an evolution family Vet,s) on C(D)
with Green's function G(t,s;x,y). Equation (1.3.3) is
then equivalent to the evolution form
(1.3.4) X(t,x) = J G(t,O;x,y)XO(y)dy D
t + J J G(t,s;x,y)R(s,X(s,y))dyds
o D
If the region D is bounded it is necessary to pres
cribe the boundary conditions.
(a) Let A = fl < 00 'VA}. Let
and COR) = {feC(IR): sup I f(x) Ie -A I x I T
Q(.), R(.) (time homogeneous) satisfy
Lipschitz and linear growth conditions and XOe C/IR).
Then there exists a pathwise unique solution which
belongs to C (IR) and is jointly continuous in (t,x). T
(b) If At=A generates a positivity preserving semi-
group {St} , R(t,x)=R(x), Q(t,x)=Q(x), R(O) ~ 0, Q(O)
= 0, and Xo ~ 0, then a.s. X(t,x) ~ 0 for all t~O
and xe D.
appeared in the literature. An excellent reference is
Walsh (1986) but also see Iwata (1987) and Kotelenez
(1989).) The following are the main ingredients in the
proof.
(a) Use Picard iteration with the L 2 -norm to get exis
tence of L 2 -solutions. Then use Kolmogorov's criterion
to obtain Holder continuity.
ness argument (cf. Kotelenez (1989».
Remarks:
(1989) to time inhomogeneous pseudo differential oper
ators A(t) satisfying certain regularity conditions on
its Fourier transform which are satisfied if
(0 A(t) =!J. := -( _!J.)a.l2 in IR, cxe(1,2]. cx
(ii) A(t) is the closure of an elliptic operator, A,
of order 2m in IRd, 2m > d which yields < -d/2m IG(t,s;x,y)I - const-It-sl .
Thus if the dimension d~2, the elliptic operator A
must have order 2m > 2, i.e. at least 4 in order to
satisfy the conditions given in example (0 above.
But an operator A which generates a positivity pre
serving semigroup must satisfy the positive maximum
principle (cf. Ethier and Kurtz (1986, Ch. 4, Theorem
2.2», and therefore it must be a second order opera
tor. Hence the class of non-negative density-valued
processes constructed from stochastic partial diffe-
rential equations involving Lipschitz coefficients and
space-time white noise is essentially restricted to
the one dimensional case.
(2) For d=l, A = I::. and Q(u) = u'1, 1 ~ '1 < 3/2, the
existence of a solution of U.3,4) (global in time) is
established in Mueller U 989), U 990).
If Q(.) and R(.) are only continuous and satisfy
linear growth conditions at infinity (again with
R(O)?::O, Q(O)=O and XO?::O), the existence of a weak so
lution can be obtained but the question of pathwise
uniqueness is open. In particular the equation t
U.3.5) <X(t),</» = <X(O),</» + J <X(s),I::.</»ds o
</> e CcoORd), with c
t + J Q(X(s,x» </>(x)W(ds,dx)
measure-valued branching processes. It is known that
this equation has a solution only if d = 1. For this
case weak existence is proved in the sense that there
exists a pair of processes (W,X) defined on a common
probability space such that W is a cylindrical
Brownian motion and (W,X) satisfy equation (1.3.5).
However even in this case pathwise uniqueness is un
known. In dimensions d?:: 2, equation U.3.5) does
not have a solution but the corresponding martingale
problem does has a measure-valued solution. As we
will see the resulting random measures are almost sur
ely singular thus explaining the absence of density-
function-valued solution. This is a consequence of
10 D. A. Dawson
that will be investigated below.
1. 4. The role Qf dimension
As we have seen above, the known examples of
stochastic parabolic partial differential equations
involving space-time white noise having non-negative
solutions are all restricted to the one-dimensional
case. To obtain some insight into the reason behind
this it is natural to investigate the continuous limit
of the approximating particle and/or lattice systems
in IRd, d>1 (cf. Dawson (1975»). In this spirit, M.
Reimers (1986) has reformulated the question of the
existence of solutions to stochastic partial differen
tial equations in IRd with d>1 in the framework of hy
perfinite stochastic partial difference equations with
continuous Q. In particular the equation considered
therehas a well-defined internal solution and the
question is whether or not it has a non-trivial stan
dard part. In the case Q(u) = ur , 0 < r <lIZ, the
solution turns out to be deterministic. On the other
hand if Q(u) = ur , lIZ < r ~ 1, he shows that the
quadratic variation is simply too large for the solu
tion to be nearstandard as a measure-valued stochastic
process and thus no existence result is to be expec
ted.
jecture that the case Q(u) = ru, that is, the bran-
ching case plays a special role in dimensions di!:2. It
suggests that in some sense the collection of measure
valued systems in higher dimensions having non-trivial
"spatially homogeneous" (Le. invariant under spatial
shifts) orthogonal martingale measures may belong to
the same "universality class" as the measure-valued
branching system. In any case it is clear that an
understanding of the structure of measure-valued bran
ching processes can provide insight into these ques
tions.
study of measure-valued branching processes in higher
dimensions can serve as an important first step in
the development of a more general theory of measure
valued processes. In particular, the phenomenon of
spatial clumping they exhibit gives some insight into
the dimension-dependent behavior of spatially distri
buted population systems. One of the main objectives
of these lectures is to provide an introduction to
this aspect of branching systems.
In order to give some intuitive feeling for these
systems we begin with some elementary calculations
without worrying about technical details. Let MF(lRd)
denote the space of finite Borel measures on IRd. We
·d . I on the lattl·ce ,,"1 11271.d c conSI er a partlc e system ~_
IRd in which each particle is assigned a mass £ = d/2
£ 1 £2· Given such a system of particles {x/t):t~O,
• 1/271.d } . d th I d JE £1 we conSI er e measure-va ue process
N£(t) = £ L Ox .(t) j J
d and for tPE Cb (IR ) let
<N (t),tP> := JtP(x) N (t,dx). £ £
Random Measures and Superprocesses
We now consider the case in which the {x .(t)} form a J
critical branching random walk on €!/2 I d with jumps
(i) Ox ~ ° 1/2' I e I =1, with rate lI€l' X+€l e
(ii) ° ~ 0 with rate '1(x)/2€ x ~ ° +0 with rate '1(x)/2€ x x .
where '1(.) is a bounded non-negative function,
13
other.
der functions on MF(IR) of the form F(Il):=
we consi-
co 2 d where fe C (IR), t/>e Cb (IR). Then
= J{ 1 \ d/2 1/2 €1 L If(<Il,t/>>+€l €2(t/>(x+€1 e)-t/>(x»
lel=l
11 denotes the Laplacian on IRd, and ~e C2(lRd).
2.2 Limiting Regimes
d/2 (1) £1"""'-+ 0, £2 = c/£ 1 ,c>O. Then we obtain a
system of branching Brownian motions in the limit.
In this case we obtain a
continuous state lattice system with generator
(2.2.1)
xe£l
where {Z (t,x):xe£~/2zd} is the solution of the sys £1
tern of stochastic differential equations
(2.2.2) dZ (t,x) = 11 Z (t,x)dt £1 £1 £1
+ / -r(x)Z (t,x) dB (t) £1 x
where the {Bx(.):xe £~/2Zd} are a family of indepen
£1 dent Brownian motions. Then the process X (t) has
limiting process.
o Z (t,X):=
We assume that as C1 ~ 0, the initial measures
Z (O,dx) c1
xec1
15
bounded density with respect to Lebesgue measure on
!Rd. We will now investigate what happens to the first
two moments of the density process as cl~ O. c
If {T 1: t~O} denotes the semigroup on t
Cb (c!/27Ld) with generator fl we can verify that the c1
mean density is given by
If
mean
1 0 d/2 measures M (t,x)(c 1 c3)
c 1 x
c b to M (0,.), then the
converge to
semigroup. But by Lemma 3.2.8 this implies that the
random measures are tight and hence we expect the
random measures Z (t,.) to converge weakly as c1 ~ c 1
O. We next consider the covariance function of the
density process:
Using Ito's lemma we can verify that
(2.2.3)
1 £1
responding to the semigroup
r (t,x,y) £1
Then the evolu-
= II P£l (t,x1,X)P£1 (t'Y1,y)r(O,X1'Y1);\1 (dX1)A£1 (dY1)
t
+ IoII P £1 (t-s,x1,x)p £1 (t-s,y l'Y)
where A ({y}) d/2 for each 1I271d and = £1 Y E £1
£1
tion.
Now consider the limit as £1 ~ O. It can be
verified that that A (dx) ~ dx and for x:l:y, £1
r (t,x,y) ~ r(t,x,y) where for x:l:y £1
r(t,x,y)
= It I p(t-s,z,x)p(t-s,z,y)r(z)MD(s,z)dzds. o
If x=y, {MD(s,. ):O~s~t} is bounded away from zero on
a neighbourhood of x, and rex) ~ cO> 0 then this
integral
~ It I p(t-s,z,x)p(t-s,Z,X)cOMD(S,Z)dZdS o
= const J: (t-s fd/2 (I p( t -s, x, z)Jl (s, z )dz 1 ds.
Therefore if x=y and d~2 nt,x,x) = + 00.
From the above calculation we conclude that the
density process is not well defined (at least in the
L 2 -sense) in the €C~O limit in dimensions ~ 2. In
d=l the L 2 -density does exist. Formally, if it exis-
18 D.A. Dawson
differential equation
where Wet) is a cylindrical Brownian motion (W(dt,dx)
is a space-time white noise). Once again the above
calculation suggests that this stochastic partial
differential equation will only have a solution if
d=1.
Remark If 'l(z) is zero except on a set of Lebesgue
measure zero it may be possible to obtain an absolute
ly continuous measure-valued process in dimensions
d>1.
Now let us return to the measure-valued process
Z (t,x). If </>(.,.)e C1•2(1R xlRd). then by ItcJ"s £1 +
lemma
-((a: + G,Jx+J~(S.XlZC1(s.clxl)dS
= exp (-I </>(t,X)Z£/t.dX»)
- {It (8 8</>( s. x) Z (s.dx) - III </>(s.x)Z (s,dx) o s £1 £1 £1
+ 1/2 I </>2(s,x)'l(X)Z £/S.dX»)
= Mt (</» is a martingale.
8u
Then
- 2 r u£ ' 1
u (O,X) = !/I(x). £1
8</>(s x) 1 2 - 8 ' = l!l </>(s,x) - - rex)</> (s,x).
s £1 2
for O:S s :S t and therefore
E( exp(-J!/I(X)Z£/t,dX»)) = exp(-Ju£/t,X)Z£/O,dX»).
19
We now consider the £1---+ ° limit. Note that if 2 d !/Ie Cb (IR ), then
112 u (t,X) = u(t,X) + 0(£1 ) £1
where
U(O,X) = !/I(x).
Then if Z (O,dx) ==> Z(O,dx) in MF(lRd), the finite £1
dimensional distributions of the measure-valued pro-
cesses Z (t,. ) converge in distribution to a time £1
d homogeneous MF(IR )-valued Markov process with La-
place transition functional
20 D. A. Dawson
Laplace functional satisfies the multiplicative pro
perty Ut,fl1+fl2,1/J) = Ut,fll'l/J)' Ut,fl2,1/J)·
For the remainder of this section we assume that d
'lex) == 'l > O. Then z(t):= Z(t,IR ) has the same law
as the solution of the stochastic differential equa
tion
when I/J(x) == 9, satisfies
and the non-extinction probability satisfies
lim t.P (z(t) > 0) = -zOu9(t)
lim t.(1- lim e ) t-+oo Zo t-+oo 9-+00
2 = r zO°
A Measure of Clumping
model. For the moment we consider the total popula
tion {z(t):t~O} when the particle mass is c and the
branching rate is -1
critical branching processes (cf. Athreya and Ney
(1977» starting with one particle at time 0 the
probability of non-extinction by time
satisfies
and conditioned on non-extinction, c -' z(tlc) t
is
z(O) = [zOIc), then the number of initial particles
having descendents alive at time t is Bernoulli B(n,p)
with parameters n = [zOIc) and p = 2clt. Conse-
quently starting with [zOIc] particles at time 0,
c· z(tlc) converges in distribution to a Poisson super
position (with Poisson mean 2z0/t) of independent
exponential random variables each having mean t/2.
Recall that if N is Poisson with parameter A,
then
and if Y is exponential with mean 1/0: = B, then
-9Y 0: 1 E(e ) = 0:+9 = I+B9 .
Hence the sum of a Poisson number (with mean A = 2/t)
of independent exponential (with mean t/2) random var
iables yields
E(ex+a 1 Yj )) = exp(~~::) = exp(l~al2)' This gives an alternative derivation of the Laplace
transform of the distribution of the infinitely
divisible random variable z(t) when z(O) = 1.
22 D.A. Dawson
Now consider the binary branching random walk on 112.."d £ 11.., d ~ 3, starting with one particle of mass
£ d/2 at each of the £ -d/2 lattice points in the unit 1/2.."d cube in £ 11... Assume that the random walk rate is
-d/2 11£ and the branching rate is £ (This means
that 1'=1, £ =1 in our earlier notation.) Let 2
N (t)
d/2 \ x£ (t,A) := £ L 1 A (x .(t». J=l J
Now consider the situation at time t=£. The proba-
bility that an initial particle has descendents at
time £ is £ (d/2)-1
a family of particles, which we call a cluster, of
size ~o £ I-d/2 where ~ is (asymptotically £-+0) as an
exponential (with mean 1/2) random variable.
Thus the number of surviving clusters is Poisson
with mean K = 0(£-1) and the number of particles per £
surviving cluster is L = ~.£1-d/2, that is, a total -d/2 £
of 0(£ ) 0 ~ particles. x., i = 1, ... ,L , 1 £
Let
denote the number of random walk steps taken by the
particles in a cluster. During the time interval [0,£]
the ith surviving particle takes a
1) number of random walk steps, e-1
e ,and therefore P(x. > n) ~ 1
Poisson (with mean X.
C e , for large n
and for some constant C. We shall obtain an upper
bound for the number of sites occupied by the cluster
by counting all sites in a box whose side is
2 o max(Xl' ... ,XL ). £
:s e1-Cd/2).p(Xl > k log lie)
1-(d/2)C -k log lie :s e .e
= C ek e1-Cd/2) «1 if k > d/2 -1.
23
Hence the total number of occupied sites in a fixed
bounded subset of IRd. 0(e1l2). is less than
const· K • (k e
Ce (k+ 1)-( d/2) .
The corresponding effective dimension
log 0(e1l2) : = --=------,,.......,.,,,..... log 1/e1l2
log e
= 2 + o(e).
Finally. between t=e and t=2e the probability that
1 t (f . G> 1-d/2) . . 0(1) I a c us er 0 SIze c>. e surVIves IS . n
fact the number of particles (and thus the number of
new clusters) to survive is Poisson with mean l5. We
end up with a critical branching random walk on
e1/2Zd with O(e -1) clusters. each cluster occupies a
number of lattice sites of order at most k log lie.
Thus the effective dimension estimates propagate in
time. In fact we can obtain such an estimate that is
true for all t = 0.e.2e •...• [tle] with arbitrarily
24 D.A. Dawson
The above calculations lead us to expect that for
t > 0 and d > 2, the limiting measure-valued process
Z(t,dx) is so highly clumped that it lives on a set
of strictly smaller dimension. Thus in particular it
is a singular random measure. This explains the
reason behind the fact that the second moment density
is singular in dimensions d > 2. A rigorous develop
ment of these ideas is given in Chapter 7.
on Polish Spaces.
3.1. State Spaces
Let E be a Polish space with metric d. M(E)
denotes the space of Radon measures on (E,fn where fS
is the Borel O"-algebra. C(E) (resp. bC(E), bCb (E),
C (E)) denotes the space of continuous functions on E c
(resp. bounded continuous, bounded continuous with
bounded support, continuous with compact support) and
bfS (resp. pbfS, pbfSb) denotes the space of bounded
measurable (resp. non-negative bounded measurable,
non-negative bounded measurable with bounded support)
functions on E. <Il,f>:= Jfdll ' lleM(E), fe Cc(E)' If
E is locally compact, we denote by CO(E) the con
tinuous functions which decrease to zero at infinity.
Some Special Cases
If E is compact then M(E) may be compactified via
the Watanabe compactification. Let
~i(E) = M(E)v{ll}
Il ~ Il e M(E) iff <Il ,f> ~ <Il,f> V fe C(E), n n
Il ~ 11 iff <Il ,1> ~ 00. n n Then M(E) is a compact metrizable space.
b. Locally finite measures.
Let fSb be the ring of bounded sets in fS. Let
(MF(E), 'l'w) denote the finite Borel measures with its
26 D. A. Dawson
T
fl ~ fl iff <fl ,f> ---7 <fl,f> V fe bC(E). n n
(MLF(E), TV) denote the collection of Borel measures
finite on gb with the vague topology T
fl ~ fl iff <fl ,f> ~ <fl,f> V fe bCb(E)' n n Both (MF(E), TW) and (MLF(E), TV) are Polish spaces.
c. p-Tempered Measures on IR d
For p>O let if> (x) := (1+ I x 12) -Pj M (lRd) = P P
{fl:<fl,if> > < oo} with the topology T defined by p P
fl ~ fl iff <fl ,f> ~ <fl,f> V fe K (lRd) n n p
where K (lRd) = {f:f = g + «AI. ge C (lRd) «e IR}. p V'p' c'
(M (lRd ), T ) is also a Polish space. p p
Let IRd: = IRdu{oo } where 00 is an isolated p p
point and extend if> to IRd by ~ (00 ) = 1. Let p p p
·d • M (IR ) = {fl:<fl,if> > < oo} with the p-vague topology p p
·d defined by fl ~ fl iff <fl ,f> ~ </J.,f> V fe K (IR )
n n .d p (defined as above). In this case M (IR ) is locally
p •
compact and sets of the form {fl:<fl,if> > ::5 k} are com p
pact (cf. Iscoe (1986a».
3.2. Random Measures
Let E, (MLF(E);l) be as above and At = ~(MLF(E» denote the Borel subsets of MLF(E)' A locally finite
random measure on E is given by a probability measure
P on (MLF(E),M).
A sequence {f n} in pblS'b is said to converge bbp
to f if f (x) ~ rex) \/x, 3 M < co such that sup f (x) n n n,x
~ M, and there exists a bounded set B such that \/n
f (x) = 0 for n
c xeB. Given H c pblS'b' the bb-
pointwise closure is the smallest collection of func-
tions containing H which is closed under bbp conver-
gence.
Lemma 3.2.1. There exists a countable set V = {f } S;; n
pbCb (E) whose bbp closure is pblS'b and which is con-
vergence determining in MLF(E).
Proof. [EK pages 111-112].
Ethier and Kurtz (1986) (abbreviated as [EK)) for
basic facts on weak convergence of measures on Polish
spaces. )
We can then define a metric on MLF(E) in terms
of V = {f } (as in Lemma 3.2.1) as follows n
d(fJ,v) := I 2-m (IAI<fJ,fm>-<V,fm>I).
Because V is convergence determining it follows that
fJ converges to fJ in the topology 't' if and only if n v
d(fJ ,fJ) ~ O. n
If E is compact we can take 1 e V. Then
<fJ ,1> ~ m < co implies that {fJ} is relatively com- n n
pact. From this we can verify that d is a complete
metric if E is compact.
28 D.A. Dawson
we begin by recalling that every Polish space E is
Borel isomorphic to a compact metric space (cf. Cohn
(1980) Theorem 8.3.6). For each positive integer m let
Bm := {x:d(x,xO)::$m} for some fixed xo. We can then put
a metric on B such that it becomes a compact metric m space and such that the Borel sets for this metric
coincide with the Borel subsets of B. We then choose m
a countable subset V' of the space of functions with m supports in B which is dense in the space of funcm tions (with supports in B ) whose restrictions to
m B
m are continuous in the new metric. Let V' = UV' c
m pbgb' Then a metric defined as above but using V' in
place of V is complete.
Lemma 3.2.2. (0 If A is a class of bounded Borel
sets in E closed under finite intersections and con-
taining a basis for the topology on E, then
At = cr{f A: AeA} where fA (IJ) = IJ(A).
(ii) Let V be as in Lemma 3.2.1. Then At = cr{<.,f>:feV}.
Proof. (0 For fe bCb(E), IJ --7 <1J,f> is continuous
and hence At-measurable. It is easy to check that {f:1J
--7 <IJ,f> is At-measurable} is closed under bbp limits.
Since pbgb is the bbp closure of bCb (E), this implies
that IJ --7 IJ(A) is At-measurable for any Ae gb and
hence At::> cr{f A: AetA}. On the other hand for each f of
n
f = L aixA. with A.e A. a. e IR. the map- 1 1
the form
i=l 1
ping Il ---7 <Il.f> is o-{f A: AeA}-measurable. By Dynkin's
class theorem for functions ([EK p. 497]) the bbp clo-
sure of this class of functions contains
Hence for fe bCb(E). Il ---7 <Il.f> is
o-{f A: AeA}-measurable which implies that At c
o-{f A:AeA}.
(ii) The proof is similar. 0
Lemma 3.2.3. Let A. V be as in Lemma 3.2.2. Then a
probability measure. X. on (MLF.At) is uniquely deter
mined by the
"finite dimensional
distributions"
of
or
by
consistent family of "finite dimensional distribu
tions" of the form
{Of f'~ 'Pf f :f1.···.f e 1····· n n 1' ...• n n where
Q = IRn and ~ = B(lRn) which satisfy the "al- f1' .... f n n
most linear" and "almost positive" properties. Then
(Q.~ .P) is obtained as the projective limit of this
family and the random measure is then obtained by
applying one of the following two theorems.
30 D. A. Dawson
Theorem 3.2.4. Let (Q,:f,P) be a probability space, E
be a compact metric space and (5 the cr-algebra of Borel
subsets of E. Let T: C(E) ~ b:f be almost linear and
almost positive, i.e.
and
f 2:: 0 implies that Tf?; 0 P-a. s. 'If.
Then there exists an MF(E)-valued random variable X
defined on (Q,:f,P) such that Tf(w) = IreX)X(W,dX)
P-a.e. w for each fe C(E).
Proof. Let 1f c C(E) be a countable vector space over
the rationals Q which contains f=l and is dense in + C(E). Let M = IITill. Let 1f = {he1f:h?;O}. Then there
exists a set Nc Q, peN) = 0, such that V we NC,
T(a.f +(3g)(w) = a.Tf(w)+(3Tg(w) , f,g e 1f, 0'.,(3 e Q,
Tf(w) ?; 0 if fe 1f+
Tf(w) ~ Mllfll, fe 1f.
Tf(w) = Tf(w) if weN
= 0 if weN.
Since 1f is dense in C(E) Tf(w) can be extended by
continuity to feC(E) for each w. Then by the Riesz
representation theorem for each w there exists a mea
sure X(W) such that Tf(w) = IreX)X(W,dX) for each
fe C(E). It is then easy to verify that X:Q ~ MF(E)
is measurable. 0
Theorem 3.2.5. Let (Q,~,P) be a probability space, E
be a Polish space and g the cr-algebra of Borel subsets
of E. Let T: bCb (E) ~ b~ be almost linear and almost
positive. Assume in addition that
(3.2.6) if {f } c bg, O::!5f ife bgb' then Tf iTf P-a.s. n n n
Then there exists a MLF(E)-valued random variable X
defined on (Q,~,P) such that Tf(w) = If(X)X(w,dX)
P-a.e. w for each fe bCb(E).
Proof. The proof uses the fact that a Polish space is
homeomorphic to a Borel subset of a compact metric
space (cf. Cohn (1980), page 260) and Theorem 3.2.4.
For details, see Getoor (1975).0
A sequence of probability measure {P } on the n
Polish space {MLF(E),'t') converges weakly to P if for
every bounded continuous function F on MLF(E),
lim I F(Il)P n (dll) = I F(Il)P( dll)· n-tco
Then Ml (MLF(E» with the weak topology is also a Po-
lish space.
Theorem 3.2.6. A subset A c MLF(E) is relatively
compact in the vague topology if and only if f or all
bounded and closed sets BeE,
(0 sup Il(B) < co, and ileA
(ii) for any E > 0, there exists a compact subset
K c B such that E
32 D. A. Dawson
Proof. This is a straightforward extension of Proho-
rov's theorem for finite measures on a Polish space.
Lemma 3.2.7. If the mean measures I (B): = n J fl(B)P n (dfl), Be l5b, are tight in (MLF(E),M), then
{P n} are relatively compact in Ml (MLF(E»'
Proof. By Prohorov's theorem it suffices to show that
for every e>O, there exists a compact subset C c e MLF(E) such that sup Pn(C~) < e. Because the mean
n measures are tight Theorem 3.2.6 implies that for
each m ~ 1 there exists a constant k and a compact m,e subset K c B := {x:d(x,xO) ::s m} c E such that
m,e m
sup J fl(B )P (dfl) < k 12 and m n m,e n
sup J fl(B \K )P (dfl) < e2/2m+l. m m,e n
n But then by Chebyshev's inequality, for all nand m,
Then
P ({fl:fl(B \K ) > e}) < e/2m. n m m,e
c sup P (C ) < e where n e n
C = {fl:V m~l, fl(B )::s 2mk Ie, fl(B \K )::s e } e m m,e m m,e
To show that C is compact note that if B is a e
bounded and closed set then there exists m ~ 1 such
that B c B . But then for e>O, m
: = BrU< satisfy the hypotheses of m,e we conclude that C is compact.o e
k = k e m,e' and K e
Theorem 3.2.6 and
3.3 Laplace Functionals. Let P be a probability mea
sure on (MLF,.At). The associated Laplace functional is
defined by
J -<Il f> Uf) := e' P(dll), fe pbgb' MLF(E)
It is easy to check that U.) is continuous under bbp
limits.
( ) ( ) . __ e-<Il,f> Lemma 3.3.1. i Let F f Il . , feY where v is as above. Then .At = CT{F f:feV}.
(ii) The space of .At-measurable functions is the bound-
ed pointwise closure of the linear span of V = {Ff:feV}.
(iii) P is uniquely determined by Uf), fe V.
Proof. 0) This follows immediately from Lemma
3.2.20i).
Oi) Sets of the form {1l:<Il,f.> eO., i=l,oo.,n} 1 1
where O. are open in IR, f.eV, form a fundamental set 1 1
of neighbourhoods in MLF(E)' Note that the pointwise
bounded closure of finite linear combinations of the -b.x
form L aie 1 contains sets of the form 10 , 0 open.
Hence .At = CT{F fL ):feV} follows by Dynkin's class
theorem.
Theorem 3.3.2 0) {F r=feV} is convergence determining
in M1 (MLF(E)), i.e., if J F f(Il)P n (dll) ~ J F f(Il)P(dll)
V fe V, then P converges weakly to P. n
Oi) Consider {P n} in Ml (MF(E))' There exists a
34 D. A. Dawson
countable set of strictly positive g-measurable func
tions V' such that if J F f(Il)P n (dll) ~ Uf) V fe V',
then there exists peM~1(MF(E» such that Uf) = J F f(ll) P(dll) V fe V'.
MF(E)
Proof. (i) The linear span of {F f:feV} is an algebra.
We will show that it is strongly separating (cf. [EK
p. 113]). Let lleMLF(E) and 0>0.
and 2-m < 0/4. If d(v,ll) > 0, then m
Let V = {f.: ieIN} 1
\ I <v,f.>-<Il,f.> I ~ 0/2 and hence L 1 1
i=1
max I <v,f.>-<Il,f.> I ~ 0/2m. 1<.< 1 1 -l-m
But then
ma~ IFf. (v)-F f. (Il) I 1 ~l~m 1 1
const 0
then follows since a strongly separating algebra of
functions is convergence determining (cf. [EK p.
113]).
Oi) We can put a new metric on E making it a compact
metric space and for which the Borel sets coincide
with g. We then choose a countable set V' of strictly
positive functions on E which are dense in the space
of nonnegative functions continuous for the new metric
and, consequently, convergence determining for MF(E)'
Recall that the terms of a sequence {P n} e M1(MF(E»,
E compact, can be viewed as elements of M1(MF (E»
where MF (E) denotes the Watanabe compactification of
MF(E). The result Oi) follows by noting that
M1 (MF (E) ) is compact, every element of M1 (MF (E) )
gives a subprobability on MF(E), recalling the conti-
nuity theorem for Laplace transforms
(0. The details are left to the reader.D
and applying
Remark. S = pbl:\ is a semigroup with the operation
of pointwise addition and is two-divisible, i. e. t=s+s
for some seS 'v'teS.
tive definite if
n
n
\ a.a .4>(s.+s.) ~ 0, n~l, {a.} ~ IR, L 1 J 1 J 1
i, j=l
n
If \ a.a .q,(s.+s.) s 0 whenever n~2, \ a l. = 0, L I J I J L
i,j=l i=l then 4> is negative definite.
Theorem 3.3.3 0) Uf), a Laplace functional, is posi
tive definite.
(i0 If L:pbgb ~ [0,1] is positive definite such
that Uf n) i Uf) whenever {f n} c pbgb' f n! f
pointwise, then there exists a unique P e MS1 (MLF(E))
such that
(( -<x,¢.» 2)
(ii) This is based on a general result on positive
definite functions on semigroups which can befound in
Berg, Christensen and Ressel (1984) and Theorem 3.2.5
(See also Fitzsimmons (1988 (A.6))).D
A random measure x is said to be infinitely
divisible if for every positive integer n, x D
X1+"'+Xn where Xl'""Xn are independent iden-
tically distributed random measures.
such that Utf) is positive definite T/ t > O. Then
the log-Laplace functional LUf):= - log Uf) ~ 0 is
negative definite. (ii) Conversely, if ",(f) ~ 0 is
negative definite on pbt5b, then e -t",(f)
is positive
definite. If in addition, ",(f n)! ",(f) if f !f point- n
wise and ",(0) = 0, then e -t",(f)
is the Laplace func-
on MLF(E).
Proof. 0) If L a. = o and exp( - t LUf)) is po- l
sitive definite, then
1 { -tLL( <1>. +<1> .) } t L L aia j (e 1 J - 1) ~ O.
Taking limits as t~O, we get
g'(O) = - \ \ a.a.LL(<1>.+<1>.} ~ O. L L 1 J 1 J
37
and the appendix of Fitzsimmons (1988).0
The Poisson random measure (with intensity mea
sure 1\(.) e MLF(E» is given by the Laplace functional
(3.3.0 L1\ (<1» = exp ( -I (1 - e -<1>(X»A(dX»), <1> e pbgb'
Let E1 and E2 be Polish spaces. The Poisson clu
ster random measure with locally finite intensity mea
sure 1\ on E1 and cluster probability law {P} E on x xe 1
MLF(E2) such that I I Il(B)P x(dll)1\(dx) < co V MLF(E2) E1
B e g2,b' is a locally finite random measure on E2
defined for <1>e pbgb (E2)
(3.3.2) L1\,{p} (<1» = exp (-IE (1 - P xe -<·,<1»)1\(dX»). x 1
Poisson random measures and Poisson cluster ran-
dom measures are infinitely divisible.
3.4 Canonical Representation of Infinitely Divisible
Random Measures.
on a Polish space E.
For n e 7L and xOeE fixed let B := + n
{x:d(x,xO)~n} c E. We can and shall put a metric p on
38 D. A. Dawson
MLF(E)\.{O} such that and M c MLF(E)\.{O} is a bounded
set iff inf Il(B ) ~ 11k for some nand k e 7L and n +
lleM such that Ilm ~ Il e MLF(E)\.{O} if and only if
P(1l ,Il) ~ O. (Thus a bounded set M does not contain m elements arbitrarily close to zero.)
Let M2,LF(E) denote the set of measures, v, on
MLF(E) such that
(ii) J (l-e -1l(A»)v(dll) < co VAe t;b' MLF(E)
If v e M2,LF(E) then for e>O, V({Il:Il(Bn ) > d) < co
and hence each measure in M2,LF(E) restricted to
MLF(E)\.{O} is in MLF(MLF(E)\.{O}).
Theorem 3.4.1 (Canonical Representation) Let X be
a random measure with values in MLF(E) with infinitely
divisible law P and with E(X(B)) < co for each B e
t;b' Then there exists a pair (M,R) e MLF(E)xM2,LF(E)
such that for each </> e pbt;b'
(3.4.1) L(</» = E(e -<X,</») = e -u(</»
where
MLF(E) (3.4.2)
functional of an infinitely divisible random measure.
«3.4.2) is called the canonical or KLM representation
of the infinitely divisible random measure.) V Proof. By infinite divisibility, X = X1+ ... +Xm,
where X1 •...• Xm are LLd.
malized empirical measure
m
j=l J
then X = ,.,. X (d,.,.). Let R := E(X ). Then for 7) J A A
m m m
Beg-b' J ,.,.(B)Rm (d,.,.) = E(X(B» < 00. and Rm ({O}) = O.
,.., Let B be a bounded closed set in
Then 3 nand kB >0 n
then
MLF(E)\{O}.
~ lIkB . But n
R cEi> ::s R ({,.,.:,.,.(B ) ~ lIkB }) ::s EX(B ). kB m m n n n n
Hence Rm (restricted to MLF(E)\{O} when appropriate)
is in MLF(MLF(E)\{O}) and for each bounded set S in
MLF(E)\{O}. sup Rm(S) < 00. In order to prove that the m
Rm are tight in MLF(MLF(E)\{O}) we are using Theorem
3.2.6. It suffices to show that for each bounded
closed set S in MLF(E)\{O} and for each £>0 we can ,.., ,..,
find a compact set K c B such that £
sup R (S\K ) < £. m £ m
Using Theorem 3.2.6 again we note that ,.., ,.., K c B is
compact if and only if for each k.n e 7L there exists +
a constant K. and a compact subset K k c B such n n. n
that
40 D. A. Dawson
sup Il(B ) < K, SUp Il(B \K k) < 2 -k. ~,n n ~ n n,
lleK lleK
Since J Il Rm (dll) = E(X) e MLF(E) by assump- MLF(E)
tion, E(X)(B ) < IX) and by Chebyshev's inequality we n
can choose K such that n
R ({1l:Il(B ) > K }) ::!5 E(X(B »/K < e/2n+1. m n n n n
Furthermore since every finite measure on B is inner n
regular we can choose a compact set K' c B such n,k n
that
Then
k n+k+l R ({1l:1l(B \K' k) ~ 112 }) < e/2 \1m. m n n,
We now define
~ k 112 \In,k}. Then by Theorem 3.2.6 K is compact in
e MLF(E) and R (]hK ) ::!5 e
m e This shows that R
m
MLF(MLF(E)\{O})' By Lemma 3.2.7 this implies that the
laws of X are tight in the weak topology on m
Ml (MLF(MLF(E)\{O}»'
We can then choose a convergent subsequence which
for convenience we also denote by {R }, i.e. R ---7 R m m in MLF(MLF(E)\{O}) as m~lX). We can extend R to MLF(E)
by setting R({O}) = O. Without loss of generality we "
can also assume that Xm ---7 XIX) in Ml(MLF(MLF(E)\{O}»
(cf. Lemma 3.2.7).
Let A!' ...• An be bounded measurable subsets of
MLF(E)'{O} with Ai"A j = 0 for i;!: j and R(8Ai )=0 for
each L Since XI' .... Xm are LLd .•
P(X (A1)=k1 •... X (A )=k ) m m n n
m! = kl!. .. k !(m-I:k.)! n 1
( R (A 1) ) kl (R (A ) ) k ( 1 n ) m-I:k. . m ... m n n 1- _ \' R (A.) 1 m m mL mi'
1=1
As m -? (XI. R (A.) -? R(A.) and we obtain mil
P(X (A1)=k1 •... X (A )=k ) (XI (XI n n
1 k. -I:R(A.) = n R(A.) 1 e i 1
k1!.··k ! n. 1 1 n 1=
By Lemma 3.2.3 these probabilities uniquely determine
the law of X as that of a Poisson random measure on (XI
MLF(E)'{O} with intensity measure R(.). In other
words
for I/J e pb~(MLF(E)'{o}) with bounded support in
MLF(E). Let gn:MLF(E)'{O} ~ [0.11 be continuous with
42 D. A. Dawson
bounded support and assume that g (fl) = 1 if fle B n n
where Bn = {fleMLF(E)'{O}: P(fl,flO) ~ n}. Then I/Jn (fl) = gn (fl)<fl,</» with </>e pbCb (E) belongs to
Cb(MLF(E)'{O}). Then
Note that for </>e pbg>b' me 7L +' and n -+ 00,
! E(exp ( - J<fl,</» Xm(dfl»)) = E(e-<X,</»)
and therefore
This implies that for Be 8 b,
J U-e -1l(B»R(d, .. t} < co, i.e. R e M2,LF(E). MLF(E)
In addition for </>epb8b,
Note that
and define M := E(X) - E(X'), i.e. for </> e pbCb(E),
<M,</» = lim n~co
m
= lim 1 im E( L <Xj'</»oU-gn(xj »). n~co m~co
j=l
Since </> e pbCb (E), for E > 0 there exists nO(E)
such that if n~nO(E), then < Eo
Then for n~nO(E),
J= It follows that
E <M,q,>.
E(e -<X,q,»
we obtain
-<X q,> ( I -<v q,> ) E(e ' ) = exp -<M,q,>- (l-e' )R(dv)
MLF(E)
This completes the proof of the first part of the
theorem.
To prove the converse we first note that u(</» of
the above form is a negative definite function on
pb&'b. But then by Lemma 3.3.40) Ln (</» :=
exp( -u(</»/n) is positive definite. Furthermore, if
</>m J, </> e pb&'b' then u(</>m) 1 u(</» by the monotone
convergence theorem. Therefore by Theorem 3.3.3(ii),
L (</» is the Laplace functional of a subprobability n
measure on MLF(E)' On the other hand lim L (9</» = 1 9-+0 n
and hence the latter
sure. But then U</»
= [L (</»]n for all n and hence n
U</» is the Laplace functional of an infinitely di-
visible random measure. 0
real-valued random variable with E(X) < 00. Then the
Laplace transform
L(a) := E(e -ax) = exp [- rna - r:+ O-e -ax)n(dx)]
where 9 ~ 0, m ~ 0, and n(dx) is a measure on (0,00)
such that 00 J x n(dx) < 00. o
Proof. This corresponds to the special case E = {e}.
Remark: The assumption of a finite first moment used
above is not essential and can be replaced by the as
sumption that X is simply a locally finite random
measure. However the first moment assumption makes
things a little easier and covers the needs of these
notes.
In this section we collect some standard results
on Markov processes and on weak convergence for pro
cesses with values in a Polish space and apply them to
the measure-valued setting. Basic references for this
material are Dynkin (1965) [DY], Ethier-Kurtz (1986)
[EK] and we will refer to these sources for the
proofs.
4.1 Transition Functions and Markov Processes.
Let (E,&') be a Polish space. , A transition function on E, is a mapping (s, t,x)
~ P(s,t,x,.)
&' -measurable
A, P(s,t,.,A) is
to
(or time inhomogeneous semigroup)
The transition operators satisfy the semigroup proper-
ty
T = T (T ) for r ~ s ~ t. r.t r.s s.t
The transition operators are said to be time homoge
neous if T t = TO t- for all 0 ~ r ~ t. r. • r
47
tion v one can contruct a consistent family of
finite dimensional distributions given by
Using Kolmogorov' s extension theorem one can construct IR
a Markov probability measure P on (E.g) + with the v
above finite dimensional IR
IR
distributions.
Then
Markov process with transition function PC •.•.•. ).
For some purposes additional structure such as sample
path regularity or strong Markov property is required
and then we consider a more restrictive classes of
processes such as Feller or right processes (cf.
Sharpe (1988)).
Let E be a locally compact. separable metric
space. Let C(E) (resp. CO(E)) denote the space of
continuous functions on E (resp. continuous func
tions which vanish at infinity) furnished with the
48 D. A. Dawson
strongly continuous, positive, time homogeneous,
contraction semigroup {Tt:t~O} satisfying T 1 = 1 t
for all t~O. The (strong, infini tesimal) generator of
the semigroup {Tt } on CO(E) is the linear operator
A defined by
where the limit is taken in the sup-norm topology.
The domain, V(A), of A is the subspace of all f
for which this limit exists. The Feller semigroup on
CO(E) is uniquely determined by its strong infinitesi
mal generator.
weak infinitesimal generator. Note that in this case
"weak convergence" is bounded pointwise convergence
(cf. [DY, p. 50].)
Theorem 4.2.1 The closure A of a linear operator A
on V(E) (E as above) is single-valued and generates a
strongly continuous, positive contraction semigroup
{Tt } on CO(E) if and only if
(a) V(A) is dense in Co(E),
(b) A satisfies the positive maximum principle, i.e.,
if fe V(A), xOe E and sup f(x) = f(xO) ~ 0, then xeE
(c) The range ~(A-A) is dense in CO(E) for some
A > O.
Proof. The proof is based on the Yosida approximation
A = A(I-n -1A)-1 (see e.g. [DY, Chapt. 2, Theorem 2.8] n
or [EK Chapt. 4, Theorem 2.2]).
A subspace D c V(A) is a core for A if the clo
sure of the restriction of A to D is equal to A.
Lemma 4.2.2 Let A be the generator of a strongly
continuous semigroup {Tt } on CO(E)' Let DO and D be
dense subspaces of CO(E) with DO cDc V(A). If
Tt:DO ~ D for all t~O, then D is a core for A.
Proof. Watanabe (1968), or [EK p. 17].
Example 4.2.3. Brownian and symmetric stable pro-
cesses in IRd.
will be used below. The first is the d-dimensional
Brownian motion with generator A = ~ Lld (where Lld is
the d-dimensional Laplacian). 00 d
C (IR) (the space of infinitely differentiable func- c
tions with compact support) is a core for Lld. The
process can be extended to the one-point compactifica
tion IRd: = IRdv{oo} (see remark at end of this sec
tion).
index a, 0 < a < 2, with semigroup {S~} and generator
A = Ll = -( -Ll )a/2 where d,a d
(4.2.1) Lld,a<l>(x) := J IRd [(x+Y)-(X)
50 D. A. Dawson
-V</>(x) . (y 1(1 + I y 12»] I y 1-( d+cx) dy
for tfJ e C2(lRd) and the latter is a core (cf. [EK c
Chapt. 8, Theorem 3.4]). The transition functions are
spatially homogeneous p(t,x,y) = p(t,y-x) and
E(eia.X(t)IX(O)=O) = exp{-leICXt} if cx < 2
= exp{-I e 12t/2} if cx = 2.
From this we obtain the scaling relation: for R e IR
(4.2.2) (SCX f)(y/R) = E(f(X(t/Rcx» I X(O)=y/R) tlRcx
= E(f(X(t)lR) I X(O)=y)
Lemma 4.2.4 t
J ° s~p (S:lB(O,r»(y)dS :s const. rcx if d > cx
:S const. r cx log + lIr if d=cx.
Proof. See Perkins (1988), Lemma 2.7.
For each Feller semigroup there exists a time
homogeneous transition function P(t,x,.) and thus
given an initial distribution v one can contruct a
time homogeneous Markov process with finite dimension
al distributions given by
=J ... .J P(tn -tn_l'Yn_I,An) .. ·P(tl'Yo,dYI)v(dyO)· AO An
However given a Feller semigroup on a locally compact
separable metric space we obtain a stronger result.
Theorem 4.2.5. Let E be locally compact and separ
able, and let {T(t)} be a Feller semigroup on CO(E).
Then for each v e Ml (E) there exists a Markov process
X corresponding to {T t} with initial distribution v
and sample paths in O([O,co);E). Morover, X is strong
Markov with respect to the filtration ~X ;= n ~X t+ e>O t+e
where ~~;= O'{X(s):s~t}. Proof. lOY] or [EK Thm. 2.7, p. 169].
Remarks. 4.2.6 One can also construct an associated
Markov process on the one-point compactification, E ;
= E u {co} and require that the point co, be absorbing
by setting Tl(co) = f(co) 't/ fe c(E). 4.2.7. If {Xt } is a Feller process with generator
(D(A),A) and q,eO(A), then
is a P -martingale for every xeE. x
4.3 Measure-valued Processes and Laplace Transition
Functionals
Let E be a compact metric space and let MF(E)
denote the space of finite Borel measures on E fur
nished with the weak topology. This space is a local
ly compact separable metric space and can be compac
tified as in 3.l(a). Let p be a complete metric on
MF(E). We can characterize a transition function on
MF(E) in terms of a Laplace transition functional
52 D. A. Dawson
Lemma 4.3.1 In order that the transition function on
MF(E) with Laplace transition functional L corres
ponds to a Feller semigroup on CO(MF(E» it is neces
sary and sufficient that
(i) for fixed t > 0, the mapping fl ~ P(t,fl,.) from
MF(E) to Ml (MF(E» be continuous, L e.,
L(t,fl,t/» is continuous for each t/> E V (where
as in Theorem 3.3.2),
V is
(ii) lim P(t,fl,.) = 0/1' Le., lim L(t,fl,l) = 0 V t~O,
fl -7/1 fl-7/1 (iii) the mapping t ~ P(t,fl,.) is uniformly stochas-
tically continuous at zero, Le. for each £ > 0,
I im sup [1 - P(t,fl,N (fl»] = ° tj,O fl £
where N (fl) := {v:P(fl,v)<£}. £
Corollary 4.3.2 Under the other hypotheses of Lemma
4.3.1 condition (iii) follows from the weaker condi
tion that P( t,.,. ) is stochastically continuous, L e.
lim P(t,fl,.) = 0 or equivalently tj, ° fl
-<fl t/» lim L(t,fl,t/» = e ' for each fl E MF(E) and t/> E V. tj,O
Proof. [OY Theorem 2.10].
tions of Lemma 4.3.1 and an initial measure fl we
can construct the canonical version of the measure-
valued process. This is given by
(D,V,(Vt+\~O,(Xt\~O'(P fJ.)fJ.eMF(E» where D =
D([O,co),MF(E», Xt:D ~ MF(E), Xt(w) = w(t), V t = CT{Xs:O~s~t}, V = vVt , P fJ. is a probability measure on
V, the Borel CT-algebra on D, and the finite dimension
al distributions of P are as above. fJ.
4.4 a Class of Inhomogeneous Markov Processes Qll
Polish Spaces
In Section 5 we will also consider not only time
homogeneous Markov processes associated with Feller
semigroups but also more general time-inhomogeneous
Markov processes whose state spaces E are Polish
spaces. An important class of time inhomogeneous
Markov processes which is less restrictive than the
class of Feller processes is given by the following
definition.
Definition. Let (E,g) be a Polish space with its A t
Borel CT-field. Let E e ~([O,co»xg and set E :=
{x:(t,x)e E}. Z = (Q,~,~[s,t+),Zt'P ) is an inho- s,z
mogeneous Borel strong Markov process with cadlag
paths in Et c E iff:
(0 (Q,~) is a measurable space and {~[s,t):s~t} is a
non-decreasing collection of sub-CT-fields of ~ indexed
by compact intervals. ~[s,t+) = co n ~[s, t+lIn) ,
n=l ~[s,co) = v ~[s,n].
n A
(i0 'rJ (s,z) e E, P is a probability measure on s,z (Q,~[s,co» such that for all Ae ~[u,co), (s,z) ~
54 D.A. Dawson
P (A) is Borel measurable on Er!([O,u]xE). s,z
(iii) 'tI t ~ 0, Zt:m,~[t,t]) -7 (E,g) is measurable
and satisfies P (Z(s)=z, Zt e Et 'tI t~s and Z is s,z •
cadlag on [S,CXI» = 1 'tI (s,z) e E. A
(iv) If (s,z) e E, I/J e b~([S,CXI)xD([S,CXI),E» and T ~ s
is a stopping time with respect to {~[s,t+]:t~s}, then
E (I/J(T,Z(T+.» I~[S,T+])(W) s,z
= ET(w),Z(THw)(I/J( T(W),Z( T(W)+.»,
We associate with the process {Z(.)} the inhomo-
geneous semigroup (or evolution operators) Stand s, transition function P(s,t,x,dy) defined by
S tf(x) := E[f(Z(t» I Z(S)=X] s,
= Jf(Y)P(S, t,x,dy),
(s,x,t) e {Ex[O,CXI):t~s}.
b~(Et) into b~(Es) and S tf(x) is jointly s,
measurable in (s,t,x).
Let {Y t} be a symmetric «-stable process in IRd.
Let {W t} be a process with state space D( [0,(0) ,lRd)
defined by
Wt := {Yt :selR} AS +
t d e D := {weD([O,CXI),1R }: w =wt 'tis}. s AS
Then {Wt } is a time inhomogeneous Borel strong Markov
process with cadlag paths in Dt (cf. [DP, Prop.
2.1.2]). We call the process {Wt } the (<<,d)-path
process.
The law of a measure-valued process is normally
given by a probability measure on the Skorohod space
D([O,oo),M(E». Consequently it is necessary to develop
tools to study the weak convergence of such probabili
ty measures. In this section we review some of the
basic criteria for proving relative compactness for
sequences of such processes.
space. Let A be the set of continuous, strictly
increasing functions from [0,00) onto itself. For i\.e A
define
( Ilog i\. ( s ) -i\. ( t ) ) ~(i\.) := 1 A su
t P I t-i\.(t) I v sup I s*t s-t
For w,w' e D([O,oo),E)
p(w,w')
i\.eA
It is easy to verify that p defines a metric on
D([O,oo),E). The resulting topology is called the
Skorohod topology on D([O,oo),E).
Theorem 4.5.1
and only if the following two conditions are satis-
fied:
56 D. A. Dawson
(a) for every £>0 and rational t~O, there exists a
compact set Kc,t such that
sup P {X(t)e (K t)c} < £, n' £
n ' (b) for every £>0 and T>O there exists 5>0 such that
sup P {w'(X,5,T)~c} :5 £ n
n where w'(x,5,T)
1
Proof. See e.g. [EK p. 128].
1 1+
D([O,oo);E»
Let (E,d) be a Polish space. Let {P } be a sequence of n
probability measures on D([O,oo),E) satisfying condi-
tion (a) of Theorem 4.5.1. Let V t = cr{X(s):s:5t} be
the canonical filtration. Assume that for some (3)0 and
each T,5>0 3 a non-negative measurable function
rT(5)~0 such that n
En ([IAd(X(t+5),X(t» ](31 V t):5 En[ r! (5)1 V t]' O:5t:5T,
and lim limsup E [rT(5)] = o. Then {P } is relatively n n n 5-+0 n-+oo
compact.
in D([O,oo),E»
Let (E,d) be a Polish space. Let f be a family of
continuous functions on E that separates points in E
and is closed under addition, i.e. f,g e f ~ f+g e f.
Given fef, f:D([O,oo),E)
measures on 13(D([O,oo);E))
family {P} of probability n
is tight iff the following
(0 for each T>O and e>O there exists a compact
KT cE such that ,e
P (D([O,T1,KT )) > l-e, n ,e
Oi) the family {P } is f-weakly tight, i.e. for each n
fef the family {P o(f)-I} of probability measures in n
D([O,oo);IR) is tight.
Finally, if the family {P} is tight, then it is n
relatively compact in the weak topology.
Proof. Jakubowski (1986).
of the relative compactness of
that for the real-valued case.
{P} on D([O,oo),E) to n
A very useful criteria
given by the following well-known criteria of Aldous.
Theorem 4.5.4 (Aldous Conditions for tightness in
D([O,oo),IR))
Let {P} be a sequence of probability measures on n
D([O,oo),IR) such that
0) for each fixed t, Pnox~1 is tight in IR,
(ii) given stopping times Tn bounded by T and 0nlO as
n~, then
58 D. A. Dawson
lim P (IX ~ -X I > £) = 0, or n -r +u -r n-+oo n n n
(ii') V 1»0 3 o,nO such that
sup sup P (I X(-r +a)-X(-r ) I > c) ~ 1).
n~nO ae[O,o] n n n
Then {P} are tight. n
Proof. Aldous (1978).
Let M1 [O(IR +,Mr(E))], M1 [D(IR +,Mp (iRd)] and
M1 [D(IR +,IR)] denote the spaces of probability measures
on the respective Skorokhod spaces with the topology
of weak convergence. Versions of the next theorem have
been used by several authors (e.g. Roelly-Coppoletta
(1986), Vaillancourt (1990) and Gorostiza and Lopez
Mimbela (1990)).
Theorem 4.6.1 (a) Assume that E is compact and IF
is a dense subset of C(E). A sequence {P} c n
M1[D([0,oo),Mr (E»] is tight if and only if {P n or;1} is
tight in M1[D([0,oo),IR)] for each 4>e IF where f 4>(Jl.) = <Jl.,4» •
(b) A sequence {P n} in M1[D([0,oo),Mp(iRd»] is tight if
~-1 and only if {P n of 4>} is tight in M1[D([0,oo),IR)] for
·d each 4>e Kp (IR ) where f 4> (Jl.) : = <Jl., 4». Proof. The proof of (a) is similar but easier than
that of (b) so that we will only prove the latter. By
hypothesis {Pnor;1} is tight in D([O,T],IR) for each p
T > O. Hence for each e>O there exists a compact K e
c D([O,T),IR) such that P or:l(K ) ~ I-e. Then by the n 'f' e
p characterization of compact sets in D([O,T),IR) there
exists k such that K e .d • e
r := {P.:P.E M (IR ): I <p.,4> > I T,e p p
c D([O,T),[ -k ,k )). e e
Let
P (D([O,T1;rT ) n ,e =
P or:1(D([O,T1;[-k ,k]) ~ l-e n 'f' e e p
and hence condition
(i) of Theorem 4.5.3 is satisfied. Now let IF denote
the class of functions on Mp (~d) of the form f 4> (p.)
:= <p.,4» with 4>E K (~d) and note that IF separates p
points and is closed under addition. The result then
follows by applying Theorem 4.5.3. 0
60 D. A. Dawson
Semigroups
state (regular) branching process if
(5.1.1) P (e -<X(t),</») s, III +112
= P (e -<X(t),</»)P (e -<X(t),</») S,111 S,112
this is differentiable in s at s=t, and (s,x) ---7
P s,o x
the multiplicative property and implies that for each
t, X(t) is an infinitely divisible random measure.
Additionally, we assume that the Laplace transition
functional has the form
-<X(t) A..> -I(V s t</>)(X)I1(dx) P ( e ' 'I' ) = e' A..e pb5' .
S,11 ' 'I'
The family of operators {V t} are in general nons, linear but satisfy two basic structural properties:
0) V (V ) = V (semigroup on pb5') r,s s,t r,t
(ii) the mapping </> ---7 V t</> is negative definite on s,
pb5' and has a canonical representation of the form
(3.4.2).
The family {V t} is called the cumulant semigroup, s, log-Laplace semigroup or iI!-semigroup. The theory of
iI!-semigroups was developed in Watanabe (1968) and Sil
verstein (1969). There is a one-to-one correspondence
between continuous state branching processes and log
Laplace-semigroups.
Let The following end result
of the work of Lamperti, Silverstein and Watanabe
provides a complete characterization of time homoge
neous continuous state branching processes in the fi
nite case and also points the way to the generaliza
tion to the case in which E is a Polish space.
Theorem 5.2.1 (a) A time homogeneous continuous state
branching process on where
M({el' ... ,ek}) (cf. Section 3.1), and for which the
point {Ll} is a trap, is given by a Feller semigroup on
Further, there exists a core con-
sisting of linear combinations of functions of the
form
2 F(, .. t) = f( <fl,c/») , c/>e pC(E), fe CO(IR) and
(5.2.1) GF(fl) = f fl(dx)c(x)ox(dy)F"(fl;x,y)
- ~ F'(II'x)] l+u ,..,
62 D. A.Dawson
semigroup {St} on C(E) (Le. a kxk matrix with non
negative off-diagonal elements and row sum zero),
d F'(II;x):= -d F(II+£~) I 0 = ,.. £,.. X £=
F(fl+£~ )-F(fl) I im x £!O
82 F"(II·x y) := F(II+£ ~ +£ ~ ) I ,.., , ,.. 1 x 2 y £ =£ =0'
8£18£2 1 2
II 2 and sup [ y n(x,dy)+n(x,[1,oo»)] < 00, 0 ~ c(x) ~ cO.
x 0 (b) Conversely, given c(.), b(.), n(.,.), A as
above, there exists a continuous state branching pro
cess with generator G given by (5.2.1).
(c) The transition Laplace functional of the process
is given by
-<X(t) ",> -I(V 0 t</J)(X)fl(dx) P fl(e ,.." ) = e' , </Je pbg',
where v(t,x) := (VO,t</J)(x) = (Vt</J)(x) is a mild so-
lution of
(log-Laplace Equation)
t v(t,x) = St</J(x) + I St_s~(.,v(s,.»(x)ds,
o where
(5.2.3) ~(x,A)
2 JOO( -AU AU) = -c(x)A + b(x)A + 0 l-e - l+u n(x,du)
and for each x, n(x,du) is a measure on (0,00) satis-
II 2 fying u n(x,du) < 00.
o Proof. This was proved for the case k = 1 by Lamperti
(1967) and for the case k = 2 by Watanabe (1969).
Watanabe's proof can be extended to the general case.c
In the remainder of this chapter we will restrict
our attention to the critical case:
(5.2.4) sup Ioo u n(x,du) + II u2n(x,du) < 00,
x 1 0 where sup c(x) < 00, and thus we can write
x
00
The restriction b == 0 is made for notational simpli-
city and is not essential; however if the condition on
n is removed new phenomena such as explosions can
occur which we do not consider.
As a consequence of the hypothesis St1 = I, it
follows that {St} is the semigroup of a Markov chain
with state space {el' ... ,ek} (which we call the motion
process) and we can verify that
E,} <X(t),</») = <1J.,St</».
Note that for each x, t(x,.) is the log-Laplace func
tion of an infinitely divisible random variable. If in 1
addition we assume that I un(x,du) < 00, then o
-t(x A) ( -A(ZI+Z2-E(Z2») e ' = E e
64 D. A. Dawson
where ZI is a normal random variable and Z2 is an
independent non-negative infinitely divisible random
variable (cf. Corollary 3.4.2). The continuous state
branching can then be viewed in terms of the corre
sponding infinitely divisible process run with a clock
speed proportional to the current mass.
The nonlinear semigroup {V t} plays a key role s, in the construction of superprocesses in the general
case. Moreover the log-Laplace equation characterizes
the process. This is a consequence of the following
result.
Lemma 5.2.2. (0 Given AO 3 K(AO) such that
I ~(x,Al)-~(x,A2) I :s K(AO) I A1- A2 1 V A1,A2 e (O,AO).
(ii) Equation (5.2.2) has a unique solution and hence
characterizes the process.
Proof. (0 If ~1 < A2,
I ~(x,Al)-~(x,A2)1 :s 2M(AO) sup I c(x) II A1- A2 1
x
(ii) This is a variation of the standard Picard uni-
queness argument. D
5.3. The (W,t)-Superprocess.
existence result of Theorem 5.2.Hb) to the infinite
dimensional case. This was first carried out by Wata
nabe (1968) in a Feller process setting and recently
generalized to a right process setting by Fitzsimmons
(1988) and to the time and spatially inhomogeneous
setting by Dynkin (1991).
processes with space-time homogeneous branching mec
hanism but time inhomogeneous motion (this will cover
both the (cx,d,{3) superprocess and historical process).
The construction is carried out by means of a bran
ching particle approximation.
Let E be a Polish space, Et c E for te IR and 0 = {weD([O,CXI),E), wteEt Vt~O} is a measurable subset of
D([O,CXI),E), the Skorohod space of cadlag functions on
IR. The process will be based on a branching mechanism
(playing the role of t in Theorem 5.2.0 and a motion
process in E (playing the role of the linear operator
A in Theorem 5.2.1).
Time Inhomogeneous Motion Process:
W = (D,V,V[s,t+],Wt,{P } (0) ES is an inho-s,x se ,CXI ,xe mogeneous Borel strong Markov process (cf. Section
4.4) with semigroup
66 D. A. Dawson
s S tf(x) = P f(Wt ), O~s~t, xeE , fe bg.
s, s,x
The function {t(A):A?!O} is assumed to be of the form
IX) (5.3.1) teA) = -c o A2 + J O-e-Au-Au)n(du)
o where n e M«O,IX)) is assumed to satisfy
{( u n(dul + (u2n(dul} < 00,
and
(Here
t' (A) = -2c 0 A - JO-e -AU)U 0 n(du) ~ 0,
t'(O) = 0,
(_1)k+1t (k)(O) ?! 0, k?!3.
denotes the first derivative and (k) denotes
kth order derivative.)
mogeneous Markov process, called the
(W,t)-superprocess, with Laplace transition functional
(5.3.3) ( -<X(t) <P» ( J ) PS,f.L e ' = exp - (Vs,t<PHx)f.L(dx) ,
for ~ E pbgt, where {V t:ossst<co} is a nonlinear evos, lution family on pbgt, and V t<l> is the unique solu-s, tion of the nonlinear integral equation
(5.3.4) V t<l>(x) s, t
= S t~(X)+ I (S t(V t<l>(' )))(x)dr. s, s,r r, s
Proof. We will begin with three lemmas.
Lemma 5.3.2 Assume that h:[O,co)xE ~ IR is such that
sup I h(t,x) IsM < co, and t,x
t h(r,x) s c1 + c2I (Sr,sh(s))(x)ds for re [O,t).
r
h(r,x) s c1'e for all re [O,t).
Proof. For each n ~ 1,
n
h(r,x) s c1 L c~ I .. .J l(r<sl< .. <sk <t)ds1"dsk k=O
+ c~+II .. .Jl(r<sl< .. ·<sn+l<t)dSr·dSn+l
• (S h(s l))(x) r,s 1 n+ n+
n
The remainder n+l n+l s M.c2 (t-r) I(n+l)! ~ 0 and
the proof is complete. []
68 D.A.Dawson
offspring generating function 00
§'(z) := L p(n)zn,
n=O where p(n)' Kdt is the rate that n offspring are
produced by a branch of a particle at time t. 00
We
assume that L n' pen) < 00. Then the branching particle
n=O system exists (i. e. no explosion) and transition
Laplace functional
._ Z (-<Zt,4») w t(x) .- P so e , r, r,u x
satisfies
r Proof. Condition on existence and time of the first
branch in the interval [s, t] (cf. Dawson and Ivanoff
(1978)).0
r,s s, r
(b) w t(x) r,
t = (S te -cf>)(x) + KJ (S §'(w t(' »)(x}ds r, r,s s,
r
r are equivalent.
Proof. Assume (b) and now calculate
t J -K' (s-r) K [e (S §'(w (. »)(x)Jds r,s s,r
r
= - KJr(e JsK[(Sr,u§'(Wu,t(')})(X}JdU)dS
t + J K[(S §'(W t(' »}(x)]ds (integration r,s s,
r by parts)
Jt -K' (s-r ){ -cf> = -K e (S W t(' )}(x) - (S te )(x) r,s s, r,
r
+ KJ\(S W t(' })(X)]dU}dS r,u u, s
t + W (x) - S te -cf>(x) + KJ [(S W t)(x)Jdu r,t r, r,u u,
r
69
-K·(t-r) = W (x) - e (S t e -cf>)(x)
r,t r,
follows by reversing the chain of equalities. D
70 D.A. Dawson
Proof of Theorem 5.3.1. Given ~ satisfying (5.3.1)
and (5.3.2) it is easy to check that for any 0<e<1,
ye(v) := a(e)-I[a(e)v - e~«(1-v)/e)] -1
with 0 < aCe) := - ~'(e ) ~ c/e + c2
is the probability generating function of a non
negative integer-valued random variable with mean one
(cf. [DP, Lemma 3.4]). In fact using (5.3.2) it is
easy to verify that ye(O) = - e~(1/e) ?! 0, y e(1) = I, e e
DY (v)l v=O = 0, DY (v) I v=1 k e
= I, and D Y (v)l v=O ?! 0
for all k ?! 2 (where D denotes differentiation with
respect to v).
branching particle system {Z~} to the (W,~)-super
process with initial measure IJ. at time r. We begin
with an initial Poisson random measure Ze with inten- r
sity IJ./e.
with offspring generating function
measure Ze and particle mass e. Combining Lemma r
5.3.3 with the Poisson cluster formula (3.3.2) we ob-
tain the Laplace functional of the random measure
as follows:
e -<Zt,et/» (
P r,1'aLo.(lJ.Ie) e = exp -
where we t is as in Lemma 5.3.3 r, ye and
e 1-w r,t
t C + I KC[e -K . (s-r)(S §'C(wC t(' )))(x)]ds.
r,s s, r
Then by Lemma 5.3.4
C w t(x) r,
t = (S te -c)(x) + I KC[(S §'c(wc t(' )))(x)]ds
r, r,s s, r
r,s s, r
ression for §'C we get
C (5.3.5) v t(x) r,
( -c ) t = S t 1 - e (X)+I [(S ~(vc t(' ))]ds
r, r,s s, C r
( -C<I»
Note that O:s vC t(x):s S 1 - e (x) since r, r,t
C
:S 0.
C!O Lemma 5.3.5 VC t(x) ~ v t(x) exists (uniformly in r, r, x) and v t(x) is the unique solution to Equation r, (5.3.4).
Proof. The uniqueness follows by a standard argument
using the fact that A ---7 ~(A) is locally Lip-
schitz (cf. Lemma 5.2.2) and Lemma 5.3.2.
Since ~(A):S ° and ~(O) = 0, the Lip
schitz property yields
o ~ ~(V) ~ - M(AO)V for OSVSAO.
But then from (5.3.5) it follows that 0 S vC t L ) S r,
AO' provided that OScf>SAO. Then using the Lipschitz
property of ~ we obtain
S II _1-_e __ _ c 1
This yields the existence of the limit v t. The fact r, that v
r,t satisfies the equation (5.3.4) then fol-
lows by a bounded convergence argument. 0
We now complete the proof of Theorem 5.3.1. C First note that the first moment measures, E(cZt (dx))
= Stll e MF(E) and hence are tight. Therefore by Lem- c
rna 3.2.7 the random measures {cZt:c > O} are also
tight. Together with Lemma 5.3.5 and Theorem 3.3.2
this implies that cZ~ converges weakly to a random
measure with Laplace functional (5.3.3) for each t.
c C-70 J Note that cZ ~ Il and that Il ---7 (V tcf>)(X)Il(dx) r s, is a ~(MF(E) )-measurable function. The fact that the
family {P (X(t) e . ):Ossst<oo; lleMF(E)} satisfies S,1l
the Chapman-Kolmogorov equation follows from the semi-
group property of {V t}. This completes the proof s, that (5.3.3) and (5.3.4) define a Laplace transition
functional. . 0
Remark: Using the results of Fitzsimmons (988) it can
be shown that there exists a version of the process
constructed above which is a time inhomogeneous Borel
strong Markov process with cadlag paths in MF(Et ) c
MF(E). For the details refer to [DP, Section 2]. In
Section 5.5 we will verify that the (O'.,d,{3)-super
process defined in the next section are Feller pro-
cesses.
5.4 Examples.
5.4.1. (O'.,d,{3)-superprocesses. In this case the mo- . y . . bl . IRd bon process IS a symmetrIc O'.-sta e process In
with semigroup {s~},generator flO'. = _(_fl)O'./2, 0<O'.:s2,
and with paths in D = D([O,oo);lRd) (cf. section 4.2).
t(A) = -'1A 1 +{3, '1>0, 0 < {3 :S 1 for tPE bpg
(spatially homogeneous)
IC = I, (3(1+{3)'1 1
n(du) = r( 1-(3) {3+2du , c(x);;; 0, if (3 < I, u
n(.) ;;; 0, c(x);;; I, if (3 = 1.
The approximating branching particle systems have off
spring generating functions
§'E(v) = v + U+(3)-IU- v)I+{3
which in this special case are independent of E. In Ell 2 the case (3 = 1 , this reduces to §' (v) = - + -v
2 2
(critical binary branching). Also ICE = '1U+(3)E -{3.
The resulting MF(lRd>-valued process is called the
(O'.,d,{3)-superprocess and is characterized by its log
Laplace functional v t = V t tP which is the unique so-
74 D. A. Dawson
5.4.2. (a,d,(3) Historical Process
tion process arises in the construction of the his
torical process. Let {W t} denote the (a, d)-path
process defined in Example 4.4.1 Since {Wt } is a
time inhomogeneous Borel strong Markov process and we
can use Theorem 5.3.1 to construct the (W,~)-super
process {H :t~O} which we call the (a,d,(3)-historical t
process (see [DP] for more details).
If A e ~(lRd), define X(t,A) = Ht({y:y(t)eA}).
Then X(t) is a version of the (a,d,(3)-superprocess,
that is, H(t) is an enriched version of X(t) which
contains infomation on the genealogy or family struc
ture of the population.
In this section we will show that the
(a,d,(3)-superprocesses can be extended to the space ·d M (IR ) and realized as cadlag strong Markov process
p es. If p > d/2, then d-dimensional Lebesgue measure
belongs to M (lRd) so that this allows us to include p
the case of spatially homogeneous initial conditions.
If a < I, we make the additional assumption p <
(d+a}/2. Let M (lRd) be defined as in Section 3.1 and p
let
C (lRd ) := {feC(lRdv{oo }): lim f(x}/4> (x) = f( {oo })} p p I x I ~oo p P
with norm IIfll .- sup I f(x) 1/4> (x) (recall that {oo } p x p P
d is an isolated point). M (IR) can be identified with
·d P·d {J.!.:J.!.e M (IR ), J.!.({oo }) = O}. M (IR ) will be furnished
p ·d P P with the C (IR ) weak topology.
p Lemma 5.5.1. If a < 1 and d/2 e V(f1 ) and I f1 4> (x) I ;s const 4> (x). p a a p p
Proof. (D Recall that 4> (x) = (1+lxI 2 )-P, and p
p (t,x) ;s constllxl 1+a if t > 0, Ixl ~ 1, a
Then we can easily verify that
J 4> (x+y)p (t,y)dy ;s const 4> (x) 't/x. d pap
IR 2 d (ii) Recall that 4> e V(f1 ) if 4> e C (IR ) and f1 4> e a a C (lRd) where
p f1 4>(x)
= J [4>(x+Y)-4>(X)-V4>(X)·(Y/(I+ lyI 2))] IYI-(d+a)dY.
IRd
Since 4> e C2 (lRd), it suffices to show that p
J [4> (x+y)-4> (x)-V4> (X)·(Y/(1+ly!2))] IYI-(d+a)dy d p p P
IR
;s constl(I+! x 12)P.
76 D. A. Dawson
For 1 xl> 2, consider the integral over each of the
three regions { 1 y 1 ~ 1 x 1 12}, {1:s 1 y 1 :s 1 x 1 12}, { I y I :sl}.
N t th t "'" . . t bl IRd d 1 I-(d+o:) . o e a ." p IS In egra e over , an y IS
integrable over I y 1 ~l. We then obtain the required
bound for the integral over the first and second re- -(d+o:) .
gions by noting that I y I :s const t/> (x) In the p
first, and t/> (x+y) :s const t/> (x) in the second. The p p
result in the third region is then obtained by noting
that It/> (x+Y)-t/> (x)-Vt/> (x)· (y/(1+ I y 12» I :s const 2 P p P
t/> (x)lyl for Iyl:sl. 0 p
In order to construct the process we proceed as d
above but in addition we find fJ.£ e MF(IR ) i fJ. e
M ORd). We consider the approximating branching par p
ticle systems as above in which the initial measure is
In this section we will show that the
laws of the approximating branching particle systems
are relatively compact in M1(D([0,OO);MpORd»). We
first need to recall some elementary inequalities.
Lemma 5.5.2.
2 -x (a) 1 - x +x 12 - e ~ 0, x~O,
-1 2-x (b) x (1 - x + x 12 - e ) ~ const, x~2,
(c) 0 ::s e -x -1 + x ::s const xl+f3, x~O, if 0<{3::S1,
1+8 JOO 8 (d) E ~ :s 1 + (1+8) r P(~~r)dr, (~ ~ 0), 8 > -1, 1
2/r (e) P(~~r)::s const r J [Uu)-I-uL'(O)]du
o -~u where Uu) = E e , I L'(O)! < 00, ~~O.
Proof. co
(d) E ~1+9 = J P(~i!: xllU+9»dx
o JI 9 JCO 9 = U+9) r P(~i!:r)dr + U+9) r P(~i!:r)dr
o I
2/r I (e) r J [Uu)-I-uL'(O)]du
o
= r(r[ J:(e -ux -l+UX)dP(~"")]dU JCO r [ -2x/r 2] = 0 x l-e - 2x/r + 1I2(2x/r) dP(~~x)
(interchange of order of integration)
JCO r [ -2x/r 2] i!: r x l-e -2x/r+1I2(2x/r) dP(~~x) (by (a))
i!: const· P(~i!:r) (by (b)). c
Theorem 5.5.3. Let £ £ X (t,du) = £ Z (t,du) where
Z£(t,du) denotes the approximating branching particle
system with Z£(O) given by a Poisson random measure
with intensity measure f..I., symmetric a-stable mo- £
tions, K£ = '1U+(3)£ -(3ds, and offspring probability
generating function
~£(v) = v + U+(3)-IU-v)I+(3.
Let p e (d/2,(d+a)/2) if 0<a<2 or (d/2,co) if a = 2. £ (a) Then the processes {X (t,.)} under
pZ t:D • ( /) with f..I. ~ f..I. e M (lRd) converge weakly r ,r.Q{4 f..I. £ £ P
£ ·d ·d in D([O,co),M (IR » to a cadlag M (IR )-valued process,
p p
PIl ( e-<X(t),</») = exp(-Jvt (X)Il(dX»),
where 1+13 8vt (x)/8t = l1ex.vt - r(vt ) ,
d vO(X) = </>(x) e Cp(1R ).
·d Ile M OR ),
P
(b) If Il({oo}) = 0, then sup X(t,{oo }) = 0, P -a.s. p t p Il
(c) If Il e M (lRd), then X is a.s. cadlag (for the d p
C (IR ) topology). p
tributions when Il e MF(lRd ) follows from Theorem 5.3.1
(using the Markov property). The convergence of the
finite dimensional distributions when Il i Il e e
M (lRd) follows by verifying the tightness of the first p
moment measures - this is a simple consequence of the
fact that S~ maps CpORd ) into itself. It thus re
mains to prove the tightness of the processes Xe in ·d D([O,oo),M (IR ». Before doing this we need to derive a
p maximal inequality.
(0,1),
(5.5.1)
where
e (e ex. 1 +13 e ex. ) UXO):= sup E <X (O),St</> > + <X (O),St</> > t~T p P
and K' is independent of e.
Further if I(X~) < I, then
(5.5.2) E sup {<X€(t),</> >1+9} ~ K". (I(X~»U+9)/u+(:n. t~T p
(b) If X~ is as above but with fJEMF(lRd), then
E sup {<X€(t),1>1+9} < CD.
t~T
E{<X€(t)'</>p>1+9 1 x€(On
CD
[ 1 9J2/r
+ <x€(t),u</>p>} I x€CO)]
By Lemma 5.5.2(e) the inner integrand is equal to
and therefore
~ I Scx(u</» - Scx(€ -1U- e -€u</») I t t
€ cx -1 -€u</> + Iv t (u</»- St (€ U-e )) I
79
t (3 1+(3 a J a a 1 +(3
:S const E u St<l> + "¥ St_s[Ss(u<l»)J ds o
(by (c) and (5.3.5))
a 1 +(3 :S const (St<l»u for t:sT.
Hence we get
E(eXp{-<xE(O)'V~(u<l>p»} - 1 + <XE(O),US~p»
1+(3 (E a 1+(3 E a ) :S const u E <X (O),St<l>p> + <X (O),St<l>p> .
and we obtain
:S 1 + const dr[r + u + duJ· I(X~) 1 0
00 1+B :S 1 + const J r 2+(3dr I(X~) < const I(X~)
1 r
It follows from the Markov property of the critical
branching particle system that t
M (t):= <XE(t), > - J <XE(s),Ll >ds E p 0 ap
is a pZ c:n • ( / )-martingale. But then r ,.r Q.U). Il E
E
E sup { <XE(t), >1+B} t:sT p
:S const ( E sup 1 M (t) 11+B <T E
t- T 1+B
+ E[ J o <XE(t),ILla<l>pl>dt] )
:S const ElM (T) 11+B + const sup E {<xE(t), >1+B} • E t:sT p
(by Doob's maximal inequality, Jensen's
inequality and
~ const sup E { <xE (t),4>p>l+e} t~T
111 4> 1 ~ const 4> ) a p p
E const (1+HXO)) < 00
and this bound does not depend on E. Since XE(O) = 1'oLo.(fJ IE) with fJ ~ fJ in M (~d), then HXOE) can be
E E P bounded by a constant independent of
(5.5.2) follows by replacing 4> by p
in (5.5.1).
(b) follows by a simple modification of the above. 0
Continuation of Proof of Theorem 5.5.3. (a) Let E ~ n
o as n ~ 00. By Theorem 4.6.1, in order to prove E
tightness of the processes {X n} it suffices to show
that for ~ e K ORd ), the family {Z (t) := p n
E
Aldous condition (Theorem 4.5.4) it then suffices to
show that
Z (-r +0 )-Z (-r ) ~ 0 in distribution as n ~ 00. n n n n n
Here 0 are positive constants converging to zero as n
n ~ and each -r is a stopping time of the process n
Z with respect to the canonical filtration, satis n
fying -r ~ T. n
By the strong Markov property applied to the pro E
cess X n

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