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Superprocesses

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Title:
Superprocesses
Physical Description:
v, 116 leaves : ; 29 cm.
Language:
English
Creator:
Šikić, Hrvoje, 1959-
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theses   ( marcgt )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 112-115).
Statement of Responsibility:
by Hrvoje Šikić.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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Full Text

SUPERPROCESSES

By

HRVOJE SIKIC

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1993

ACKNOWLEDGEMENTS

Grateful acknowledgements are due to persons whose help and support made

this work possible. First, and foremost, my advisor Prof. Murali Rao is thanked

for his guidance throughout my study. His original insight into mathematics and

his humanity are a constant inspiration for me. Prof. Zoran R. Pop-Stojanovi6

already directed some of my work in Zagreb. His vast knowledge made mathematical

discussions particularly enjoyable. Prof. Joseph Glover helped me very efficiently on

numerous occasions, ranging from research questions to organizational problems. I

also wish to thank Prof. James K. Brooks and Prof. Andrew J. Rosalsky for their

service on my committee.

I wish to thank faculty members of the Department of Mathematics of the Uni-

versity of Zagreb for giving me a strong mathematical foundation. In particular, I

am grateful to Prof. Nikola Sarapa, who introduced me to probability and led me

to magister degree, and to my longtime friend and colleague Dr. Zoran Vondracek,

with whom I worked from our first mathematical steps until today.

And last, but not least, I wish to express my gratitude to the members of my

family. My parents, whose working attitude and understanding of criticism as the

basic approach to science, are lasting examples for me. My wife Sonja, who, be-

ing a mathematician herself, was a supportive and understanding companion in the

sometimes very demanding life of a mathematician.

ACKNOWLEDGEMENTS

ABSTRACT ........................

CHAPTERS
1 INTRODUCTION ...............................

2 GENERAL RESULTS ..... .......................

2.1 Nonlinear Perturbations of Positive Semigroups
2.2 Supersemigroups and Superprocesses .
2.3 Kac Semigroup and Linear Case . .
2.4 Feller Condition and Infinitesimal Generator .
2.5 Branching Property. . ..
3 ONE-DIMENSIONAL CASE . .

3.1 Supersemigroups and Infinitesimal Generators .
3.2 Time Change and Zero-One Law .
3.3 Measures on Rays .................
3.4 (Idt, IF )-Supersemigroups . .

4 TRANSITION PROBABILITIES OF SUPERPROCESSES ........

4.1 Trotter-Kato Formula ...........................
4.2 Superprocesses over Markov Chains ... .................
4.3 Superprocesses over Deterministic Markov Processes ..........

5 CONCLUSION ................... ..............

REFERENCES ...................................

BIOGRAPHICAL SKETCH ............................

. . iv

88

88
95
100

107

112

116

11111

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SUPERPROCESSES

By

HRVOJE SIKIC

May 1993

Chairman: Dr. Murali Rao
Major Department: Mathematics

Dawson-Watanabe superprocesses (Xt), over a Markov process (t), are usually

defined by transition supersemigroups Qt(p, dv). The Laplace transform of Qt is

determined by a nonlinear semigroup (V,), which is obtained as the perturbation of

Markov semigroup (Tt) (given by () by a branching mechanism T.

In the first part of this work we establish a general analytical framework for

nonlinear perturbations of positive semigroups, which preserve positivity. The semi-

groups involved are only weakly measurable with respect to the subspace M of the

space of linear functionals. The well-known results on superprocesses by Fitzsimmons

and Watanabe are obtained as special cases. In this fashion (Xt) can be defined as

M-valued process, in general. Very useful and new, Fubini-type lemma is applied

several times in this part; among other things to estimate (Vt) from below and above

in terms of the Kac semigroup of (Tt).

In the second part we analyze (Xt) through (Vt) and (Qt). We start by detailed

account of one-dimensional superprocesses (studied by Lamperti in the late sixties).

Some new formulas for (Vt) and (Qt) are derived. These results are then used to give

iv

detailed description of "component processes" (XT) (which depends only on ') and

(X4) (which depends only on 1). The new notion of a measure on rays is introduced
to obtain the description of (Xt"). It is shown that (Qt) can be approximated, by
Trotter-Kato type formula, using (Xt) and (X\). Some explicit examples are given.

It is shown that in the case of Markov chain (), the local behaviour of (Xt) can be
obtained, by means of Cameron-Martin formula, from (X*). Finally, for deterministic

((t) the explicit formulas for (Xe) are obtained, which show that "the mass" of (Xt)
is governed by T, while the "spatial movement" of (Xt) is governed by (&). In
section 3.2 a Zero-One law for Brownian motion is obtained, which complements
Engelbert-Schmidt Zero-One law.

CHAPTER 1
INTRODUCTION

Discrete-valued branching processes were studied in the 19th century as math-

ematical models in a discussion of the survival of family names. In this century

applications broadened toward studies of the survival of mutant genes, nuclear chain

reactions, and many others. Considering the problem of population growth, it was

suggested by William Feller that in the case of large populations continuous models

work better than discrete ones (which are successful in the case of relatively small

populations). He presented his programme to the mathematical public in 1951 [26].

In this paper Feller introduced a class of diffusion processes, with values in the set of

positive real numbers, which can be obtained as limits of discrete-valued branching

processes. Feller also computed the transition probabilities of these diffusions and

characterized them in terms of corresponding differential equations.

The consequences of Feller's work [26] are extensive, and we will not attempt to

follow all of them. However, two lines of developments coming from Feller's work [26]

are important to us. We will see that this two lines later merged.

The first of these two lines, and rather central to us (although it appeared later

than the second one), started with the pioneering work [8] of Donald Dawson in

1975. Dawson followed the line of extension of Feller's consideration to the space-

time context. Namely, although Feller's diffusions are continuum valued, they are

models for the number of individuals (or total mass) of a population. The idea was

later to find models for populations distributed in space, which are changing through

time. Such processes will have to be measure-valued, and, as it was already noticed

before 1975, will be described by nonlinear stochastic evolution equation. Dawson's

idea was to consider n individuals of equal mass located at n points yi,..., y, in the

space E C Rd. Each individual has an exponentially distributed lifetime with mean

1/7, and at the end of its lifetime it is replaced by r similar offspring with probability

pr, all initially located at the final position of their parent. Each individual moves
independently through space E, according to the given law of a Markov process

(.t), whose infinitesimal generator we denote by A. If N(t) is the total number of
individuals at the moment t, then the mathematical model for the situation above is

the branching diffusion
N(t,w)
X"(w) = E 6,(w) (1.1)
i=I
where S6 is the point-mass at y E E. The continuous analog of (Xn/) led Dawson to

the stochastic evolution equation

dX(t) = (A + a)X(t)dt + 2-yX(t)dB(t) (1.2)

where B(t) is, so called, Brownian motion with spatially correlated increments. The

main technical difficulty, in giving a mathematical meaning to equation 1.2, is the

appearance of the non-Lipschitz factor 2-X(t). By treating the characteristic func-

tionals of (Xf) through the application of semigroup theory, Dawson succeed in

defining rigorously measure-valued process (Xt),. which can be understood as the so-

lution of 1.2. He also showed that, by properly adjusting the parameters, (Xt) is the

limiting case of branching diffusions (Xy) as n -+ +oo. It was done in the case

when (6() is a standard Brownian motion, and po = 1/2 2a/n, P2 = 1/2 + 2a/n.

The process (Xt) is the example of what was later to be named superprocess. In

doing so Dawson not only offered one of the main ideas for construction of superpro-

cesses, but also gave the intuitive meaning to (Xt) as the mathematical model for

the propagation of a group of individuals (or cloud of particles, for example) through

space and time.

In the succeeding few years Dawson outlined main directions in the study of

multiplicative (i.e. branching) stochastic measure diffusion processes, as he called

them then, or superprocesses, as most of the authors call them today. His programme

was carried through most of the eighties. At the end of the eighties and the beginning

of the nineties several new ideas emerged, but we will say more about it later.

Except for the paper [8], two other papers in the late seventies were crucial for the

development of the theory of superprocesses. In 1977 Dawson, in his paper [9], studied

the limiting behaviour of superprocesses, i.e., the behaviour of Xt as t -) +oo. In

particular, the questions of extinction and of the existence of invariant distribution

were studied. This line of study was continued and developed [11], [19], [32]. I. Iscoe

[34] introduced weighted occupation time processes and investigated their behaviour

when t --- +oo.

Another important early paper was published in 1979 by Dawson and Hochberg

[12]. In this paper local properties were studied, i.e., if we fix t > 0, then we can ask

questions about the nature of the measure Xt. In many cases it is a singular measure,

so the questions about the support of Xt naturally came into consideration. This

type of questions were further studied by Sylvie Roelly-Coppoletta in her work [52]

and other works [58] and [43]. In 1988 Edwin Perkins [48] considered the question

of the nature of the random measure over time intervals (i.e. not just for a fixed

t > 0). Special consideration was given to the case of the superbrownian motion

(superprocess over standard Brownian motion (at)), and to the study of its paths

[13]. All this led to the development of an important new idea, i.e., the idea of a

historical process. It is a superprocess with values in the space of measures on the

space D(E) of E-valued cadlag paths on R+ = [0, +oo). Intuitively, the historical

process takes into consideration the "genealogy" of the starting superprocess (Xt).

For detailed study we refer to Dawson and Perkins [15]. Let us also mention the

paper [25] by Steven Evans and E. Perkins in which they introduce the notion of

a collision measure. They use it to "gauge" the overlap of a fixed measure and a

superprocess.

Throughout all that period one of the main problems was the problem of the

construction of superprocesses. Dawson's starting idea, given in 1975, was later for-

malized in terms of a martingale problem (developed by Stroock and Varadhan). We

refer to [14], [52] for these results, and to [24] for general treatment. In [52] the mar-

tingale properties of superprocesses were emphasized. In the following years more

general processes (6t), more general conditions on branching {p,}, and more general

initial conditions were considered. For construction of superprocesses on p-tempered

distributions see [34], for further formalization in terms of stochastic differential equa-

tions see [43], and, together with further study of martingale properties, [46]. Already

in the eighties superprocesses (Xt) were characterized as measure-valued processes,

such that their Laplace transforms were given by

E [e- f )x] = e-(vf)( ; X0o -= (1.3)

where f > 0 is bounded, continuous function on E, and (Vt) is a nonlinear semigroup

which satisfies the equation

Vtf = Ttf + j T,t[,(V,f)]ds (1.4)

where (Tt) is the Markov semigroup of (6t) and T(A) = --y2. These formulas sug-

gested strong connections with the work on branching processes done by S. Watanabe

in 1968. It was further exploited by P. Fitzsimmons and E. Dynkin. But before we

go to their work, let us exploit the other line of development related to Feller's work

mentioned at the beginning.

In 1951, when Feller offered his programme, the notion of branching processes

meant only the discrete-valued processes. Miloslav Jifina, in his paper 1958 [36],

developed the notion of branching property to processes with continuous state space.

Since additive structure of state space is needed, measure-valued processes rep-

resent an important example. However, it was not known how to characterize

continuous-valued processes as branching. Of course, the simplest case was the case of

R+ = [0, +oo). Real-valued branching processes were characterized as time-changed

Levy processes by John Lamperti in 1967 [44]. It turned out that R+-valued branch-

ing diffusions are exactly the diffusions obtained by Feller in 1951. The next step,

i.e., the characterization of R+-valued branching processes was carried out by Shinzo

Watanabe in 1969 [57]. To simplify notation, he worked on the case n = 2. Watan-

abe characterized completely these processes in terms of their infinitesimal generators,

and, in the case of diffusions, analyzed the behaviour on the boundary of R'. Inde-

pendently, same results for infinitesimal generators were obtained by Yu.M. Ryzhov

and A.V. Skorokhod in 1970 [51]. More general branching processes with immigra-

tion were treated in another work [40]. An account of branching semigroups on the

space of finite measures on compact separable metric space was given in 1969 by

M.L. Silverstein [55]. Most recently, Dynkin, Kuznetsov, and Skorokhod studied the

structure of general branching measure-valued processes [22]. However, the most im-

portant work for our consideration is the paper [56] by S. Watanabe, published in

1968. Watanabe constructed the class of measure-valued branching processes (Xt),

characterized by 1.3 and 1.4, where E is a separable compact metric space, and T is,

so called branching mechanism, given by

S(A) = ao + a A a2A2 e-Au -1 + n(du) (1.5)
Jo0 1 + u )

where A E (0, +oo), ao, al, a2 E R, ao, a2 > 0, and n is a measure on (0, +oo) such

that

0(0 U du)< +oo (1.6)

His assumption was that (,) is a Feller process, and then he proved that (Xt) is also

Feller and characterized its infinitesimal generator.

Now we can go back to the story of superprocesses. In 1988 Patrick Fitzsimmons

offered a new way of constructing superprocesses, following Watanabe's approach. He

considered a very general class of Markov processes (st), and very general form of xF

with nonconstant coefficients. He showed [28], [29] that the corresponding, i.e., ((, I)-

superprocess can be constructed and that it satisfies nice regularity properties. The

superprocess is now defined so that it satisfies 1.3 and 1.4. Similar approach was taken

by Eugene B. Dynkin [18]. He considered branching mechanism T(A) = -A2, and

time nonhomogeneous case. Also, his methods are different from Fitzsimmons'. In

this way Watanabe's and Dawson's theories were formally unified. Dynkin is also re-

sponsible for the introduction of the term superprocess over Markov process (t) with

branching mechanism I'. We will follow this terminology and mostly Fitzsimmons'

line of approach in this work. We will actually use the term Dawson-Watanabe super-

process, since there are other types of processes (Fleming-Viot, Ornstein-Uhlenbeck)

sometimes called superprocesses, which will not be considered in this work.

Let us finish this brief survey of the theory of superprocesses by mentioning some

other recent developments, which otherwise will not be treated in this work. The

third, very different, construction of the superbrownian motion was recently offered

by Jean-Francois Le Gall [30]. The connections between the theory of superprocesses

and PDE-s has been investigated by Dynkin. The systematic treatment of these

problems is given in [20]. Part III of this paper contains a nice historical survey of

the theory of superprocesses. The LUvy-Hincin representation for superprocesses is

analysed in [39].

Despite all these achievements and thorough analysis of some particular cases,

like superbrownian motion, it appears that the "nature" of superprocesses is still

somewhat hidden from us. For example, even in the case of Markov chain (&t),

where the corresponding superprocess (Xt) is an R"-valued branching process, and

we can characterize the infinitesimal generator of (Xt) completely (see [57]), it is

still not completely clear which part of the behaviour of (Xt) comes from (&t) and

which from 'k. Intuition would suggest that (it) influences "spatial motion," while V

governes "changing of the mass." Can we describe mechanisms which govern these

behaviours? And what about other type of processes, in more general spaces? It

seems that one of the main difficulties is in the fact that equation 1.4 is nonlinear,

and in general we can not expect to have explicit solutions of 1.4. However, if we

concentrate only on "components" of superprocesses we can hope to characterize,

through explicit formulas even, these "components" rather completely, and then to

use them to recover some information from the most general cases. After establishing

superprocesses in a very general situation, we will follow this line of approach, i.e.,

we will try to give as explicit formulas for the "components" as we can, and then to

recover general formulas from these, if possible. We will say what we can in general

cases, but we will concentrate on the simplest examples of ((i). Although we can

not completely answer the questions mentioned above, we hope that this work offers

some improvement and clarification in this direction.

The thesis is organized into five chapters, where the first chapter is this intro-

duction, and the fifth chapter is a conclusion. Hence, the results are contained in

chapters 2,3, and 4.

In chapter 2 we deal with the problem of the foundation of superprocesses. We

take an approach through the semigroup theory of Markov processes. Hence, we have

to solve a nonlinear integral equation first, and then to show that the result is the

exponent of the Laplace transform of the transition probability of the superprocess.

This method was used by both Watanabe and Fitzsimmons, but the first dealt with

the Feller semigroup on a compact state space, while the second dealt with more

general Lusin state space. In section 2.1 we offer a unified approach, so that both

Watanabe's and Fitzsimmons' result are special cases of our general method, as is

shown in sections 2.2 and 2.4. Actually our method solves the problem of positivity

preserving nonlinear perturbation of positive semigroups, and the class of semigroups

included is very large. Particularly useful is Lemma 2.1.3, which we will apply to

deal with linear case and Kac semigroup in section 2.3. In section 2.5 we mention

the branching property of superprocesses, and how it simplifies our dealing with the

transition probabilities of superprocesses.

In chapters 3 and 4 we analyze the transition probability for particular superpro-

cesses. We start by describing the simplest, one-dimensional superprocesses, which

were studied first by Lamperti. We compute some new explicit formulas in these

cases. In section 3.2 we also prove a zero-one law for Brownian motion, which com-

plements the Engelbert-Schmidt zero-one law. In section 3.3 we introduce a notion

of measures on rays; these are example of measures on the space of measures. Using

this notation we can precisely describe transition probabilities for superprocesses over

(, where t = Jo, denoted by (X'X). This will be done in section 3.4.

In chapter 4 we use results from the previous chapter to interpret superprocesses

in terms of their "component processes" X" and X, which are much simpler. X is

obtained when IF = 0 and is a deterministic process, which is easy to describe. In the

9

general case we can only approximate the transition probability of X through tran-

sition probabilities of the "component processes" (see section 4.1). However, in some

special cases we can have more explicit interpretations. The case of deterministic (t)

is rather completely resolved in section 4.3, where we obtain nice explicit formulas.

The case of Markov chains is already more complicated and we were able only to

describe the local behaviour of X through the local behaviour of X*, by application

of Cameron-Martin formula. This will be shown in section 4.2.

CHAPTER 2
GENERAL RESULTS

2.1 Nonlinear Perturbations of Positive Semigroups

In this section we consider the problem of nonlinear perturbation of a linear posi-

tive semigroup, in such a way that the positivity of semigroup is preserved. Although

our approach relys on standard iteration method and is specially suited for our pur-

poses, we believe that the fact that we consider a very general class of semigroups

and nonlinear operators, makes it important even in an abstract framework of math-

ematical analysis. A good reference for the standard case, i.e., linear perturbation of

not necessarily positive and Co-semigroups is [47], and for the case of perturbation

by branching mechanism (see the next section) the references are [56], [28], and [18].

Let (B, |II <) be a Banach space and a linear lattice with closed positive cone.

More precisely, (B, II ) is a Banach space and (B, <) is a partially ordered set with

the following properties:

for every x,y,z E B x < y ==- x + z < y + z (2.1)

for every x,y E B and real A > 0. x < y == Ax < Ay (2.2)

for every X B there exists sup{z, 0}, denoted by x V 0 or xz (2.3)

B = {x E B I x > 0 } is closed in the norm topology (2.4)

It can be shown (see for example [53]) that in (B, <) x V y and x A y exist, for

every x,y E B, and that xVy + xAy = x+y. Then we define x- to be (-) V0,

and zIx to be x+ + x-. Some of the properties that x+, x-, and lzx satisfy are:

S= x+ x-, i.e.,B=B+ B+ (2.5)
10

x+ v x- = xV (-x) = xI X+ A x- = 0 (2.6)

Ix+yi y < xl + yl I = I XI Ij (2.7)

Remark 2.1.1 Notice that (B, 11 I|, <), as defined above, is not necessarily a Banach
lattice. Banach lattice is a partially ordered Banach space which satisfies 2.1-2.3, and
the property that, for every z,y E B,

I1 _< Yl == \\lx < Ilyll (2.8)

If B is a Banach lattice then the mappings x -,* x+, x x-, (x,y) x V y are
uniformly continuous, and 2.4 is satisfied. See [53], 11.5. for details. Therefore, the
class of Banach spaces that we consider is bigger than the class of Banach lattices.
Typical example of the space which satisfies 2.1-2.4 is a Sobolev space H'((a, b)) with
the relation of partial order defined by ( f < g if f(x) < g(x) for every x E (a, b)). But

H'((a, b)) is not a Banach lattice. It is easy to construct a sequence of nonnegative
functions {f,} in Hl((a,b)), such that 0 < f, < 1 and lf,\llL2 +oo. Thus

H'((a, b)) does not satisfy 2.8.0

As usual, we will say that an operator (not necessarily linear) O : B B is
positive if O(B+) C B+.
Let B* be the dual space of B, i.e., the space of all bounded linear functionals on
B. We define the cone of positive functionals B*,+ by

B*+ = {f E B* I f(x) >0, for every x E B+ (2.9)

It is true that, if f(x) > 0, for every f E B*'', then x C B+. This statement follows
from the fact that B+ is closed, see [41] p.226.
We will slightly generalize the notions of weak measurability and weak integration

(see the examples below for the cases which require such generalizations). Let M C

B* be a nonempty subset of B* with the following properties:

(M, | II) is a Banach space, where 11 is the operator norm of B* (2.10)

m(x) = m(y), for every m 6 M ==Z x= y (2.11)

m(x)> 0, for every m M = M B*'+ x E B+ (2.12)

1 =sup ; meM \ {0} (2.13)

Notice that, if M = B*, then M satisfies 2.10-2.13. We define a-algebra M on
M to be the smallest a-algebra such that, for every x E B, the mappings m m(x)
are measurable. Let I C R be an interval and B(I) the a-algebra of Borel sets on I.

Definition 2.1.1 A vector-valued function x(s) : I -- B is:
M-weakly measurable ifm o x is B(I)/B(R) measurable, for every m E M.
M-measurable if (m, s) i- m[x(s)] is M ( B(I)/B(R) measurable.
M-Pettis integrable if there exists an element xj E B, such that, for every m E M,
mo x is Lebesgue integrable and

m(xi) = m[x(s)]ds (2.14)

In this case xj is called M-Pettis integral of x(s) and is denoted by

xi = (MP) f x(s)ds (2.15)

If M = B*, then these are the standard notions of weak measurability and Pettis
integrability (see, for example, Chapter III in [33]), and in this case we will denote
Pettis integral x, by

(P) j x(s)ds (2.16)

The notion of M-measurability requires additional explanation in some cases (see
Proposition 2.1.1 and examples below).

Definition 2.1.2 A linear bounded operator B : B --- B is M-invariant if, for every
mE M, moBE M.

The basic properties of M-Pettis integrability are given in the following theorem.
Notice that ||I(s)]I is not necessarily measurable, even if x(s) is M-Pettis integrable.
Various formulations of Fubini theorem are possible. We will state the formulation
which serves our purposes.

Theorem 2.1.1 Let x(s) and y(s) be M-Pettis integrable on I, and a, f real numbers.
Then the following is true:

The M Pettis integral of x(s) is unique. (2.17)

If, for every s I, x(s) E B+, == (MP) x(s)ds c B+. (2.18)

If M1 C M and Mx satisfies 2.10-2.13, then x(s) is M1-Pettis integrable and two
integrals coincide. In particular, if x(s) is Bochner or Pettis integrable, then x(s) is
M-Pettis integrable and

(B) x(s)ds = (P) x(s)ds = (MP) x(s)ds (2.19)

where the first integral is Bochner integral.

ax(s) + py(s) is M Pettis integrable and the integral is linear. (2.20)

If a(s) : I -- [0, +oo) is Lebesgue integrable and, for every s E I, I x(s) II a(s),
then

(MP) x(s)ds i a(s)ds (2.21)

If B is M-invariant, then B[x(s)] is M-Pettis integrable and

B [(MP) x(s)ds = (MP) B[x(s)]ds (2.22)

If x(s,t) : I x I -- B is such that s -* z(s,t) and t I-* z(s,t) are M-Pettis inte-
grable, and, for every m E M, m[x(s, t)] is B(I) 0 B(I) measurable and integrable on
I x I, and t i-* (MP) f, z(s,t)ds is M-Pettis integrable, then s i- (MP) f, z(s,t)dt
is M-Pettis integrable and

(MP) dt (MP) ds r(s, t) = (MP) ds (MP)f dt x(s,t) (2.23)

Proof. The statement 2.17 follows from 2.11, 2.18 from 2.12, and 2.19 by defi-
nition and the fact that every Bochner integrable function is Pettis integrable and
two integrals coincide (see [33], p.80). The statement 2.20 follows directly from the
definition, and 2.21 from the definition and 2.13. The statement 2.22 follows from
the definition and the fact that, for every m E M, m o B E M and (since x(s) is
M-Pettis integrable)

(m o B) ((MP)J x(s)ds) = (mo B)(x(s))ds

To prove 2.23 notice that there exists a vector xo E B such that

m(Xo) = di m [(MP)Jr ds x(s,t) = J dt J ds m[x(s,t)]

Recall that m[x(s, t)] is integrable on I x I and real-valued. Thus, by Fubini theorem
for real-valued functions,

m(Xo) = ds I dt m[x(s,t)] = ds m [(MP)r dtx(s,t) ,

where the second equality follows from M-Pettis integrability of t -4 xz(s, t). No-
tice that s F-+ f dtm[x(s,t)] = m[(MP) f z(s,t)dt] is Lebesgue integrable and
the integral is equal to m(xo), for every m E M. By definition, it means that
s i-* (MP) fJ z(s, t)dt is M-Pettis integrable and

Xo = (MP) ds (MP) dt (s, t)
*//0

Notice that the consequence of 2.18 and 2.20 is that M-Pettis integral is monotone,

i.e., if x(s) and y(s) are M-Pettis integrable, and, for every s E I, x(s) < y(s), then

(MP) x(s)ds < (MPJ y(s)ds

Q.E.D.

In the following text we will consider only operators which preserve M-measurabi-

lity. An operator O : 1(0) C B --) B (not necessarily linear) is M measurable if

t O[x(t)] is M-measurable for every M-measurable function t i-* (t) with values

in D(0).

Lemma 2.1.1 Every M-invariant operator B : B --- B is M-measurable.

Proof. Let x(t) : I B be M-measurable. It means that the mapping

F(t,m) : I x M R, F(t,m) = m[x(t)] is B(I) (9 M/B(R) measurable. Consider

a mapping H(t, m) : I x M I x M defined by H(t, m) = (t, m o B). Notice that

H(t, m) E I x M, since B is M-invariant. We claim that H is B(I) 0 M4/B(I) 0 M

measurable. It is enough to prove that (t, m) 1- m o B is B(I) 0 M/IM measurable,

i.e., it is enough to prove that m F-, mo B is MIM measurable. By the definition

of M, it is enough to prove that, for every x E B, m '-, (m o B)(x) is M/1B(R)

measurable. Since Bx E B, it is obviously true. Now, (t, m) '-* m[B(x(t))] = F o H

finishes the proof.

Q.E.D.

Let us specify the class of semigroups that we will consider in this section.

Definition 2.1.3 A semigroup (Tt; t > 0), To = id, of bounded linear operators on

B is a positive, M-Pettis integrable semigroup if it satisfies the following properties:

Tt is M invariant and positive, for every t > 0 .

(2.24)

There exist D > 1 and w > 0, such that liTtil < D e", for every t > 0. (2.25)

For every x(t) : (0, +oo) B, M-measurable and bounded on bounded subinter-
vals, the mapping (m, s, t) '- m[T.[x(t)]] is M 0 B((0, +oo)) 0 B((0, +oo))/B(L)
measurable and, on every bounded subinterval, the mapping

s i- T,[x(s)] is M Pettis integrable. (2.26)

Conditions 2.24 and 2.25 are standard. Condition 2.26, which is rather technical,
enables us to use Fubini-type theorem and weak integration. We will see in examples
that 2.26 is fulfilled under more standard assumptions.

Example 2.1.1 Let (B, I[ 1, <) be a separable Banach space and a linear lattice with
closed positive cone. Among such spaces are LP(I), 1 < p < +oo, where 1L is a a-finite
measure defined on a countably generated a-algebra, Co(X), where X is a locally

compact Hausdorff space with a countable base, and Sobolev spaces W"P(I), where

1 < p < +00. In all these spaces < is inherited from the order on the real line. Let
M = B*. Notice that in this case every bounded linear operator is M-invariant. Let

(TI; t > 0) be any positive semigroup of bounded.linear operators, which satisfies 2.25
and such that t Ttx is weakly measurable. Since B is separable, t "- Ttx is strongly
measurable, and, since (Tt) is a semigroup, t (0, +oo) '- Ttz is strongly continuous

(see [33], Theorem 10.2.3.). Consider now a mapping (s, f) (0, +oo) x B* --- B*
given by (s, f) f o T,. For every x E B, (s, f) f(Tx) is M-measurable in the

second coordinate, and continuous in the first coordinate s G (0, +oo). Since (0, +oo)

is a separable metric space, (s,f) -} f(Tbx) is B((O, +oo)) (9 M/B(R) measurable.

Therefore (s, f) F- fo T, is B((0, +oo)) M/M measurable. If x(t) is M-measurable
and bounded on bounded subintervals, then

(f, s, t) (f o T,,t) (f o T)(x(t))

is M (0 B((0, +oo)) 0 B((0, +oo))/B(R) measurable. In particular, (f, s) (- f (f o

T,)(x(s)) is M 0 ((0, oo))/B(R) measurable. Since M = B*, s -* T,[x(s)] is
weakly measurable, and s jjIT[x(s)]|| is bounded on bounded subintervals. B is
separable implies that s --* T,[x(s)] is strongly measurable, and, therefore, Bochner
integrable on bounded subintervals. It shows that (Tt) satisfies 2.26.0

Example 2.1.2 Let (E, ) be a measurable space, and (B, I1 |) a Banach space of
bounded E-measurable real-valued functions with the sup norm. Then we have a
natural partial order on B, inherited from the real line. Such a space B satisfies
2.1-2.4, and, even more 2.8. Let M be a set of all finite signed measures on (E, E).
Then M with the total variation norm satisfies 2.10-2.13, and M 7 B*. Let pt(x, dy)
be a positive kernel on E x such that (t, x) pt(x, dy) is measurable, and there
exist D > 1, w > 0, such that, for every t > 0 and x E E, pt(x, E) < D exp(wt). We
define Tt on B by

(Ttf)(x) = f(y)pt(x,dy) (2.27)

Typical example of such a semigroup is a Markov transition function pt(x, dy) and
its semigroup, when it is a transition function of a Borel right process with the state
space (E, B(E)), where E is a Lusin topological space and B(E) the a-algebra of
Borel sets (see [54], pp.13, 105, and App.2). In this case w = 0 and D = 1.
Let us show that (Tt), defined by 2.27, satisfies all the requirements of Definition
2.1.3. Since pt(x,dy) is a positive kernel and IITt l = sup{pt(x,E);x E E}, 2.24 and
2.25 follows immediately from our assumptions. Let us denote JE f(x)1.(dx) by (f, p),
where f E B and y, E M. Consider a function F(s,x): (0, +oo) x E -- R defined
by F(s,x) = 1B(s) 1H(), where B 6 B((0,+oo)), H E Then (F(s,.),A) =

1B(s) j(H), and, therefore, (s, A) '-* (F(s, -), A) is B((0, +oo)) 0 M/L/(;.) measur-
able. Using the monotone class argument, it follows that (s,' ) (F(s,.), ) is

B((0, +oo)) 0 M/3B() measurable for every bounded B((0, +oo)) /B(R) measur-
able F(s, x). By our assumption, F(s, x) = (Tf)(x) is such a function, for every
f E B. Hence, for every H E E and B E B((0, +oo)),

(1B(t) 1H('), AT.) = lB(t) (T.(1H), /)

which implies that (IL, t, s) '-4 (G(t, -), IT,) is jointly measurable, for G(t, y) = 1H(y)*
1B(t). By monotone class argument, the same is true for any jointly measurable and
bounded function G(t,y). Notice that we can identify (E, ) with ({6,; x E E}, M n
{6,; x E}), where 6, is a point mass at x. Therefore, if f(s) : (0, +oo) -- B is M-
measurable, then (s, x) F- f(s)(x) = (f(s), ,) is B((0, +oo)) /B(lR) measurable
and bounded on any bounded subinterval. It follows that

(I, s, t) F- (T,[f (t)], J) (f (t), JT,)

is M ( B((0, +oo)) 0 B((0, +oo))/B(R) measurable. In particular, it shows that

(s, x) f T,[f(s)](x) is B((0, +oo)) 0 EI/B(IR) measurable and bounded on bounded
subintervals. It follows, by Fubini theorem, that

fo() = f,)T,[f(s)](x)ds

exists, for every x E E, and defines a bounded E-measurable function fo. By Fubini
theorem, for every p E M,

(fo, ) = ,)(T, [f(s)], ) ds

which shows that s H- T,[f(s)] is M-Pettis integrable on every interval (0, t). Hence,
(Tt) satisfies all the requirements of Definition 2.1.3.0

The main problem of this section is the perturbation of a positive, M-Pettis in-
tegrable (linear) semigroup. Let (Tt) be a linear, positive, M-Pettis integrable semi-
group (as defined in Definition 2.1.3). Let L : B+ B be an M-measurable (not

necessarily linear) operator such that L[x(t)] is bounded on bounded subintervals,

whenever x(t) E B+ is bounded on bounded subintervals.

We would like to solve an integral equation

Vtx = Ttx + (MP) (o T,[L(Vtx)]ds (2.28)

where x E B+. When we say that (Vt) is the solution of 2.28, we mean that (Vt; t > 0)

is a family of operators Vt : B+ B+, such that t -4 Vtx; is M-measurable and

bounded on bounded subintervals, and (Vt) satisfies 2.28. Notice that under these

assumptions s '-* T,[L(Vt_,x)] is M-Pettis integrable (see 2.26). Thus, 2.28 makes

sense.

We will solve the problem by iteration method, which relies on well-known Gron-

wall's inequality. In the case of superprocesses the method has been used in the works

of Watanabe [56], Fitzsimmons [28], and Dynkin [18]. Gronwall's inequality states

that if a : [0, +oo) [0, +oo) is a bounded measurable function with the property

that, for every t > 0,

a(t) a+ b a(s)ds a,b> (2.29)

then, for every t > 0,

a(t)< a ebt (2.30)

In particular, if a = 0 then a = 0.

The problem is that some of the functions that we will consider are not measur-

able. However, we can easily adjust the standard proof of the Gronwall's inequality,

to obtain the following, say nonmeasurable, version of the Gronwall's inequality.

Lemma 2.1.2 Let a : (0,) --- [0, c] be a bounded mapping, where 0 < r < +oo,

0 < c < +oo. If there is a family of measurable mappings {/31; -7 E t E (0, )},

(0 : (0,t) [0, +oo), such that, for every t e (0, 7),

and, for every s (0, t),

ft
a(t) < a + sup Pt(s)ds

(s) < b a(s) ,

(2.31)

(2.32)

where a > 0, b > 0 are constants, then, for every t E (0, r),

a(t) < a e b

(2.33)

Proof. Notice that a(t) is bounded, but is not necessarily measurable. We claim

that, for every n E N,
( -1 (bt) (bt)"
a(t)_ l -- + c-n ,
k=0
and 2.33 follows then directly.

a(t) < a + sup /f t(si)ds,
-1i 0

(2.34)

3 i, (s) < b .a(si) ab + sup bf (s,)ds,

I I I

- (sn-1) < b a(si) < ab + sup b )
Tn

Hence, .-(,Sn-1) < ab + fn"- b2cdsn, and, going backwards with these inequalities,

we obtain

a(t) < a + abds, + ab2ds2ds +...+
Jo Jo Jo

+ /'-- ".
o J0

b"c ds ... ds ,

which implies 2.34.

Q.E.D.

Let us give sufficient conditions on L, such that the solution of 2.28 exists, is

bounded and unique, and has a semigroup property. We will start with boundedness.

Theorem 2.1.2 (Boundedness) If (Vt; t 0) is a solution of 2.28 and L satisfies one
of the following two conditions:

a) there exists the smallest nonnegative number, say IIL II, such that, for every

x E B+,

IILxzl < IILI l\x\ (2.35)

b) B is a Banach lattice, x i-4 x+ is M-measurable operator, and there exists the
smallest nonnegative number, say IIL+II, such that, for every x E B+,

II(Lx)+Il < IIL+l Il x (2.36)

then there is a constant K > 0, such that, for every x E B+ and t > 0,

IIVfxll < De"t lxl (2.37)

In the case a) K = w + IILI|D, and in the case b) K = w + IIL+IID, where w and D

are defined by 2.25.

Proof. Let us prove the case b). The case a) is analogous, but simpler than

the case b). Notice first, that, since 2x+I = xz < x + x- = I||, the operator
x '-* (Lx)+ is M-measurable and (L[z(t)])+ is bounded on bounded subintervals,

whenever x(t) E B+ is bounded on bounded subintervals. Since (Vt) is the solution

of 2.28 it follows that s T,[(L(Vt_,x))+] and s Tt_,[(L(VSx))+] are M-Pettis

integrable and that, for every x E B+,

0 < Vtx = Ttx + (MP) j(,t Tt-.[L(V)x)ds

Tt_.[(L(Vz))+]ds (MP) 0Ot)

Since, for every s E (0, t), Tt_, is positive, we obtain

0 < Vtx < Tt + (MP) 0) T,_[(L(V.a))+]ds

B is a Banach lattice, thus, by 2.8, we obtain

IIVt I < IITtxI + (MP)

(o,t)

(2.38)

Let 0 < T < +oo be fixed. By assumption t E (0,r) i- IIVtxll is bounded. Let
a(t) = exp(-wt)1lVtxll Then a : (0,r) -- [0,+00) is bounded, and, by 2.25 and

2.38,

a(t) < DIIxll + sup
mEM\{0}

< Djllj +

m {(MP) f(o,t Tt-, [(L(Vx))+]ds}\
im Ie"

supM\ e-t J m(Tt-|[(L(Vax))+])d|
mMEM\o} 01m1

Therefore, we choose

S( a e-m(T-_,[(L(V,x))+])|
Pm(s) = e-

Then, the conditions of Lemma 2.1.2 are fulfilled, since

3 (s) < De-'3[[(L(Vx))+I <_ DIIL e-"'IIVx=I = DjIL+ Ia(s)

Lemma 2.1.2 implies that, for every t E (0, 7),

e-^wtllVtx < DjIIxjeDIL+I*

Since r was arbitrary, D > 1, and Voa = Tox = x, 2.37 is satisfied for every t > 0
and x E B+.

Q.E.D.

= Ttx + (MP) 1(0,)

Tt-.[(L(V.x))-]ds

Tt-[(L(V,x))]ds .

For the existence and uniqueness of the solution we need Lipschitz condition. We
will say that L : B+ B is locally Lipschitz if L(O) = 0, and, for every K > 0,
there exists C(K) > 0, such that

II Lx Ly II < C(K) x y \ (2.39)

whenever x,y G B+, and Ijj1|| K, IlyI < K.
Notice that, if x(t) : I -) B+ is bounded, i.e., there exists K > 0 such that,
for every t E I, jIx(t)|l < K, then, by 2.39, L[x(t)] is bounded (by C(K) K) on
I. Therefore, if L is locally Lipschitz, then it is enough to keep as our starting
assumptions (see 2.28) that L is M-measurable. Of course, every locally Lipschitz
operator is also continuous.

Theorem 2.1.3 (Uniqueness and semigroup property) If L is locally Lipschitz, then
there is no more than one solution of 2.28. If ( Vt; t > 0) is a solution of 2.28 and
L is locally Lipschitz, then (Vt) is a semigroup.

Proof. Assume that there are two solutions, say (Vt) and (Ut), of 2.28. We fix
xz B+ and 7 E (0, +oo). Then there exists K > 0, such that max{ IVtx;|, IIUtxll} <
K, for every t E [0, -]. By 2.28, 2.20 and linearity of T, we obtain, for every t E [0, 7],

Vt UX = (MP) J) T,[L(Vt-,x) L(Ut-,x)] ds =

= (MP) ft) Tt_,[L(Vx) L(U,x)] ds

It follows that

tf (m o Tt_,)[L(Vx) L(Ux)] ds <
II|| V UtX || = sup -.
mEM\{o} tlim-
t (m o Tt_.,)[L(Vx) L(Ux)] ds (2.40)
Ssup ds (2.40)
meM\{0} o |Iml|

Consider now a(t) = jI Vtx Utx |I on (0, 7), and : (0, t) [0, +oo), where

= (mo Tt-_)[L(Vx) L(U,x)]|
Ilmll
By 2.40 we get a(t) < C(K)-2K-. -Dexp(wr), i.e., a is bounded. Since, by Lipschitz
condition and 2.25,

,(s)< | |,-,ITt.I. II L(Vax) L(Ux) II <

< De(-') C(K) I V,x Ux I < De" C(K) a(s)

Lemma 2.1.2 implies that a(t) = 0, for every t E (0,7). Since Vox = Uox = x, and,
x and 7 were arbitrary, we obtain Vt = Ut, for every t > 0.
A similar idea, with a little more computation, leads to the semigroup property
of (Vt). Consider x E B+, and s,t E [0, +oo). By 2.22 and 2.24, we get, for every
SE [0, t],
T,(V,x) =, T. (T + (MP) ) T[L(V.x)] dv =

= T,T, + (MP) 0,,) T,.+[L(V.,x)]dv =

= T,,,x + (MP) fu T[L(V+,_,-x)]dw

where the last equality follows from the fact that it is true for every m c M. It
follows that, for every u E [0, t],

Vx VuVX = (MP) 0 Tz[L(Vu+,z_,)]dz-

-(MP) ,,) T,[L(V,,_.x)]dw (MP) f,) T,[L(V.-(V.x))]dy =

= (MP) J) Tz[L(V(+,,) L( (V.x))]dz =

= (MP) J0,) Tu-.[L(V.+,x) L(V,(Vx))]dw

We are now in a similar position as in the case of uniqueness. We consider functions
a : (,t) -- [0, +oo), a(u) = I V.+,x V,(V.x) I, and 3, : [0, u] [0, +oo),

(m o T-.) [L(V.+ax) L(V,(Vzx))]I
Ilmll
By using K = max{max{ lV,+,xll, IIVw(Va)|}; w E [0,t]} and Lemma 2.1.2, we
obtain, as we did before, that a(u) < C(K)-2K-t-D exp(wt) and ,8,(w) < Dexp(wt).
C(K) a(w). Hence, a 0. In particular Vt+,x = VtVax. Since s,t and a were
arbitrary, (Vt) has the semigroup property.
Q.E.D.
Let us show that the perturbed semigroup (Vt) behaves nicely with respect to
further linear perturbation. This fact will have an impact on some of our results later,
but the special case of this property is needed to prove the existence of the solution of
2.28. Let ( St; t > 0) be a linear, M-Pettis integrable semigroup (it means that (St)
satisfies all the conditions of Definition 2.1.3 except, maybe, positivity) such that, for
every x E B,
Stx = Ttx + (MP) t) S,[B(T,_,x)]ds (2.41)

where B is M-invariant bounded linear operator. Recall (Lemma 2.1.1) that such an
operator is M-measurable. Thus L B is M-measurable operator on B+, such that
(L B)[x(t)] is bounded on bounded subintervals, whenever x(t) E B+ is bounded
on bounded subintervals. It shows that the following statement makes sense.

Lemma 2.1.3 If ( V; t > 0) is a solution of 2.28, then, for every x E B+ and t > 0,

Vt = St + (MP) 0) S,[(L B)(Vt.x)]ds (2.42)

Proof. By our assumptions (see 2.26) s -, S,[L(Vt_,x)] is M-Pettis integrable.
Thus, we consider the following integrals

(MP) (,t) S,[L(Vt.,x)]ds = (MP) Jo) T,[L(V_,x)lds+

+(MP) f() (MP)f(o S[B(T._(L(Vt-,x)))]du" ds ,
where equality follows from 2.41. The condition 2.26 guarantees that (m, u, ) -s
m{S,[B(T,_,(L(Vt.x)))]} is M (0 B((0, +oo)) ( B((0, +oo))/B(R) measurable, and
is bounded on (0, t) x (0, t). Hence, the requirements for Fubini theorem 2.23 are
fulfilled. Together with the fact that the first integral above appears in 2.28 we
obtain
(MP) ft) S.[L(Vtx)]ds = Vtx Ttx+

+(MP) J0,) [(MP) ,t) S,[B(T,-,(L(Vt,,x)))]ds] du
Notice that S, o B is an M-invariant bounded linear operator. Thus we can apply
2.22 on the integral on the right hand side, to get that it is equal to

(MP) J(t)(S. o B) (MP) i) T_,,[L(Vt-,x)]ds du =

=(MP) Jt)(S o B) [(MP) f(ot) T,[L(V(t,.x)]dw du=

= by 2.28 = (MP) t)(S, o B) [Vtx Tx] du = by linearity of S o B =

= (MP) ft)(Su o B)(Vtx)du (MP) J) (, o B)(T_,x)du =

= by 2.41 = (MP) J (S, o B)(Vtux)du Six + Ttx
Finally, it shows that

(MP) Jt) S,[L(Vt-,x)]ds = Vtx Ttx+

+(MP) ot) S,[B(Vt_,x)]du Stx + T ,
which, since S, is linear, finishes the proof.

Q.E.D.

For any a R, an operator B, given by B(s) = ax, is M-invariant, bounded

and linear. Also, St = ea tT is a linear M-Pettis integrable semigroup, which is also

positive. Even more, (St) satisfies, for every x E B,

(MP) 0,t) e"T,[B(t-,x)]ds = (MP) 0t) ae"Tx ds =

= aea"ds Ttx = eatTtS Ttx = Stx Ttx

If we apply Lemma 2.1.3 on a and -a, then we obtain the following statement.

Corollary 2.1.1 A semigroup (Vt; t > 0) is a solution of 2.28 if and only if it is a

solution of

Wtx = e-"tTz + (MP) 0,t)(e-"' T)[(L + a)(Wt_,x)]ds (2.43)

where a E R.

In the case of superprocesses, Corollary 2.1.1 has been proved, using integration

by parts, in [10], p.58.

We will use Corollary 2.1.1 to prove the positivity of the solution of 2.28. To

prove the existence of the solution of 2.28, we require several properties on L. We

will assign a special name to such L.

Definition 2.1.4 An operator L : B+ -- B is a SOLP-operator (solvable with posi-

tive solution) if it is an M-measurable, locally Lipschitz operator which satisfies either

condition a) or condition b) from Theorem 2.1.2, and such that, for every K > 0,

there exists a = a(K) > 0, such that

Lx + ax E B+ (2.44)

whenever x E B+ and II||I < K.

Theorem 2.1.4 If (Tt; t > 0) is a positive, M-Pettis integrable semigroup, and L is
a SOLP-operator, then there is one and only one solution to 2.28. The solution is a
semigroup and satisfies boundedness condition 2.37.

Proof. It is enough to prove the existence of the solution of 2.28. We will prove it
under the assumption b) from Theorem 2.1.2. The case of condition a) is analogous,
but simpler than the case b).
Without loss of generality we can assume that D = 1. Recall that there is a norm

|| I' on B, defined by
111' = sup Ile-WTtll ,
t>o
such that I xaI I 11xi1' < Dlix i, i.e., II I and II 1' are equivalent norms. It follows that

IITt I' < exp(wt), and, for every m e M, IIm l' < Ilmll < DImll'. All the requirements
in this section are of topological or measure-theoretical nature, except for 2.8 and
2.13. Thus, they will not change when we change to the equivalent norm. Also,
we will use 2.8 only in case 0 < x < y. Therefore, to prove that we can restrict
ourselves to the case D = 1, we have to prove that II 11' satisfies 2.8 for 0 < x < y,
and 2.13. If 0 < x < y then, for every t > 0, 0 < exp(-wt)Ttx < exp(-wt)Tty, since
Tt is positive. By 2.8 for the norm II i we obtain I1 exp(-wt)Ttxll Il exp(-wt)Tty l,
which implies that jIx' < Ilyll'. 2.13 is equivalent to

Ix||i' = sup{lm(x)I; m E M, Ilmll' < 1}

Notice that |lxij' > sup{lm(x)i; Ilmll' < 1}, since Im(x)l 5< lmjl' l|xl', for every
bounded linear functional. Using 2.13 for i II we obtain

ix|||' = sup Ile-wtTtll = sup sup e-w(m o Tt)(x)l
t>0 t>0 IlmiS
For every t > 0 and m e M, Ilmll < 1, h = exp(-wt)(m o T() is in M, since Tt is
M-invariant and M is a vector space. Using the equivalence relations for the operator

norms i| || and II |' we get

Ilhll' < e- wllmll' |IT|t' < e-" 1 e" = 1

Hence

11x\' < sup{jh(x); h E M, \\h|\' < 1} ,

which proves 2.13 for || ||'. Therefore, we can restrict our proof to the case D = 1.
Fix a strictly positive number 7 and vector x e B+. Let K be equal to

iix\ e ,

where IIL+Il is defined by 2.36. Let a = a(K), defined by 2.44. By Corollary 2.1.1
it is enough to prove the existence of the solution of 2.43. We define inductively

Vox 0, and, for every k E N U {0},

S tTtx + (MP) t)e-'T,[(L + a) Vt,z]ds (2.45)

We claim that, for every k N U {0}, and, for every t E [0, ], Vkx is well-
defined, 0 < Vkx, 1Vtkz < \x\ exp((w + H|L+)t) < K, and [0,7] 9 t -- Vtk
is M-measurable. Obviously, for k = 0 the statement is correct. Assume that the
statement is true for k. Since L + a is M-measurable and locally Lipschitz, it follows
that

(m, t, s) t- m[e-"T,[(L + a)Vtz]]

is M 0 B([0, 7]) ( B([O, t])/B(IR) measurable, and M-Pettis integrable with respect
to s. Therefore, the M-Pettis integral in 2.45 exists and Vtk+lx is well-defined. By
Fubini theorem

(m, t) m Vkt1

is M i B([0,r])/B(i~) measurable. Since 1IVt8x| < K, and Vt,x > 0, it follows, by
2.44, that (L + a)(V,k,x) > 0. By 2.18 and the positivity of T, we obtain Vktx, > 0.

Using the positivity of T, one more time we obtain

0 < Vk+x < e-Tt + (MP) 0,e-"aT[(LVa.x)+ + aVtx]ds

By 2.8 for positive vectors we obtain

iVtk+lx I < e-IITtx + II(MP) 0t) e-asT.LV x), + aV, ds

First, we estimate |Ie-a'T.[(LVt. )+ +aVt,xzj to be less or equal to e-a -ew"(IL+II +
a)| x|I exp[(w + 1IL+ 1)(t s)], and then apply 2.21 and 2.25 with D = 1, to obtain

IIVk+'Xll 11:-11

(w--a)t + e(+IIL+II)t (IIL+ + a)e-(IIL+II+a)ds) =

= II| [e(w-a)t + e(w+IIL+ll)t (-e-(l'L+ +a)I)] =

= II lle(W + Ill)

By mathematical induction, the statement is true for every k > 0.
Let C = C(K) + a, where C(K) is a Lipschitz constant defined by 2.39. By 2.45,
we obtain

VII Vkl tk I =

= (MP) f(t)e-"aT,[(L + a)Vtx (L + a)V t-x]ds

The norm
ee-"T,[(L + a)V~.x (L + a)Vt-.x] <

e-"-' C Vtk Vxt I ,

since all the vectors involved are bounded by K, and L is locally Lipschitz. Let K,
be equal to exp(|w air). Applying 2.21 recursively k times we obtain

II Vk+x Vtkx I < K Kk Ck.

S ds / d.s2 dsk = K KkCk
Jo Jo 0 k!
Therefore { Vtk; k > 0 } is a Cauchy sequence in B+ C B. B+ is a closed subset of a

Banach space B, thus there exists a limit (in the norm sense)

Vtx = lim Vkx B+
k-00oo

Since IVtkzll < K, and m[Vtx] = limkj-om[Vtkx], it follows that t Vtx is M-

measurable and bounded on [0, ]. In particular, it shows that s -* e-a'T,[(L +

a)Vt_,x] is M-Pettis integrable, and, for every m E M,

m [(MP) /) e-'T,[(L + a)Vt.,x]ds =

= Jm [e-T,[(L + a)Vt,x] ds

By the dominated convergence theorem, the integral on the right hand side is the

limit of

[ m [e-aT,[(L + a)Vkxl] ds ,

for every m e M. Since {Vtk+1l e-tTtx} converges strongly to Vtx e-tTte, and

M separates the points in B, it follows that Vtx satisfies 2.28 for exp(-at)Tt and

L + a. By Corollary 2.1.1 t V- Vtx is the solution of 2.28 on [0, T], for x E B+.

Since the uniqueness theorem (Theorem 2.1.3) is valid for every x E B+ and every

0 < 7 < +00, we can define Vtx for every x E B+ and every t > 0, so that Vtx satisfies

2.28, Vtx E B+, t -* Vtx is M-measurable and bounded on bounded subintervals.

Q.E.D.

Notice that if the condition a) from Theorem 2.1.2 is satisfied, and condition 2.44

is not necessarily satisfied, the same proof as above will show the existence of the

solution of 2.28, even when (Tt) is not positive. However, in such a case we can not

obtain the positive solution, in general.

Let (Tt; t > 0) be an M-Pettis integrable semigroup, i.e., (Tt) satisfies conditions
in Definition 2.1.3, except positivity. Let B : B B be an M-invariant bounded
linear operator. Then B restricted on B+ is locally Lipschitz (actually Lipschitz),

M-measurable, and satisfies the condition a) from Theorem 2.1.2. It implies, as we

just said in the paragraph above, that there is a solution (St) of 2.28, which is the

limit of approximations (Sf), given by 2.45, for a = 0 (and therefore Corollary 2.1.1 is

not needed in this case) and L = B, but (St) is not necessarily positive. Since all the
operators involved are linear, then St is linear, and therefore St is linear, too. Thus, it
can be extended on B = B+ B+. It shows that there exists a semigroup ( St; t > 0)

of linear operators such that, for every x E B, (m, t) m[Stx] is M 0 B(R+)/B(R)

measurable, 'lS, l < Dexp((w + IIB|I)t), and,

Stx = TtX + (MP) o) T,[B(St-,x)ds (2.46)

Also, Stx = limk-_o Sx, where S, -= 0, and

S+lx = Ttx + (MP) 0,t) T,[B(SLx)]ds (2.47)

Let us restrict our attention to the special case, when B is separable and M = B*.

As we have seen in Example 2.1.1, in such a case all the functions involved are actually

strongly measurable, and, therefore, Bochner integrable. In particular, our semigroup

(St), given by 2.46, is a semigroup of bounded linear operators which satisfies 2.25
and t t-4 Stx is weakly measurable. As we have shown in Example 2.1.1, it implies
that (Se) satisfies all the requirements of Definition 2.1.3, except, maybe, positivity.

Therefore, there is a solution (Wt; t > 0) of the equation

WtX = Sx + j) S,[(-B)(Wet-x)]ds (2.48)

where the integral involved is the Bochner integral. If we rewrite 2.46, using Bochner
integral and linearity of the operators involved, we obtain that

Ttx = Stx + ,t) T.[(-B)(St,.x)]ds

These are exactly the conditions of Lemma 2.1.3. Since (-B) (-B) = 0, we get,
for every x E B+, WtX = Ttx, and, because of linearity, it follows Wt = Tt. It shows
that

S.= To + S,[B(T,_,w)]ds ,
Stx = Ttx + ot) S.[B(Tt-)x)]ds

which is exactly 2.41. Hence, we just proved the following statement:

Corollary 2.1.2 If B is separable and (Tt; t > 0) a semigroup of bounded linear
operators on B, which satisfies 2.25, and t i-4 Ttx is weakly measurable, then, for
every bounded linear operator B : B -- B, there exists a unique semigroup ( St; t >
0) of bounded linear operators on B, such that t '-+ Stx is weakly measurable, and,
for every t > 0,

IIStll < D e(W+llI)t (2.49)

and (St) satisfies, for every x E B,

Stx = Tx + o S,[B(Tt-_x)]ds=

= Tzt + o T[B(St,x)]ds (2.50)

where all the integrals involved are Bochner integrals. Moreover, if (Tt) is also posi-
tive, and B satisfies 2.44, then ( St; t > 0) is positive.

Let us show that some of the measurability conditions given in this section are
much simpler in the separable case, then in general case. Recall that in the separable
case the c-algebra of Borel sets with respect to the norm topology is equal to the

smallest a-algebra on B such that all mappings.zx f(x), f E B*, are measurable.

Consider a mapping (z, f) .- f(x) on B x B*. It is continuous with respect to the

first variable, and M measurable with respect to the second variable. Since B is

separable metric, the mapping (x, f) f(x) is B(B) 0 M /B(R) measurable. This

fact enables us to prove the following proposition.

Proposition 2.1.1 Let B be a separable Banach space and M = B*. Then a vector-

valued function t x(t) is M-measurable if and only if it is weakly measurable.

If L : D(L) C B -- B is a continuous operator then it is M-measurable. In

particular, every locally Lipschitz operator is M-measurable, and, if B is a Banach

lattice, z x+ is M-measurable.

Proof. If t z(t) is weakly measurable then it is B(I)/B(B) measurable. Then

(t, f) (x(t), f) is B(I) M/B(B) 0 M measurable. Finally, (t, f) (x(t), f)
f(x(t)) is a composition of measurable functions, i.e., it is B(I)(M /B1(R) measurable.

The reverse statement is always true.

If t -4 z(t) E V(L) is M-measurable, then it is weakly measurable, i.e., it is

B(I)/D(L) n B(B) measurable. Since L is continuous, it is D(L) n B(B)/B(B)

measurable, which shows that t -2 L(z(t)) is weakly measurable, and, therefore,

M-measurable. The last statement follows since every locally Lipschitz operator is

continuous, and z z+ is continuous in Banach lattice.

Q.E.D.

We will finish this section with a remark on condition 2.44. As we have seen

it is a sufficient condition for the positivity of a perturbed semigroup. Notice that

for some classes of operators this is also a necessary condition. More precisely, let

B = R', where a = (zx,...,an) < y = (yl,...,y n) if x, < y( for i = 1,...,n. Let

A : R" -- R" be a linear operator, i.e., A is a matrix (a j),J=1. Then there exists a

uniformly continuous semigroup

A (tA)k
S = etA = k!
k=O
which satisfies the equality

Stx = x + A(St-,x)ds

We claim that the following three statements are equivalent:

i) St is positive, for every t > 0

ii) A satisfies 2.44

iii) for every i,j E {1,...,n}, i f j, ai, > 0

By Theorem 2.1.4, ii) implies i). iii) implies ii), since a = maxi{la il} will give

Ax + ax > 0, for every x > 0. Let us prove that i) implies iii). Suppose contrary

to the claim that there exist io and jo in {1,..., n} such that io # jo and aojo < 0.

We consider x = (x1,..., x,) such that xi = 0, if i Z jo, and a0 = 1/(-aj00). Then

zio = 0 and (Ax),o = -1. We define f(t) = (Stx)i,. Then f(t) is a differentiable

function, f(0) = xi, = 0, f'(0) = (Ax)i0 = -1, and f'(t) is continuous. Hence, there

is e > 0 such that on [0,e] f'(t) < 0, and, since f(0) = 0, we must have f(t) < 0 on

[0, e]. This contradicts i). Hence, the proof is complete and all three statements are

equivalent.

2.2 Supersemigroups and Superprocesses

In this section we will define the main notion of this work. We will follow the

approach developed by P. Fitzsimmons and E.B.. Dynkin, and show that their results

are special cases of results from section 2.1.

Let (B, jj ||, <) be a Banach space and a linear lattice with norm-closed positive

cone. Let M C B* satisfies 2.10-2.13, and consider a measurable space (M+, M+),

where M+ = M n B*'+, and M+ = M+ n, M. Following Dynkin [18] we will say, for

every measure Q on (M+, M+), that

x e B+ e-m(x)Q(dm) (2.51)
JM+

is the Laplace transform of the measure Q.

Let (Tt; t > 0) be a positive, M-Pettis integrable semigroup, as defined in Def-
inition 2.1.3, and let L : B+ B be a SOLP-operator (see Definition 2.1.4).

Then there is a unique solution (Vt; t > 0) of 2.28, which is a positive, (in general

nonlinear) semigroup (see Theorem 2.1.4).

Definition 2.2.1 A family of kernels Qt(m,dn) on the measurable space (M+,M+)

is a (Tt, L)-supersemigroup if, for every m E M+ and x E B+,

/M e-()Qt(m,dn) = e-m(v' (2.52)

The existence and the uniqueness of such semigroups in the generality of Definition
2.2.1 is not known. However, in some special cases it has been studied by several au-

thors (see [10] for detailed bibliography). The case of (Tt, L)-supersemigroup, where

(Tt) is a Markov semigroup, and L is, so called, branching mechanism has been stud-
ied by Watanabe, Dawson, Dynkin, Fitzsimmons and others. We will restrict our

consideration to this situation.

Let (E, E) be a standard measurable space, i.e., there is a Polish space (Z, B(Z)),

with Borel a-algebra, which is Borel isomorphic to (E, ). Recall ([6], Theorem

8.3.6.) that any two Polish spaces of the same cardinality are Borel isomorphic, and

that a Polish space is either countable or has the cardinality of the continuum. It

follows that a measurable space is standard if and only if it is Borel isomorphic to a

compact metric space with Borel a-algebra. Therefore, we can introduce a metric on
E, such that E becomes compact and is the a-algebra of Borel sets with respect

to the metric. Notice that our notion of a standard space is equivalent to the notion

of Lusinian measurable space in [54] (our terminology is from [6] or [18]).

We will denote the set of (respectively, bounded, bounded nonnegative) real-

valued E-measurable functions on E by (respectively, bW, bpC) E. Then B = bW

with the supremum norm, and with the partial order inherited from the real line is

a Banach lattice, where B+ = bp. If M is the space of all finite signed measures on

(E, E), then M satisfies 2.10-2.13, and M+ is the set of all finite (positive) measures
on (E, E). See Example 2.1.2 for all these results. For f E b and p E M we will

denote the integral fE fd1i by (i, f).

Remark 2.2.1 There exists a metric d on M+ (so called Prohorov metric) such that

d(i,, It) -+ 0 if and only if i,, )L; where t,, -'-- pt means weak (Bernoulli)
convergence, i.e., for every bounded and continuous f, (fn, f) (ip, f).

It is important that we understand statements about (M+, M+) properly. If we

talk about measurability properties of M+ then no topology (or metric) is needed,

since the definition of M+ is purely measure-theoretic. But, if we talk about the

Prohorov metric (or weak convergence) then we assume certain topology on E. If

we do not specify this topology, it means that we consider E as a compact metric

space described above. Notice that in this case M+ with Prohorov metric is a locally

compact, complete, separable metric space and its Borel a-algebra is exactly M+. In

particular, it shows that (M+, M+) is a standard measurable space. For the proofs

of results mentioned above see, for example, [7], pp.607-638.

However, sometimes we can consider E with its "natural" topology (if it exists,

of course), which is not necessarily compact. In such a case, we have to deal with

topological properties of M+ separately.

Let = = (t; t > 0) be a time-homogeneous Markov process, with state space

(E, ), transition function pt(x, dy), and corresponding semigroup (Tt; t > 0). We
assume that, for every A E (t, z) i- pt(x, A) is measurable. (Tt) is a contrac-
tion semigroup, and, as we have seen in Example 2.1.2, (Tt) is a positive, M-Pettis
integrable semigroup.
Let b E b, c E bpE, and n(x, du) be a positive kernel from (E, E) to the measur-
able space of positive real numbers ((0, +oo), B((O, +oo))) such that

x- -- un(x, du) bp (2.53)

In particular, 2.53 shows that there is a constant k > 0 such that, for every x E E
and e > 0,
k
n(x, [e, +oo)) -k (2.54)

We define a branching mechanism as a function T : E x R+ R of the form

(x, A) = -b(x)A c(x)A2 + 0(1 e-A)n(x, du) (2.55)

For f E bpg we define If E by

(I f)(x) = (x, f (x)) (2.56)

Using 2.53 and the fact that, for z > 0, 1 -e-z < z, we obtain that, for every f E bpE,

1f11 11 bl + c. 11ft11i + un(x, du)\) (2.57)

Notice that the factor |c||l Ilf| prevents us from concluding that k is bounded, but
2.57 does show that T : bp -- bE, and T0 = 0. Notice also that directly from
the definition 2.55 follows that T is M-measurable. Let us prove that I' is locally
Lipschitz. Applying the mean value theorem on z exp(-z), we obtain

I (Tf)(x) (g)( ) < I f(x) g(x) I-

(Ib(x)) + c() f(x)+g(x) i+ 0 un(x, du))

which, together with 2.53, shows that 'I is locally Lipschitz.
It is possible to rewrite a branching mechanism in the form
/+00
(z, A) = -b(x)A c(z)A2 + (1- e-X" Au)n(x, du) (2.58)

where
/+00
b(x) = b(x) j un(x,du) (2.59)

Notice that the assumption 2.53 was crucial here, and that it shows that b E bE.
Since 1 e-^" Au < 0, for A > 0, it follows that, for every f E bp&,

(f) = (b)-- f (2.60)

This, together with our choice of Banach lattice bW, shows that an operator 'I satisfies
condition b) from Theorem 2.1.2, and that I+|II = l- l.
Let K > 0 be a positive number. The fact that the derivative

d (x, A) < |Ib + 2AI|cI + i +0 un(x, du)

implies that
a =a(K) sup {--(,) A)
xEE,AE[0,K] (dA
is finite and nonnegative. Then, for every x e E, A H-* (zx, A) + aA is nonnegative
and increasing on [0, K], which implies that T satisfies condition 2.44. Hence, we
proved that k is a SOLP-operator. It follows now, from Theorem 2.1.4, that there
exists one and only one solution Vt : bpE bpE of an integral equation
t
(Vf)(x) = (Tf)(x) + T[TV_,.fI(x) ds (2.61)

and that (Vt; t > 0) is a semigroup, (t, z) (Vtf)(z) is measurable, and, for every
f e bpE,

(2.62)

IV1fll < eill-l"lflI .

This result was obtained by Fitzsimmons, [28], using similar method as our it-

eration method in section 2.1., but applied only to the case of bp and %k, and by

Dynkin, [18], in the case of \$(x, A) = -A2, and by different method. Here we just

proved that their results are special cases of our general result given in section 2.1.

However, the semigroups of this type were studied in connection with measure-valued

processes for the first time in Watanabe, [56], and in several works by Dawson (see

[10] for complete bibliography). It must be mentioned that the basic ideas for such

development were already given in Jifina, [36]. Following Dynkin's terminology we

will give a special name to a (Tt, ')-supersemigroup.

Definition 2.2.2 A (Tt, %)-supersemigroup, where %I is a branching mechanism, and

(Tt) is a semigroup which corresponds to a Markov process (&t) is called a Dawson-
Watanabe supersemigroup.

Markov process (Xt; t > 0) with the state space (M+,.'M+) is called a Dawson-

Watanabe superprocess over (It), with branching mechanism T, if its semigroup is a

Dawson- Watanabe supersemigroup.

We will use the same name (Dawson-Watanabe supersemigroup) to indicate the

transition function Qt(,, dv) on (M+, M+), and the actual semigroup of operators

(QtF)(p) IM F(v)Qt(A, dv) (2.63)

Also, when it is clear what 4 and ((t) are, we will refer to (Xt) as to a superprocess

only.

The question of the existence and uniqueness of the superprocess comes immedi-

ately into consideration. Notice that once we have a supersemigroup Qt(Gi, dv), then,

since (M+, M+) is a standard space, Kolmogorov existence theorem guarantees the

existence of the superprocess (Xt). Of course, for a moment we do not say any-

thing about regularity properties of (Xt). We will come to this in the next section.

Therefore, the question is the existence and uniqueness of Qt(tI, dv), given by 2.52.

Notice that the Markov property of Qt(~p, dv) is the consequence of the semigroup

property of (Vt). Notice also that /L i- exp(-(jz,Vtf)) is M+-measurable. An ap-
plication of the monotone class theorem for the multiplicative systems of functions

shows that i Qt(p, dv) is M+-measurable i.e., Qt(u, dv) must be a kernel, if
it exists. VtO = 0 shows that every Qt(p, dv) is a probability measure. That the

Laplace transform determines a measure uniquely (on a standard space) has been

proved in [18], pp.264-265. Therefore, it is enough to prove that for any p and t > 0
there exists a measure Qt(I, dv) which satisfies 2.52. The solution to this problem
has been given in [28]. However, some parts of the construction from [28] are more

demanding than it may look at first sight. This and the wish for completeness of the

work led us to include these proofs here.
Notice that (B+, +) forms a two-divisible semigroup, i.e., for every x E B+, there

exists y E B+, such that x = y + y (of course, y = (1/2)x). Hence, the Laplace

transform is a positive definite function on the semigroup B+ (see 2.51). The theory

of positive definite functions on a two-divisible semigroup (S, +) is developed in [2]. It
has been proved there that for every bounded positive definite function p : S R

there exists a unique positive Radon measure 7 on S, such that

s(s) = p(s)7(de) (2.64)

where S is the set of bounded semicharacters, i.e., of the functions e : S -* [0,1]

such that e(0) = 1, and e(si + s2) = e(si) e(s2). Moreover, if p(0) = 1, then

7 is a probability measure. Recall also ([2], Theorem 3.2.2.) that positive definite
functions are negative exponentials of negative definite functions, i.e., of the functions

S: S -- R, such that

E aiajO(s, + s,) < 0 (2.65)
t,=l

whenever {ai,...,a,,} C R sum to zero, {sl,... ,,Sn} C S, and n > 2. Therefore,
in application of these theories on the solution (Vt) of 2.28, one has to prove first
that x m[Vtx] is negative definite. In terms of branching mechanism and in the

notations of bp, one has to prove that f (Vtf)(x) is negative definite, for every
x e E (do not confuse x E E and x E B !).

Remark 2.2.2 The statement about negative definitness of f F-+ (Vtf)(x) is given in

[28], p.341, and it is suggested there that the proof is standard. Through private
communication with the author of [28] we pointed out some possible difficulties in
the proof. After some discussion both sides offered correct proof (which happened to

be the same on both sides). This is still a standard proof, but different from the first
proof that was suggested. We present the proof below.o

Proposition 2.2.1 Let B+ = bp, and L = T. If (Vt; t > 0) is a solution of 2.28,
then, for every t > 0 and x E E,

f -* (Vtf)(z) (2.66)

is negative definite on (bpE, +).

Proof. Consider, for every m E N, the branching mechanism

'.(, A) = -b(x)A + (1 e-"A )ii(x, du) (2.67)

where

f(x, du) = n(x, du) + 2m2c(x)6)1_ (2.68)

where Sa} is the point-mass at a. Notice that, for every x and A, Im(x, A) -Q I(z, A),
when m -t +oo. Hence, for every f and x, there is a subsequence {mk} such that

(Wtkf)(x) (Vtf)(x), where (Wm; t > 0) is the solution of 2.28 for L = mQ.

Therefore, it is enough to prove negative definiteness of the solution of 2.28 where

L is the branching mechanism of the form 2.67. Notice that, because of 2.53, such

branching mechanism can be written in the form

(x, A) -b(x)A + J (1 e-A")9(x, du) (2.69)

For a fixed x E E, the function above is a Bernstein function (see [3]). The com-

position of a Bernstein function with a negative definite function is again negative

definite (see [3], and [2] p.114). If ( Wt; t > 0) is the solution of 2.28 with the branch-

ing mechanism of the form 2.69, then (Wtf)(x) = lirrm oo(Wtf)(x), where W' are

defined inductively by 2.45. Since Wo 0 is negative definite, and linear function
(Ttf)(x) is negative definite, we prove inductively that (WT'f)(x) is negative definite,

for every I (applying the above fact about Bernstein function).

Q.E.D.

Remark 2.2.3 There is a possibility for an easy mistake in the proof above. In some

cases a function, say 0, can be regarded as a function on a group and on a semigroup,

as well. Typical example is the function 0(x) = -x2. 0 is the negative definite

function in the sense of a semigroup (R+,+), but 0 is not the negative definite

function in the sense of a group (R, +). However, when we compose 0 the situation

changes. 0 composed with the negative definite function in the sense of a semigroup

(R+, +) is not necessarily a negative definite function on (R+, +). For example,
x F-4 ln(1 + x) is a negative definite function on (R+, +) (to prove that consider the

Laplace transform of the r-distribution), but x -* -[ln(1 + x)]2 is not.O

The last part of the construction of Qt(l, dv) is given in [28]. For the sake of

completeness we will present it here (in some nonessential details our argument will

be different).

Notice that, for every x E E, limn_.o(Vt;)(x) = 0, by 2.62. By Proposition 2.2.1
and 2.64 there is a probability measure 7 on S (where S = bpE), such that

= P(f)-y() (2.70)

and is unique. Since p(l/n) = e(1)1/", it follows that 7 is concentrated on

S+ = {e gS (s) > 0, for every s S }

If f, \ 0, f, E bpE, then (Vtf,)(x) \ g(t) > 0, since Vt is monotone, by 2.70 (in the
sense that if f,g E bpE, f < g, then Vtf < Vtg). Using 2.28 we get

g(t)= T,['g(t- s)]ds

By the uniqueness of the solution of 2.28, we obtain g(t) 0, i.e., Vtf, \ 0. By 2.70,
e(fn) / 1 7-(a.e.).
If we fix f E bp, g E S+, and, for every t > 0, we denote e(tf) by g(t), then
g(0) = 1, g : [0, +oo) -- (0,1], and g(t + s) = g(t)g(s). Hence, g(t) = exp(-at),
where a = In o(f) > 0.
Now, let us define T: bpE -- bB(S+) by

(Tf)()= -Ine (f) (2.71)

where B(S+) is the o-algebra of Borel sets on S,. We just proved that T is additive,
homogeneous for positive real numbers, and, if f, \ 0, then Tf, \ 0 7-(a.e.). Very
simple adjustment of the theorem in [31] for the case of standard spaces, shows that
there exists a kernel K on S+ x bounded and positive, such that, for every f e bp,

Kf = Tf 7-(a.e.). (2.72)

Hence, for every g E S+ there is a measure K, in M+, and, by definition of M+,
e K, is B(S+)/M+-measurable. Let Q be the image measure of under Lp Kp.

Then, by 2.70 and 2.72,

e-(U>f) = + e-(vf)Q(dv) (2.73)

and the construction of a Markov transition function Qt(i, dv), which is also a su-

persemigroup, is complete. In other words we proved the following theorem:

Theorem 2.2.1 (Dynkin, Fitzsimmons) Let (E, E) be a standard space. For every

time-homogeneous Markov process ( on E, such that (t, x) pt(z, dy) is measurable,

and, for every branching mechanism 9, which satisfies 2.53, there exists one and only

one supersemigroup Qt(I, dv).

Remark 2.2.4 Locally Lipschitz property was crucial in the construction of the su-

perprocesses. We were using 2.53 to obtain it. However, it is worth mentioning that

locally Lipschitz property of T follows under weaker condition than 2.53, i.e., under

the condition

u n(x, du) + un(x, du) e bpS
0O 1
But in this case we can not transform 2.55 into 2.58 and vice versa. Therefore, it

leads to different constructions where our general approach, developed in section 2.1.,

cannot be applied so easily. For the construction of superprocesses under this weaker

condition we refer to [10], Chapter 3.

Let us also mention that in [28] the author assumes 2.53 and one more condition,

i.e., that

x u2n(x, du) E bpE ,

but he uses it only for the regularity properties of the superprocesses. For the con-

struction of the supersemigroup it is not needed.0

Let us mention that there is a nice (and important, as we will see later), proba-

bilistic interpretation of the branching mechanism T. Notice that, for every x E E,

A W- qI(z, A) is a Log-Laplace function of an infinitely divisible distribution (see [27],

XIII.7.). More precisely,

e-('') = E (e-(D.+N.+P)) (2.74)

where D,, N,, P, are independent random variables with values in R, such that D,
is a point-mass at -b(z), N, is a centered normal distribution with Var(N,) = 2c(x),
and P, is a positive infinitely divisible distribution with finite expectation

E(P.) = un(x,du) (2.75)

Recall also (see [35] for example), that there is a unique temporally homogeneous
Levy process, i.e., a process with stationary independent increments with no point
of fixed discontinuity and whose sample paths are cadlag, say (Lk(")), such that

e = E e- L')) .= (2.76)

If T(z,A) = -A then (Lt (")) is a uniform motion to the left. If T(x,A) = -A2

then (Lt ("')) is a standard Brownian motion. If {(x, A) = 1 e- then (L*( '')) is a
Poisson process.

We will apply Lemma 2.1.3 on superprocesses to obtain an interesting result,
which suggests that in some cases we can study always the same (and extremely sim-

ple) Markov process, and change the perturbing operator accordingly, to get Dawson-

Watanabe superprocess.
Let us denote the trivial semigroup, which consists only of identity operators, by

(Idt; t > 0). This semigroup is defined on bE, but also it is defined on any subspace
of b. To this semigroup corresponds a trivial Markov process on (E, ), which "stays
forever at the point". Let us denote this Markov process by (dt; t > 0). Then, for
every t > 0, dt do.

Theorem 2.2.2 Let ( be a Markov process, such that there exists an M-invariant,
bounded, linear operator A, such that, for every t > 0, Tt = exp(tA). Then there
exists a Dawson-Watanabe supersemigroup over (, with branching mechanism T, and
is equal to a (Idt, E + A)-supersemigroup.

Proof. By our assumptions, for every G E E,

pt(x, G) = P(( E G) = (TlG)(x) = E -(A~nl)(X)
n=.O "
which shows that (t, z) '-4 pt(x, dy) is measurable. By Theorem 2.2.1, there ex-
ists one and only one supersemigroup Qt(L, dv), such that its Laplace transform is

exp(-(A, Vtf)), where, for every f E bp,

Vf = Ttf + (MP) (ot) T.[W(Vtf)]ds

As we have seen in Example 2.1.2 (Tt) is a positive, M-Pettis integrable semigroup,
and
Ttf = Idtf + (MP) Jt) T,[A(Idt_,f)]ds

which implies, since A and T, commute,

Idtf = Ttf + (MP) j) Id,[(-A)(Tt,f)]ds

Since A is M-invariant, we are in the situation of Lemma 2.1.3, which implies that

Vtf = Idtf + (MP) 0t) Id,(i (-A))(V-,)]ds =

=f + (MP) [0t)(T + A)(Vf)ds (2.77)

Q.E.D.
An example of a Markov process which satisfies Theorem 2.2.2 is a Poisson process.
Another interesting class of processes which satisfy Theorem 2.2.2 consists of Markov

chains with differentiable transition probabilities. We will describe this case more

carefully here.
Let E = {1,2,...,n} and E = 2E. In this case B = b is equal to R", and

B+ = bpE is equal to

R+ ={(x,...,xn) E R" Ix > 0, for every}

However M = B* is also equal to Rn, and M+ = R+. Notice that M = B(R") and

M+ = B(nR"). To distinguish between measures and functions we will use notation

f, g, .., f = (fi, fn) for functions, and ,., = (L1,..-, in) for measures. Of
course I = I({i}), and f, = f(i), and (i, f) = =1C ifi. In this case Tt is a matrix

with elements pi(t), i,j E E, and the measurability condition, on (t, x) pt(z, dy),

is equivalent to the condition that, for every i,j, t -4 pij(t) is measurable. By

Theorem 2.2.1, the Dawson-Watanabe supersemigroup exists already in this case,

but we would like to consider the cases in which requirements for Theorem 2.2.2 are

fulfilled. Since we are in a finite dimensional case, these requirements are equivalent

to the condition that (Tt) is a Feller semigroup, which is equivalent to the condition

that, for every i,j,

limp,,(t) = 6j. (2.78)

In the following we will restrict our attention to such semigroups (Tt). In such a

case there exists a matrix A, an infinitesimal generator, such that A = (aj)n and

Tt = exp(tA), and

a, > 0, for i Z j, Eai = (2.79)
j=1
Compare this condition to the end of section 2.1.

A good reference for these results is [5]. In [5] a transition matrix (Tt), which

satisfies our conditions, is called a standard transition matrix, and the matrix A is

called a Q-matrix. We will refer to (Ti) as a standard transition matrix, but to A as
an infinitesimal generator (we use Q for supersemigroups).
Typical example of a Markov chain which has a supersemigroup, but does not
have a standard transition matrix (and, therefore, does not satisfy Theorem 2.2.2)
is, for n = 2,
Tt= 0 ; t> (2.80)

2.3 Kac Semigroup and Linear Case

In this section we will describe Dawson-Watanabe superprocesses over very gen-
eral class of Markov processes with linear branching mechanism. In this case Dawson-
Watanabe supersemigroups are related to a familiar notion of Kac semigroup, and
they can be used as upper and lower bounds (in some sense) for the general case of
branching mechanism.
Let (() be a Borel right Markov process on a Lusin state space (E, ) with
transition probability pt(x, dy) (see [54]). In this case E is the a-algebra of Borel
sets, ((t) is a.s. right continuous, and (t, x) pt(x, dy) is measurable. We will also
assume that, for the corresponding Markov semigroup (Tt), Ttl = 1. Let ck bC be
a bounded measurable function (not necessarily nonnegative). We define, for every
GE 9,
kt(x, G)= EX [IG(t) *e- fk()du (2.81)

Then, by monotone class arguments, kt(x, dy) is'a positive kernel, such that (t, x) -
kt(x, dy) is measurable. This kernel defines a semigroup ( Kt; t > 0) given, for every
f bpE, by

(K f)(x)= f (y)k(x, dy) = E' [f( () -e- fo k()d] (2.82)

Notice that (Kt) is not necessarily Markovian, since k can be negative, but (Kt) is
bounded by exp(I|kl t) and is positive. From Example 2.1.2 we obtain that (Tt) and

(Kt) are both positive, M-Pettis integrable semigroups (in the sense of Definition
2.1.3), where M is the set of all finite signed measures on (E, E).
Recall (see, for example, [49], Chapter 3) that, in the case of Brownian motion
and k > 0, the semigroup (Kt) satisfies, so called, Kac (or Feynman-Kac) formula.
In this work we will refer to the semigroup (Kt), defined by 2.82, as a Kac semigroup.
Consider now a linear operator B : bE -- bC, defined by

(Bf)(x) -k(x) f(x) (2.83)

It is obvious that B is an M-invariant operator and an SOLP-operator (see Definition
2.1.4). We claim that (Kt) and (T() satisfy the conditions of Lemma 2.1.3. It is enough
to prove the following lemma:

Lemma 2.S.1 For every f E bE and x E E,

(Kf)(x) = (Tf)(s) + j K,[B(T.,f)](x)ds (2.84)

Proof. Using 2.82 and Markov property, we obtain

(Kf)(x) = E' [f(t) e- ()du ]

= Ex [f(6) (I k(u() e- fo )d du)] =

(Tf)(x) + E [-k.(() f (te) e- lo'(d du=

= (TfX)(x + ) E [- k(G) f () e- ufo ( )/,] du =

= (Tf)(x) + E E- [-k() E' [f(&-u)] e-fo(.)d] du =

= (Tf)() + 'K.[-k. (T.._f)](x)du
Jo

Q.E.D.

Consider now a branching mechanism %Pl(x, A) = -b(x)A, where b E bV. Let

us denote the corresponding solution of 2.61 by (Wt; t > 0). Let (Kt; t > 0) be
the Kac semigroup, where k = b. By Lemma 2.3.1, we can apply Lemma 2.1.3 on

these semigroups. Since T1 (-bf) = 0, we get Wt = Kt, for every t > 0, i.e., Kac

semigroup is a solution of 2.61.

Remark 2.3.1 One can think that we have proved already in Lemma 2.3.1 that (Kt)
is a solution for 2.61. But it is not so. Compare 2.61 (i.e., 2.28) with 2.84 (i.e., 2.41),

and notice that semigroups appear in different "order" under integral. In many cases

it will be the same solution, but one has to prove it.

However, the solution of 2.61 for l1(x, A) = -b(x)A can be computed directly,
and this is a known result (see [28]). We just wanted to emphasize that it is also a

consequence of our general method developed in section 2.1.0

Once we know that (Kt) is a solution of 2.61, it is easy to construct the correspond-

ing superprocess. We will introduce some notation first. For a measure p E M+,

we denote by A, a point mass at p. Of course, A, is a probability measure on

(M+,M+). As usual, the point mass at x E E we will denote by 6,. So, 6, is a

measure on (E, ), i.e., 6, E M+, and it make sense to consider A6,. For t Ec M+

we denote by iKt a measure on (E, E), defined by,

(Kt,f) = (. ,Ktf) = fE E [f(t) e- ob(')du] (dx) (2.85)

Obviously, pKt M+. Formula 2.85 immediately shows that we can describe the

linear case, %1l(x, A) = -b(x)A, completely, i.e., that we proved the following theorem.

Theorem 2.3.1 (linear case) A Dawson-Watanabe superprocess, over (t) with branch-

ing mechanism @1(a, A) = -b(x)A, is a deterministic Markov process (Xt), defined

by Xt = XoKt, where (Kt) is a Kac semigroup (with k = b). The supersemigroup is
given by the formula
Qt(, dv) = AKt (dv) .(2.86)

Consider now a general form of branching mechanism fi(x, A), given by 2.55. It
can be rewritten in the form 2.58. Let XI2(X, A) = b-(x)A be negative of the positive
part of T. Let (Kt) be the Kac semigroup, where k = -i-. Let (Vt) be the solution
of 2.61 for i, and Qt(p, dv) the corresponding supersemigroup. It has been proved
in [28], p.342, that the first moment of Qt(,, dv) is given by

f f)Q ,d) = (,E [ 0f( te- f0(S)ds]) (2.87)

These facts lead to one more application of Lemma 2.1.3, which gives results stronger
than 2.62 on bounds of (Vt).

Theorem 2.3.2 For every t > 0, x E E, and f E bpE, the following properties are
satisfied:
0 < (Vf)(x) < (Ktf)(z) = E" [f( t) eo-()d'] (2.88)

(Vtf)(x)= 0 -= (Ttf)(z) = 0 (Ktf)(x) = (2.89)

In particular, if b- = 0, then 0 < (Vtf)(x) < (Ttf)(x).

Proof. Consider T %I2. It is a branching mechanism again, but with the property
that, for every x E E and A > 0, (i I2)(X, A) < 0. By Lemma 2.3.1 (Tt) and (Kt)
satisfy the conditions of Lemma 2.1.3, which implies that

(Vtf)(x) = (KWtf)(x) IK,[-( '2)(V_,uf)](x)ds (2.90)

Since K,, -(%P %i2), and Vt_, are all positive, it follows that (Vtf)(x) < (Ktf)(x),
which proves 2.88.

To prove 2.89 we should notice first that (Vtf)(x) < (Kf)(x) < (Ttf)(x)
exp(1b- 11 t), which proves that

(Ttf)(x) = 0 =, (Ktf)(x) = 0 ==, (Vtf)(x) = 0

We have to prove only that ((Vtf)(x) = 0 == (Ttf)(x) = 0). If (V f)(x) = 0, then,
by 2.52, (v,f) = 0, Qt(6,, dv) (a.e.) (recall that (6,,Vtf) = (Vtf)(x)). It follows
that the first moment of Qt(6,, dv) is zero. By 2.87, we obtain that

E' [f(6t)e- fo'(,)d] =0 .

Since b is bounded, the exponential factor is strictly positive, and

f(t) e- f~6(')da = 0, P' (a.e.)

It follows that f((t) = 0, P_ (a.e.), i.e.,

0 = Ex [f(6,)] = (Ttf)(x)

Q.E.D.

Remark 2.3.2 It is a good place to comment on additional requirements on superpro-
cesses assumed by some authors. In some of his papers E.B. Dynkin (see for example

[18]) require, what seems to be a "natural" probabilistic condition, that, for every
BE ,
Probability{(t E B} = Expectation[Xt(B)]

Recall that (Xt) is a measure-valued, so above formula makes sense. By monotone
class argument, it is equivalent to

(~, Ttf) = M(, f)Qt(p,dv)

However, formula 2.87 shows that it is the case if and only if b = 0, but not always.
In particular, it is true for T(x, A) = -A2, which is the case often studied by Dynkin.
Despite these facts we will continue to study the general form of %.01

2.4 Feller Condition and Infinitesimal Generator

Regularity properties of superprocesses have been studied by many authors. In

the case of a compact metric space E, and a branching mechanism T with constant

coefficients, the problem has been treated already by Watanabe [56]. His approach

deals with Feller properties of semigroups involved. Since M+ is, in this case, a locally

compact space, we can deal with Feller property in a nice way. However, very often

we deal with Markov processes that are defined on a locally compact Hausdorff space

with a countable base. In such a case, M+ is not necessarily locally compact. Several

authors offered ways to resolve this situation (see [9], [34], and [52], for example).

Usually authors adjust the conditions according to their special needs.

In the following text it is general enough, for us, to treat the case when (t) satisfies

Feller property. We believe that the most "natural" way is to extend our process onto

the one-point compactification of the state space E, to develop a superprocess over

it which has Feller property, and to show that such superprocess actually "lives" on

the set of measures on E. Of course, the Feller property of the superprocess on the

one-point compactification is the result of Watanabe. But, it follows also from our

general method, developed in section 2.1., as a special case.

Let us mention that the problem of regularity properties has been treated in a

very general form in [28]. It is shown there that the superprocess is a right Borel

Markov process, whenever (6t) is a right Borel Markov process with Lusin state space

(E, 8), and T satisfies condition on the second moment (see Remark 2.2.4). The

temporally non-homogeneous case was treated in [17], and generalized even more in

[21]. We do not need so general a Markov process (6t), and, on the other hand,

it is useful to have Feller property for superprocesses, that is why we will take the

approach suggested above.

Let E be a locally compact Hausdorff space with a countable base, and E = B(E)

a-algebra of Borel sets. Notice that (E, E) is a standard space whose topology is
specified and is not necessarily compact. Let EA = E U {A} be the one-point
compactification of E, and A = B(Ea) a-algebra of Borel sets (we leave E intact
if it is already compact). On both E and EA we consider metric topologies. Recall
that E = EnSa.
As usual, Co(E) is the set of all continuous functions on E which vanish at infinity,
and C(Ea) is the set of all continuous functions on Ea. Co(E) can be regarded as a
subset of C(Ea), and, for every f E C(Ea), f f(A) E Co(E).
Let M be the space of all finite signed measures on (E, E), and MA the space
of all finite signed measures on (Ea, Ea). By the Riesz theorem M = Co(E)*, and

MA = C(E,)*. We will be interested in M+ and M'. Then M+ is a locally compact
Hausdorff space with a countable base, with respect to Bernoulli convergence, and
MA = B(MA) (see Remark 2.2.1). However, M+ is not locally compact (it is Polish
in the topology of vague convergence). We can regard M+ as a subset of M+ in the

sense that

M+ = {E M+ I (f{A}) = 0 (2.91)

Actually M+ is a Ga-subset of M+, and M+ = M+ n MA.

We assume throughout this section that (t) satisfies a Feller property, i.e., that

its Markov semigroup (Tt) is a Co-semigroup on the Banach space Co(E), and that

TtlE -- 1. Then ( can be regarded as a Hunt process (0, T, .F, t, Ot, P)) with infinite
life-time, and A as a cemetery (see [4] pp.44-50). Recall that in this situation we

extend (Tt) to C(Ea), by

(Tf)( (Tt f/E)(x) x E
(t ) f(A) X (2.92)

Notice that (Ti) is a Co-semigroup on C(EA). Using Example 2.1.1, we conclude that
(Tt) ((Tt)) is a positive, M (Ma)-Pettis integrable semigroup.
We have to restrict our T, too. We assume that b(x) and c(x) are continuous func-
tions which have limits at infinity, i.e., they can be extended to continuous functions
b(z), 6(z) on Ea. We assume also that there exists a kernel fi(x, du) from (Ea, EA)
to ((0, +oo),B((0, +oo))) such that, for every x E E, i(x, du) = n(x, du), and, for
every a : (0, +oo) --- [0, +oo), a(u) < u, and a continuous, the function

R+ x E 3 (A, x) a(Au)ia(x,du) (2.93)

is continuous. If we take a(u) = u, we get that 2.53 is a continuous function on EA.
If we take a(u) = 1 e-", we obtain that, for every f E C(EA)+, ', defined by,

(f)(x) = -b(x)f(x) (x)[f(x)]2 + (1 e-f(")")u(, du) (2.94)

is continuous on EA. It shows that T (W) is a SOLP-operator on Co(E)+ (C(EA)+).
Notice that in the case when b, c, and n do not depend on x, all these conditions are
satisfied.
By Theorem 2.1.4 there exist (Vt), Vt : Co(E)+ Co(E)+, and (Vt) a solution
of 2.28 for I, and (V), Vt : C(Ea)+ -. C(Ea)+ and (V) a solution of 2.28 for 4.
By the uniqueness Theorem 2.1.3 and 2.92 both (Vt) and (V) are restrictions of the
solution of 2.28 for 4' on bpE, and they are related by formula
= (Vt f/E)(x) x E (295
)x) = f(A) + f '(A, (V8,(f)(A))ds x = A

In particular, it shows that (V(f)(A) = (VOg)(A), whenever f(A) = g(A), and
(Vtf)(A) = 0, whenever f(A) = 0. Also, Example 2.1.1 shows that all the functions
involved in 2.28 are Bochner integrable, and, therefore, for every f E C(Ea)+,

Vtf = Ttf + (B) (ot) T ,[(,8-f)]ds (2.96)

Hence, the mapping t -+ Vtf is strongly continuous, and, by 2.94, t '-4 Vtf, for
f E Co(E)+, is also strongly continuous.
Using Theorem 2.2.1 we know that there exists one and only one supersemigroup

Qt(p, dv) which corresponds to the solution of 2.28 on bpA. Since C(Ena) C bpEa,
it is true that, for every f E C(E)+,

M+ e-()Qt(I, dv) = e-(') .(2.97)

Notice that in 2.97 it was important that M' is equal to M+ which corresponds to
bpn, as it was defined in section 2.2.
Consider the functions Ff : M+ R, f E C(Ea)+, defined by,

Ff(J) = e-(^'f (2.98)

and the family of functions Exp(M+), defined as the linear span of the family

{ Ff I f C(Ea)+, f strictly positive } (2.99)

Since f is strictly positive, Ff E Co(M+) (which makes sense, since M+ is a locally
compact space). Since Ff F, = Ff+, Exp(M+) is a nonempty algebra, which sep-
arates points in M+, and Exp(M+) C Co(M+). By the Stone-WeierstraB3 theorem,
the closure of Exp(M+) is Co(M+).
Recall that, by Theorem 2.3.2 b), Ttf strictly positive implies that Vtf must be
strictly positive. If f G C(EA)+ is strictly positive, then, since Ea is compact, there
exists a constant a > 0, such that, f > a. Tt is monotone, implies that Ttf > Tta a.
Thus, Ttf is strictly positive, and, therefore, Vtf is strictly positive, too. In particular,
if Ff E Exp(M+), then F1,! e Exp(MA). Using 2.97 we obtain

Qt(Ff) = Ff.f E Exp(M+)

(2.100)

Since Qt is a bounded linear operator, and Exp(M+) is dense in Co(M+), we conclude

that

Qt(Co(M )) Co(M ) (2.101)

Recall that Vf --- f strongly, when t -- 0+. It follows that, for every p. MA,

lim Qt(Ff)( l) = lim (FV,)(p) = F () (2.102)
t- O+ t-0+

Since (Qt) is a contraction semigroup, (QtF)(/I) ---- F(), for every F E Co(M+).

In other words, we have proved that (Qt) satisfies the Feller property. In the case of

E compact, and b, c, n constants this result has been proved in Watanabe [56]. Sev-

eral authors proved the same result later under various assumptions, but essentially

Watanabe's method inspired most of these proofs. In this work, we just proved that

Watanabe's result is a consequence of our general perturbation theory from section

2.1. and Theorem 2.2.1. However, in section 2.1.we were also inspired by Watanabe's

method. Our version of Watanabe's theorem says the following:

Theorem 2.4.1 If a Markov process ( with a locally compact Hausdorff state space

E with a countable base satisfies Feller condition, and a branching mechanism 41

satisfies 2.93 and 2.94, then (ft, j)-supersemigroup (Qt) satisfies Feller condition on
a locally compact Hausdorff space M+, which has a countable base.

Notice that this result guarantees the existence of a superprocess (Xt) with nice

regularity properties. By the basic construction from the theory of Markov processes

(see [4], pp.44-50), there exists a Hunt process (Xt) with state space MA, which is a
superprocess over (t) on EA with branching mechanism '. (Xt) has infinite lifetime,

since

Qtl,+ = 1

In some sense (Xe) is a superprocess over ({) and %F. But, it has values in
MA. We would like to restrict it to M+ somehow. Notice that there is a (Tt, T)-
supersemigroup Qt(j, dv) on M+, when we consider (Tt) and T on bpE. By 2.91
pL E M+ can be regarded as Az E M+, with Cj({A}) = 0. Thus, for every 1L E M+
and f E Co(E)+,

IM e-("f)Qt(p, dv) = e-('f) = by 2.95 =

= e-(',v'-)= e-(v')Qt(p, dv)

Since the Laplace transform uniquely determines measure Qt(P, dv), it follows that,
for every i E M+ and t > 0,

Qt(p,M+) = 1, and Qt(pl,dv)/M+ = Qt,(, dv) (2.103)

In particular, it shows that, for every t E M+ and t > 0,

P" [Xt M+]= 1 (2.104)

Although (Xt) is right continuous, we cannot conclude immediately that the super-
process (Xt) "lives in M+, if it starts there", since M+ is not closed in M'. However,
the statement is true, as we will show in the following theorem.

Theorem 2.4.2 For every yL E M+,

P" Xt E M+, for every t > 0 = 1 (2.105)

Proof. By our assumptions 2.93 and 2.94, b is a bounded function, thus, there
exists / = [Ibll > 0. Let f = la be a characteristic function of {A} C EA. Then F,
defined by F(() = (t, f) = ((Az) is in Mt. Using 2.87 we compute

(e-'tt)(F)(p) = e- *(t,EX [f/( f)e- f(()d.]) <

< (i,,E" [f(6t)]) = (/, TtlA) = ( L, 1A) = F() .

By the dominated convergence theorem

lim E" [f( )e- fo' =

= E" [f(o)] = (TtlA)(x) = l (x)

Notice that f(t) = 1(t=A), {A} is closed, and A is a cemetery. It shows that

lim f((t) = f(0o) (a.e.),
t-0O+

which implies the statement about the limit above. By the dominated convergence

theorem

lim (e-PtQ)(F)(j) = F(p)
t-*O+
Hence, F is /-excessive (relative to (Xe)). It follows, see [4], Theorem 2.12 b), that

almost surely the mapping

t F(Xt)= (Xt, 1) = Xt({A}) (2.106)

is right continuous. By 2.104, we get X,({A}) = 0 a.e., for every q Q+. Using

2.106 we obtain that almost surely the mapping

t Xt({A})

is identically zero, which proves the theorem.

Q.E.D.
The fact that (Tt) satisfies Feller condition, i.e., that (Tt) is a Co-semigroup on

Co(E), and (Ti) is a Co-semigroup on C(EZ), introduces another powerful "tool" into
consideration. It is the infinitesimal generator. Let A be the infinitesimal generator

of (Tt), defined on its domain D(A) C Co(E), and A the infinitesimal generator of

(Tt), defined on its domain D(A) C C(EA). Recall that f E D(A) if and only if
f f(A) E D(A), and

(Af)(x) ( A(f f(A))/E(x) x E E
(Af)() A(2.107)

Denote the infinitesimal generator of (Qt) by 9. Consider the family of functions
Exp(M+, D(A)), defined as the linear span of the family

{ Ff f E C(Ea)+ n 9(A), f strictly positive }. (2.108)

In [56] Theorem 2.4 and Lemma 2.2, Watanabe proved that (in the case of constant
coefficients b, c, n) Exp(M', D(A)) is a core for 9, and he also computed G(Ff).
Since exactly the same proof works in our case without any difficulties, there is no
need to repeat the proof here. We conclude that the following statements are true.

Exp(MA, D(A)) is a core for (2.109)

For every f e C(Ea)+ n 7D(A), f strictly positive, and for every t > 0,

Vtf e C(Ea)+ n 7D(A), Vtf is strictly positive. (2.110)

For every f E C(Ez)+ n D(A), f strictly positive,

g(F )(A) = -e-,,). (A, Af + f) =

=-(Ff)() (p, Af -bf f+ (1 e-f")fi(, du)) (2.111)

For the martingalee problem"-type characterization of the infinitesimal generator see
[28] pp.338 and 354.
It is now an easy Corollary of these results, to give a complete characterization
of infinitesimal generators of superprocesses over finite Markov chains with standard
transition matrix. Let ((t) be a Markov chain with state space { 1,2,... ,n} and a

standard transition matrix (Tt) with infinitesimal generator A, described by 2.79.

Notice that our state space is already compact, so there is no need to use one-point

compactification. Our branching mechanism is determined by b, c, and n, where now

b = (bl,...,bn) E R", c = (cl,...,c,) E R+, and n = (nm(du),...,nn(du)) where

each measure ni(du) satisfies condition

Suni(du) < +oo (2.112)

There are no other restrictions on %.

In this case A = A and 2D(A) = R". Notice that, for every f E R+, and a function

Ff :R+ n R+,

(Fi)(yx) (,, Af) = e- CZSift fj ajk) =
j=1 \k=1

j=1 k=1 j= J \ ,=1k 8
= -(p, A VF) .

Similarly we obtain

aF
(Ff)() (, bf) jb-- = -(, be VFf)
j=1 "91Lj
and

(F)(,Z) Ic/2)= nC," 2 F
j=1 "lJ
The last term in 2.111 can be rewritten as,

(F)(). (, + (1 e-f")n(i, du) =

= E 1 (F1(A + ue,) Ff(A)) ni(du)

where e = ( 0,...,0, 1, 0,...,0). Using Theorem 2.2.2, results about Kac semi-

group, and results in this section (recall that exponential functions form a core for

G; and G is determined on its core), we can summarize properties of superprocesses
over Markov chains in the following theorem.

Theorem 2.4.3 A superprocess (Xt) over Markov chain (t) with branching mecha-
nism VT is a Markov process which satisfies Feller condition, and is actually a Hunt
process with infinite life-time whose state space is R The Laplace transform of a
supersemigroup (Qt(/i, dv)) is given by

R e- E=1F IQ(('i,... ,), (dvi,... d.. )) = e- s(Vl)., (2.113)

where

(Vtf) = p,i(t)fj+
j=1

+ j P(t s) -b(Vf)j c(VCf) + (1 e-( "),)n,(du) ds =

ot "
= f, + ds E pj(t s) (2.114)
0 j=1
+o00 n
[-b(Vf), c(Vf) + (1 e-(Vf)i")n,(du) + E aj,(V f)k
i 10 ck=1
The infinitesimal generator g of (Xt) is given by

(GF)(i) = (2.115)
[ r 2F 9F +0
= b3 + (A VF) + [F(p + uj) F(p)]nj(du) .

In the special case, when T = -bA, (Vt) is the Kac semigroup and the infinitesimal
generator is

(GF)(I) = (l, A VF b VF) (2.116)

Let us show, in the example below, that even on the level of Markov chains, the
superprocess (Xt) over (t) does not necessarily satisfy Feller condition, if (&t) does
not satisfy Feller condition.

Example 2.4.1 We consider n = 2, and (Tt) given by 2.80, i.e.,

Tt=[ 0 1]
0 1

Let ('9f); = -bfi, i = 1,2; where b E R. Using results about Kac semigroup it is
easy to check that
(Vtf)i = e-b(Ttf)i = e-btf (2.117)

and

Qt(((A, 1,2); (dv1, dV2)) = A(o,(Al+,2)e-b)(ddv, dv2) (2.118)

which gives a superprocess Xt = (X'X,X2) to be

(Xt, X2) = (0, (XJ + Xo)e-bt) (2.119)

and it is obvious that such a Markov process does not satisfy Feller condition.0

2.5 Branching Property

In this section we will present some easy consequences of the fact that super-
processes satisfy a branching property. Branching property in this form, i.e., in the
continuous state space has been introduced by M. Jifina [36].
Notice that the additive structure of M+ enables us to introduce a convolution
operation for measures on (M+, M+). If Q1 and Q2 are probability measures on
(M+, M+), then Q x Q2 is a probability measure on (M+ x M+, M+ 09 M+). Since

(/, v) pt + v is M+ M +/IM+ measurable, there is a unique measure, say Q1 Q2
on (M+, M+), defined by

(Ql Q2)(H) = M+ lx+ H( + V)(Qi x Q2)(d,dv) (2.120)

where H E M+. As usual, we will call the measure Qi Q2 the convolution of
measures Q1 and Q2. Notice that it is easy to check that the Laplace transform of

Q1 Q2 is the product of Laplace transforms of Q1 and Q2. Also, if X1 and X2
are independent M+-valued random variables with distributions Qi and Q2, then

X1 + X2 has a distribution Q1 Q2.

Using the fact that (I1I + Y2, Vtf) = (l, Vtf) + (Z2, Vtf) we conclude that a

Dawson-Watanabe supersemigroup (Qt(p/, dv)) satisfies the property

Qt(GI + 2, dv) = Qt(A, dv) Qt(P2, dv) (2.121)

The consequence is that, if (XA) and (X2) are two independent copies of Dawson-

Watanabe superprocesses over (t) with branching mechanism T, where X0 = k1 and

X02 = A2 then

Xt = X' + X2 is a Dawson Watanabe superprocess, Xo = 1I + Z2. (2.122)

Property 2.121 (and 2.122 as well) is a branching property, i.e., every Dawson-
Watanabe superprocess is a branching process.

Let us fix Az E M+. Then, for every a > 0, -ap E M+. The branching property

2.121 implies that a family

{Qt(al, dv); a 0} (2.123)

is a convolution semigroup (notice that t and p are fixed, and a E R+ is a parameter).

It shows that, once we know Qt(6, dv) for every x E E, we can use 2.123 to find

Qt(a6., dv), and then 2.121 to find Qt(I, dv) for every 1L of the form

.= > aS, (2.124)
i=l
Let us also mention here that in the case of standard space (E, E) and correspond-

ing space (M+, MA), it has been proven in [18], pp.264-265, that, if CQ,(f) -- (f)

as n -* +oo, for every f E bps, then there exists a subprobability measure Q, such

66

that CQ = L, where denotes the Laplace transform. In particular, if (L,t, Vtf)

(/i, Vtf), for every f E bpE, then

CQt(m,dv) Qt(p,Idv) (2.125)

CHAPTER 3
ONE-DIMENSIONAL CASE

In this chapter we describe the simplest possible superprocesses, i.e., the super-

processes over Markov chain with one state space. Since these superprocesses are

one-dimensional branching processes, they are recruited among processes which were

studied before. We use one-dimensional superprocesses then to describe general su-

perprocesses over deterministic Markov processes which stay in the initial state for-

ever (but the state space of the Markov process is arbitrary). In this way we obtain

the "behaviour" of the superprocess which is "influenced" by the branching mecha-

nism part, only. Then, in the next chapter, we show how in some cases of Markov

processes we can get a description of the corresponding superprocesses by using the

results of this chapter. It should be emphasized that our main tool is the analysis of

the supertransition function.

3.1 Supersemigroups and Infinitesimal Generators

We restrict our attention to the one-dimensional case in this and the following

section. Hence, we consider the case when E =. {1}. Then a branching mechanism

'! depends on A only, i.e.,

1(A) =-b cA2 + (1 -e-e")n(du) (3.1)

where b E R, c E R+, and f+'" un(du) < +oo. In this case we do not have much

choice for (st), i.e., (t = (0 = 1, Tt = Id,, and A = 0. Also M+ = R+, and JL and f

are nonnegative real numbers. Using now the results from section 2.4. in this special

case, we conclude that a Dawson-Watanabe superprocess over (t) with branching
67

mechanism T is an R+-valued Hunt process (Xt) whose infinitesimal generator g is
given by
( cd2F bdF f+
(GF)(dP) = 2 c F -b + [F( + u)- F(p)]n(duj) (3.2)

The supersemigroup (Qt(j, dv); / > 0) is a convolution semigroup with respect to
1L, whose Laplace transform is determined by (Vt; t > 0), where, for every f > 0,

(Vf) = f+ f [-b(Vff) c(Vf)2 + f (1 e-(f)u)n(du) ds. (3.3)

Since we know that Vtf exists, 3.3 shows that t Vtf is a continuous function, and,
again by 3.3, t -* Vtf is a cntinuosly differentiable function on R+, which satisfies
the differential equation

f d() = -b(Vtf) c(Vtf)2 + fo+( i e-(V'f))n(du)
(Vof) f
Let us consider some cases in which we can explicitly solve 3.4.
Case T(A) = -bA cA2; c > 0, b E R.
In this case the differential equation 3.4 is a Riccati type equation

Sy' = -by cy2(3.5)
y(0) = f 3
which can be easily solved, by separation of variables, to give

f
Vtf =ebt (3.6)
e6' + fc '
where (ebt 1)/b is defined by

ebt t { 1 etif bi 0
b (3.7)
b limb-o 1 = t if b = 0
Hence, in the case when b = 0 we have

f
Vtf (3.8)
1 +fct

Using the fact that Vtf describes the Laplace transform of Qt((, dv), and by following

simple computation

-(, Vf) = /)
est + fCe
bt

c t eb1
b
c-X- c----_ f C+

we obtain that (recall that 4 and v are points in R+ here),

-" Fe^ Ga)+(dv) (3.9)
n=I "1
Qt (, dv)=e -z- 6o(dv)+Z

where

~ EXP eb- (3.10)

where, as usual, EXP(A) denotes the exponential distribution with parameter A > 0.

The infinitesimal generator of this process is (by 3.2)

d2 d
c p2 by (3.11)

If we compare these results to Corollary 2 of [44] we get the following simple conse-

quence:

Corollary 3.1.1 (Xt) is a one-dimensional superprocess with continuous paths if and

only if it is a one-dimensional continuous branching diffusion. In this case (Xt) has

a transition probability given by 3.9 and infinitesimal generator given by 3.11.

Remark 3.1.1 It should be noted that these processes have been studied before, and

it was observed that they represent superprocesses. Here we have shown how these

results can be derived in very simple and elementary way following our approach.

Also, the transition function is not usually mentioned in the form 3.9 in the literature,

and in the literature on superprocesses is usually not mentioned at all. However, in

our line of approach we will rely heavily on a transition function.

The diffusions with infinitesimal generator 3.11 were studied first by W. Feller in

1939 and especially in 1951 [26]. At that time continuous state branching processes

were not introduced yet, and Feller has shown that these diffusions are the limits of

discrete parameter branching processes. He treated them as (0, +oo)-valued diffusions

with 0 as an absorbing boundary. We include 0 in the state space, and because of

that we have a singular first term in 3.9. Feller computed also a density part of the

transition function 3.9 in terms of Bessel functions. The case c = 1/2, b = 0 is treated

in [42] pp.100-102, again from the diffusion point of view. In this case the transition

function has a nice form

Qt(I, dv) = e t o(dv) + 2e- t 1(4 ) (3.12)
t t V

where I1 is the modified Bessel function of the first kind. We will not use 3.12 any

more. It is mentioned here just to complete the description of the processes above.

In 1958 M. Jirina introduced continuous state branching processes in [36]. In 1967

J. Lamperti covered the one-dimensional case in [44] and [45]. He gave us formula 3.6

(for the first time as far as we can tell). A nice brief account of these processes from

the point of view of branching processes is given in [1] pp.257-261. Several authors

on superprocesses, like D. Dawson, S. Roelly-Coppoletta, etc., emphasized that these

diffusions (or some of them) are special cases of superprocesses. Almost exclusively

they deal with the infinitesimal generators. O

Case k(A) = m(1 e-a ); m > 0, a > 0.

This is the case when n(du) = m8,. Again, the differential equation 3.4 is easy

to solve, since it is in a form

y'=m(1- e-) (3.13)
y(0) = f

Separation of variables gives the solution

Vtf = In [ea(eaf- 1) (3.14)
a

Simple computation shows that

e_(,vt = = e-m f[ e-f(- -
[eamt(eaf 1) + 1]/a eMte [1+ e- 1]-

Since e-af(e-am 1) E (-1, 1) we can apply Newton's binomial formula to get

e-(,'f) = e-te-~mt e-af(-amt 1) =
n=O

n ( a ) e-J^mt/-amt 1) en-f(*+na)
It follows that the supertransition function is given by

Qt(, dv) = ( e-m't(e-am' 1)"6,+,(dv) (3.15)
n=0
The infinitesimal generator of this process is (by 3.2)

(CF)(I) = m4[ F(I + a) F() ] (3.16)

Remark 3.1.2 There are descriptions of these processes in [44] and [1], but they are
treated mostly as time-changed Poisson processes (see more about it in the next

section). However, we are not aware of formulas 3.14 and 3.15 in the literature. Here

they follow easily from the general approach on superprocesses. Let us also mention
that in the case a = m = 1, and = 1, the supertransition function 3.15 is given by

Q(1l, dv) = E e-'(1 e-t)"6i+n(dv) (3.17)
n=0

which is a geometric distribution

It is a good place to clarify certain facts about simple transformations of superpro-
cesses. Let (Xe) be a Dawson-Watanabe superprocess over a Markov process ()) on

(E, E) with branching mechanism W(x, A). We consider general LCCB space (E, E).
Let f > 0 be a bounded, nonnegative, E-measurable function. Then the stochastic
process (Yt), defined by

Yt = (Xt,f) (3.18)

is an R+-valued stochastic process. One may hope that (Yt) is a one-dimensional
superprocess somehow related to 9. This is particularly so in the case when f = 1,
i.e., (Yt) is the, so called, total-mass process. However, this is not always the case.
The following two examples indicate typical difficulties.

Example 3.1.1 Let ((t) be the uniform motion to the right on the real line, and
T(x, A) = -A. Let f = 111,+0). Using Theorem 2.3.1 and formula 2.85 we obtain

Y = (X, f) = XXo(dx) E- [f(t)e- foI()du

= Xo(dx) (e-f(x + t)) = e-' Xo([l t, +oo))

Hence, for every Xo =p_ / 0, where supp(/p) C (-oo, 0), we will have that Yt = 0,
for 0 < t < 1, and Yt > 0, for t large enough. But, there is no one-dimensional
superprocess with such a behaviour. For every one-dimensional superprocess it is
true that if it starts at zero, it stays there forever, since the Laplace transform of

Qt(O, dv) is identically one.O

Example 3.1.2 Let (t) be again the uniform motion to the right on the real line, and
k (x, A)= -b(x)A, where

--1; x <.-1
b(x)= xi ; x (-1,1)
1; x>l 1

Let f 1. Using Theorem 2.3.1 and formula 2.85 again, we obtain

Yt = (X,,1) = (Xo, E [e- f'b()d .

If Xo =8 a R, then

Yt = E" e- fo b(])du = e- b(a+u)du

It shows that Yt is a deterministic process, and for a = 0 it is equal to
{ e-t2/2 ; t [0, ] (3.19)
y e e-1/2 + (t ) ; t> >191

However, there is no such one-dimensional superprocess. As we can see from 3.2

the only deterministic one-dimensional superprocesses are the processes of the form

Xt = Xoe-bt.O

Notice that in the first example the coefficients of T are constant, but f \$ 1,

while in the second the coefficients of q are not constant. Several authors considered

constant coefficients and f 1 (see, for example [52]). In this case (Xt, 1) "behaves"

as we would initially expect. For the sake of completeness let us prove this result.

Proposition 3.1.1 Let (E, E) be a standard space, and (Xt) a Dawson-Watanabe su-

perprocess over ((t) with state space (E, E), and with branching mechanism T1, where

T has constant coefficients, i.e., b(x) = b, c(x) = c, and n(z) n. Then (Xt, 1) is a

one-dimensional superprocess with the same branching mechanism T.

Proof. Notice that the statement makes sense, since 'I can be regarded as a

branching mechanism over {1}, as well. Let (Vt) be a nonlinear semigroup which

corresponds to (Xt). We claim that for every A > 0, Vt(Al)(x) is a constant function.

Recall, by construction, that Vt is obtained as a limit of

V,"+(Al)(x) = T(Al)(X) + T_.,[WP(Vn(A))](x)ds
Jo

where Vo = 0, and Tt(A1) = ATtl A. Assuming that V,"(A1) is a constant, we
get that I(V,"(A1)) is a constant function, since T has constant coefficients. Then

Tt_.,[(Vn(A1))] is a constant, too, since Tt-,l = 1. It follows that V"+1(A1) is a
constant function. By mathematical induction we obtain that x i- Vt(Al)(x) is a
constant function, for every A > 0, and it satisfies

Vt(Al) = A + + '[V8(Al)]ds

If we define WtA = Vt(Al), then Wt satisfies 3.3 (by the equality above). Since, for
A > 0 and Xo = L,

I+ e-A(x,1)dP = M e-(XtAl)dP = e-(,u'v(A))

= e-((E).Vt(Al) e(Xo,1)'Vt( ) e-(Xo,1)'Wt

we have shown that (Xe, 1) is a one-dimensional superprocess with branching mech-
anism T.

Q.E.D.

Remark 3.1.3 It is possible to study (Xt, f) also by using martingale theory approach.
We will not enter this subject. Let us only mention some references like [10] and [52].
Several results on semimartingale properties of

exp(-(X, f)) ,

where f is in the domain of the infinitesimal generator of (4e), are given in Chapter
5 of [10].E

3.2 Time Change and Zero-One Law

We have seen in the previous section that one-dimensional superprocesses are
actually one-dimensional branching processes studied by J. Lamperti in the late six-
ties. J. Lamperti has also shown that these processes can be obtained as a time

change of Levy processes. Here we will emphasize Lamperti's result and show that

related to this is an interesting zero-one law for Brownian motion, which in some

sense complement Engelbert-Schmidt zero-one law (see [23]).

Let k(A) be given by 3.1, and (L L) a corresponding Levy process given by 2.76.

Then we have

Lf = -bt + VcBt + Jt (3.20)

where (Bt) and (Jr) are independent copies of standard Brownian motion and pure

jump process with positive jumps, respectively (see 2.76 and [35] for details). In
particular, (Lf) is a strong Markov process with the infinitesimal generator given by

d d2 r0+m
b +cd + +[ F(x+ u) F(x)] n(du) (3.21)
dx dx2 Jo

Consider now only the cases when L =. a > 0, and let (Ltf) be the process (Lt)

stopped when it reaches zero. Then (L" ) is a Markov process with state space R+,

whose infinitesimal generator is given by 3.21 for positive x, and is equal to zero for

x = 0.

Consider the additive functional (A*) defined by

A? = f (3.22)

Notice that (A") is a continuous additive functional, except maybe at the point

o = inf{t > 0 : f = 0 }, which is equal to inf{t > 0 : L = 0 }. Obviously

AZ+ = +oo, but it is possible that A*_ < +oo. We will see later that it is so

indeed in many cases. Therefore, we define A' to be equal to A,0_. Hence, t A'

is continuous and strictly increasing on [0, T], and the same is true for its inverse.

We define the inverse of t A* to be equal to T0, for every s > A*. Once this is

settled the standard results on time-changed process apply. Therefore, let (Xt) be a

process obtained from (L'*) by a random time change, corresponding to the additive

functional (A'). Using Theorem 10.12 from [16] we obtain that (X/) is a Markov
process whose infinitesimal generator is given by 3.2, i.e., (Xlt) is a one-dimensional
superprocess described in section 3.1.

Remark 3.2.1 The result above belongs to J. Lamperti. It was announced in [44],
but as far as we know its proof was not published in author's later papers. Of course,
the proof is more or less standard application of random time change, and it was
outlined in [44]. One only has to be more careful with the possible discontinuity at
70, as we explained above. O

Let us investigate some cases in which A"_ is finite. Consider the standard
Brownian motion (Bt) on the real line. Let 70 = inf{t > 0 : Bt = 0 }. A consequence
of the following theorem is that A_ is finite if T(A) = -A2.

Theorem 3.2.1 (Zero-One Law) For every a > 0
a [ ds < oD ; if 0 < a <2
P"[J B< B < 0 ifa>2
Remark 3.2.2 Notice that Theorem 3.2.1 is an interesting result in itself. Recall that

the Engelbert-Schmidt 0 1 law implies (see [23])

po t ds < + ;Vt >0 1 if0 o IB-" 1B;1I = 0 ; if a > 1
which says something about the behaviour of the Brownian trajectory when it leaves
zero (or on the right hand side of zero). Our result 3.23 complements Engelbert-
Schmidt's result in the sense that it compares what is happening when Brownian
motion enters zero (or on the left hand side of zero). Obviously, the behaviour is
different. Notice also that our result is not the consequence of general criteria in [23],
since these result only show that for a > 1, and a > 0,

P1. I < +oo;Vt > 0 =0

which does not imply anything about the integral up to 70.3

Proof. Let Tb =inf{t > 0 : Bt = b} and, for n > 1, an = a/n. Then we have,

since pa(Bo = a) = 1,

o ds "=+i ds.
Jo d I pa a.e.,
a n=1 Ba'n

and the sum on the right hand side is the sum of nonnegative, independent (with

respect to P) random variables. Hence, it is P" a.e. finite or P" a.e. infinite.

Consider 0 < a < 2 first. If s < ,,Tl then B, > an+I P- a.e. It follows that

s < (1 7,+- n) pa a.e.
o Ba n=1 an+1
If we denote (1/a )( -,+ ra, ) by Zn, then { Z,; n > 1 } is the sequence of non-

negative independent random variables. Applying Kolmogorov three series theorem

we obtain that

Pa 1 Zn < +00 = 1 ,

n=l 1
if

^P (Z,> 1) <+oo ,

and

ZE [Zn z n=l
Recall that P" distribution of Zn is equal to Po distribution of (1/aI+1).Tr/(n(n+l)).

It follows that

pa (Z +00 n(n^e) n^
Z> =(n) -e 2s ds <
P"(Z,,> 1)= C +;), 2 e _"27rS
a -+ ds 2a (n + 1)"/2
Sv27n(n + ) J() s/2 v n(n + 1) aa/2
and the expectation

E Z8 -(. ] = a s + e 2. ds <

a(n + 1)" S d = i2- a(n + 1)' a'12
2n(n + 1)a s3/2 s V n(n + 1)a (n + 1)F/2
Since a < 2, then a/2 = 1 e, for some c > 0, and

0 (n + 1)1-' "0 1
: =- <+ 00
n=l n(n+ 1) += n(n +l)"

Hence, the two series mentioned above are convergent, and we finished the proof of

the first part of the theorem.

To prove the second part, it is enough to prove that

-ro( ) ds
B() 00

for w E F, where Pa(F) > 0. Using the well-known L6vy's modulus of continuity

P" limsup sup I B,- 1 = 1 ,
-o o_<,
and the fact that pa( ro < 1) > 0, we can choose F to be the set of such w that

ro(w) < 1, Bo(w) = a, and there exists 8(w) > 0, such that for every e < 6

sup B -B <2
O
If t2 = r7o() and tj = t2 e, then it follows, since Bt,(w) = 0, that, for every e < 8,

()_ <2(w) 2
/2e ln(l/e)

Hence, we obtain
-*(o) ds o(W) ds
B(Wj) (w)-S(W) B

1 f( ) de 1
S c-n(/e) [- ln(ln(l/e))] I(= +00

for every w E F.

Q.E.D.

With these observations we finished our description of one-dimensional superpro-

cesses. Our idea is to use these results in describing more general cases. First we will

describe (Idt, ))-supersemigroups over the general space (E, E). To do so we have

to introduce some special measures on the space of measures on (E, ). This is the

content of the following section.

3.3 Measures on Rays

Let (E, ) be a measurable space, and M the space of all finite signed measures

equipped with a-algebra M, which is the smallest a-algebra such that, for every f

bounded E-measurable, the mapping pt -* (tz, f) is measurable. M+ is the cone of

(positive) finite measures, and M+ = M+ n M. As before, bE (bpE) denote the set

of bounded (bounded and positive) E-measurable real-valued functions.

Consider t,(da), a positive kernel from (E, ) to (R+, B(IR+)), i.e., for every G

B(R,), x i- ( t(G) is E-measurable, and, for every x E E, t,(da) is a a-finite (positive)
measure on (R+,B(R+)). Assume also that for every compact set K C R+, x

t,(K) is bounded. Using t,(da) we define i : E x M+ -- [0, +oo] by

i(2, N) = t{ a E R+ I a, E N } (3.24)

for every x E E, N E M+, where S, E M+ is the point mass at x. Assuming that 9

is a kernel, we can define, for every P E M+, a measure 6, on (M+,.M+) by

(N) = (x, N) (dx) (3.25)

The following proposition shows that this way we obtain a positive a-finite measure

on (M+,M+).

Proposition 3.3.1 1 is a positive kernel, and E, is a positive a-finite measure on
(M+,M+). For every F : M+ [-oo,+oo], which is either positive M+-
measurable or bounded M+-measurable,

I F(v)8(dv) = f F(a6,) t(da) )(dx) (3.26)

Proof. Consider the function H : R+ x E -+ M+ given by

H(a, x) = a6,

We claim that H is B(R+) C/M+-measurable. It is enough to prove that, for every
f E bps, (H, f)(a, x) = a f(x) is B(R+) 0 EC/B(R+)-measurable, which is obviously
true. Fix N E M Then 9(x, N) = t,{a E R+ (a, x) E H-1(N) }, i.e.,

Si- 9(x, N) = JR+ lH-(N)(a,x)t,(da) ,

which is E+-measurable, since t,(da) is a kernel and H-'(N) E B(R+) The fact
that i(x, -) is a positive u-finite measure follows directly from 3.24 and properties of
tx(da). It implies immediately that O, is well-defined positive measure on (M+, M+).
To prove that O, is u-finite, let us consider a sequence Nk = {v E M+ I v(E) < k },
k E N. Obviously Nk E M+, and (Nk; k > 1) covers M+. Notice that

)(x, Nk)= t([0, k]) .

Boundedness of x t,( [0, k]) implies that Q,(Nk) is finite (recall that p is a finite
measure).
Let F = 1N, N E M+. Since i9 is a kernel we obtain

S1(u),(d ) = J 4(x, N) p(dx) =

= J 0 lN(abx) t,(da) P(dx)
JJo

Standard measure-theoretic argument shows that 3.26 is valid for any F, which is
either positive M+-measurable or bounded M+-measurable.
Q.E.D.

Definition 3.3.1 Measure e,, defined by 3.25, is called a measure on rays with pa-
rameter iL, and with respect to a kernel t,(da).

Notice that 3.26 applied to F(v) = exp( -(v, f) ), f E bp', gives the formula for
the Laplace transform in the form

M+ e-^vf) ,(d) = +0 e-"6' t,(da) p(dx) (3.27)

We will be interested in some special measures on rays. Let A(x, N) be given by
3.24, where tx(da) is Lebesgue measure, for every x E E. Then the corresponding
measure on rays,
A,(N)= A(x, N),(dx) (3.28)

is called the Lebesgue measure on rays.

Remark 3.3.1 In the case E = 1 }, A, is a multiple of the standard Lebesgue measure
on R+. However, already for E = {1,...,n}, n > 2, A,, is a measure which "lives"
only on the coordinate axes. Similarly, if E is a topological space, A, "lives" only on
rays { a6x : a > 0 } through 65, where x E supp(4).0I

Assuming usual conventions 1/ + oo = 0, 1/0 = +oo, 0 +oo = 0, we obtain the
Laplace transform of A, in the following form,
J + d d J (dx)
Se- A,(d = e- da (dx)= (3.29)
M+ JE0 JE f(x)
where f E bpE. Using A, we define the exponential measure on rays exp,(h), where
h E bpE, given by

exp,(h)(dv) = e-("'hA,(dv) .

(3.30)

Obviously exp,(h) is absolutely continuous with respect to A,, and its Laplace trans-
form is given by

JM+ e-("f)exp,(h)(dv) = fM e-') e-^(h)A,(dv) =

S (dx) (3.31)
~JEf(x) + h(x)
3.4 (Idt, )-Supersemigroups

Let (E, E) be a standard space and (M+, M+) the measurable space of all finite
(positive) measures, defined as in the previous section. Let I'(x, A) : E x R+ --- R
be a branching mechanism, defined by 2.55. If we consider such a Markov process

((t) on E, that 6t = 6o, then the corresponding semigroup (Tt) is actually (Idt).
As we have seen in section 2.2., there exists a unique (Idt, ')-supersemigroup, de-
noted by Q' (p, dv), and determined by its Laplace transform. We will denote the
corresponding nonlinear semigroup by (Vt') and (Idt, T)-superprocess by (X,).
The purpose of this section is to describe Qt(y, dv). We will use one-dimensional

superprocesses and supersemigroups in this description. After that, in the next chap-
ter, we will use (Idt, t)-supersemigroups as "building blocks" in the description of
the general case of superprocesses. Recall the notation from section 2.3., where we
denoted the point mass at x E E by 6x, and the point mass at fi E M+ by A,, i.e.,
AA is an example of a probability measure on (M+, M+).
Notice that for every x E E, A -* Q(x, A) is a branching mechanism on the
one-point set {1}. We denote it by T,. Then the corresponding one-dimensional
superprocess, denoted by ( f"), is the time-changed process of the L6vy process
(LZ') (see 3.20) stopped at zero. We denote the transition probability of (e*') by

qt,,(a, db), and corresponding non-linear semigroup (which is actually a function in
one-dimensional case) by vt,x(a). Then vt,, : R+ --- R+ is a function which satisfies

the integral equation

vt,.(a) = a + (x, v,, (a))ds (3.32)

or, equivalently, the differential equation
of (v,,(a) = '(x, vt,.(a)) (333)
vo,(a) = a
The Laplace transform of qt,=(a, db) is given by

0 e-cqt,x(a, db) = e-av .(c) (3.34)

Consider (Vt~). Recall that, for every f E bpE,

(Vt'f)(x) = f(x) + f P(x,(V f)(x))ds (3.35)

The uniqueness of the solution of 3.35, and of 3.32, shows that, for every f E bp\$
and x E E,
(V, f)() = vt,,(f(x)) (3.36)

which implies immediately that

IM e-"')Qf(L, dv) = exp (- vE ,(f(x))(d) (3.37)
In the special case when 1 = m 6,, m E R+, z E E, we obtain

IM e-<.'f)Q(m6s,,dv) = e-m"",,(f())=
JM+
= e- f(z)bqt,z(m, db) (3.38)

which proves the following simple result.

Proposition 3.4.1 If (X) is a (Idt, ')-superprocess with Xo =- m6z, then

X,' = -. 6. (3.39)

where I -= m.

Remark 3.4.1 If we think of superprocesses as the distribution of a population of
particles, then 3.39 says that, if the entire population has mass m and is positioned
at z, then at the moment t, the entire population is still at z, but its mass changes
with the law of one-dimensional branching process which we described in sections
3.1. and 3.2. Notice that (t) in this case is trivial, i.e., does not change the starting
position z. We will see later that this, intuitively very acceptable behaviour (where
((t) affects the position of the population independently from 9, which affects only
the mass of the population), appears in the cases where (6t) is "simple". However,
in general case the behaviour is more complicated and is not clear.0

Formula 3.38 can be used to get Qf(pt, dv) for some other 1. The idea is to use the
branching property 2.121, and limits of the type 2.125, to calculate Q"(p,, dv). We
can not obtain explicit formulas by this procedure in the most general case. However,
in some special cases we can use measures on rays (see section 3.3) to obtain nice
explicit formulas for Q (p,, dv). The procedure is described below.
Assume that, for every t > 0, there is a positive kernel Tt,,(da) from (E, E) to
(R+, B(R+)), such that x i- Tt,,(K) is bounded for every compact K, and, that there
is a function ht : E --- R+, such that

vt,X(c) = e-cbTt,(db) h (3.40)

Using the results from section 3.3 we obtain that, for every t > 0, there is a measure
on rays OE,t such that its Laplace transform is

JM+ e-("f)O ,t(dv) = 1E j+o e-bf()Tt,z(db)p(d) =

= -(, ,vt,((f(x))) + (,ht(x)) (3.41)

Recall now that the Laplace transform of the convolution of two measures on the
space (M+, M+) is the product of their Laplace transforms (see section 2.5). Then

3.41 and 3.37 imply that

Q(, dv) = e-^,h )). ,( (3.42)
n=1 n
Let us consider now particular branching mechanisms and show how we can apply
results obtained in this section.
Case '(x,A) = -b(x) c(s)A'; b bE, cE bp, c > 0
Using 3.36 and 3.6 we obtain, for every f E bpf,

(V,~ f)(x) = f(x) (3.43)
e) + f()C()c3)43)

where the quotient (eb(-)t 1)/b(x) is defined by 3.7. This is the example which
satisfies 3.42. Let O,,t ~ exp,,(g), where exp,(g) is defined by 3.30, and 77 and g are
given by
eb(x)t eb(x)t
(dx) = cb)t-1 2(d) and g(x) = C eb )-1 (3.44)
b() bc( x)
Let a function ht be given by

1
ht(x) = c(1. (3.45)
C(X) b(x)

Using 3.31 and 3.41, we obtain that Q (it, dv) is given by 3.42, since

-(, ht(x)) + JM e-()O t(dv) =

eb( )t
= j(d x))
b(x) ( j \)

S() b( (c(x)e j-) (f()c(x) b()-) + eb(=))

E eb()t+ f (Xeb()
^ (x) {d b(x)
J~^ f(~ x}^ *,tib5

Simple computation shows that in the language of random measures O,,t is the
L6vy (canonical) measure of Q"f(, dv) (see [37], chapter 6). Notice that, when

c(x) = 0, then

(Vtf)(x) = f(x)e-b(z)t

(3.46)

and

Qt (p, dv) = Ae-(-.).t(d)(dv)

(3.47)

which is a special case of 2.86.
Case %(z, A) = n(x)( 1 e-"()^ ); n, a E bp,, a > 0
Notice that for every a > 0, n > 0, real numbers, it is not possible that there

exists a measure T on R+ such that

Se-AbT(db) = In [e(ea 1) + 1-h ,
0 a

since A = 0 would imply T(R+) = -h < +oo, and A -- +00 would imply T({0}) =
+oo. Therefore, in this case we can only use formulas 3.36 and 3.38 to get

(Vtf)(z) = 1a ln [en()a(0(e () 1) +
a(x)

(3.48)

and

Qf(mS, dv) = .() e-mn(2)t (e-n(z)(z)t 1) A(+ )),(dv) (3.49)
k=o -k
Let us also mention the case of E = {1,...',n}. In that case T is determined

by 1,..-., 2, and every p E M+ = R+ is = (/i,... ,n). Hence, if we denote

Qt (lzi8{, dv) by Q'(Li,dv), then

Q?(P, dv) = Q(j~i, dv) ... Q(n, dv) (3.50)

Therefore, the (Idt, I)-superprocess (Xt) which starts at y E R' is given by

X, =(e ,...,I ,

(3.51)

87

where (d4') are independent copies of one-dimensional Ii-superprocesses, such that
(Lt') starts at pi E R+. The infinitesimal generator 9 of (Xl) is given by 2.115, i.e.,

(f)(x) = E c bi + [f(x + eiu) f(x)]ni(du) (3.52)

Notice that one of the characteristics of the process (Xt) is that for almost all
trajectories w
(X )(o) = 0 = (X,+);(w) = 0, Vh > 0. (3.53)

We will use this observation later.

CHAPTER 4
TRANSITION PROBABILITIES OF SUPERPROCESSES

We will apply results from Chapter 3, to describe supersemigroups of general

superprocesses. In the case of "simple" Markov processes, like the uniform motion to

the right we obtain explicit formulas. However, in more general cases we are not able

to say much. Already in the case of Markov chains we can describe supersemigroups

only locally with the application of Cameron-Martin formula.

4.1 Trotter-Kato Formula

In this section we restrict ourselves to the assumptions of section 2.4, i.e., E is a

locally compact Hausdorff space with a countable base, EA the one-point compact-

ification of E, (t) a Markov process with state space (E, B(E)) (A is a cemetery),

whose semigroup (Tt) satisfies the Feller property, and a branching mechanism T that

satisfies 2.93 and 2.94.

We have seen, in section 2.4, that under these assumptions the superprocess (Xe)

is a Feller process with state space M', and (Xt) actually "lives" on M+. We

denoted its infinitesimal generator by g, and 2.111 shows how g is defined on its core

Exp(M TD(A)) (see 2.109), where A is the infinitesimal generator of (Tt) extended

to C(Ea) (see 2.92).

However, in section 3.4 we described (Idt, I)-superprocess for every standard

space (E, \$). In particular, we have the description of (Idt, 1I)-superprocess under

the assumption of this section. Let us denote (Idt, ')-superprocess by (X*t), and

its supersemigroup by (Q(). Of course, (X)) is also a Feller process on M+ and

Exp(M+, D(0)) is a core for the infinitesimal generator of (X,). Since D(0) =
C(EA), it follows that for every A,

Exp(MA, D(A)) C Exp(MA, )(0)) = Exp(M+, C(EL))

Let us denote the infinitesimal generator of (Xt) by 9'. Then, by 2.111, for every
f E C(Ea)+ and pt E MA,

["^(F)] (p) = -[F,(p)] (p, ~f) (4.1)

Consider now the case when _= 0, i.e., the case of (Tt, 0)-superprocess. Following
the notation of the section 2.4 we have that i 0, and Vt = Tt. Since Tt is linear
and positive we have that pTt, defined by

(tT,, f) = (', Tf) (4.2)

is a measure in M+. Therefore, we have an explicit formula for (it, 0)-supersemigroup.
If we denote (t, 0)-supersemigroup by (Qf) and (Tr, 0)-superprocess by (Xt), we ob-
tain that
Qt(A,,dv) = A't,(dv) (4.3)

and

Xt = Xtt (4.4)

Let us denote the infinitesimal generator of (Qf) by G9. Again, section 2.4 shows
that Exp(M+, D(Ai)) is a core for GE, and, for every f E C(Ea)+ n D(A), f strictly
positive, and 1 E M ,A

[ (F,) () = --[F,(2)] (, f) .A (4.5)

Using 4.1, 4.5, and 2.111, we obtain that

g = g' + gt

(4.6)

on Exp(M 2D(A)). Since we have the description of (Q ) (section 3.4), and the
description of (QS) is simple (see 4.3), the equality 4.6 indicates that we can use the
well-known consequence of Trotter-Kato formula (see, for example, [47], Corollary
5.5) to obtain an approximation formula for (Qt).

Theorem 4.1.1 For every F E Co(M+), and, for every t > 0,

QtF = lim Q' F=lim (Q Q/ n F (4.7)

and the limit is uniform on bounded t intervals.

Proof. The proof is now almost straightforward application of Corollary 5.5 in
[47]. We only have to be careful about the fact that we are dealing with cores for
infinitesimal generators, and not with their domains. All three semigroups (Qt), (Q?),
and (Qi) are Co semigroups on Banach space Co(MA), and all three are contractions.

Since D(g~) n D(ge) contains Exp(M+, D(A)), it is dense in Co(MA ). The image of
Exp(M+, D(A)) with respect to the operator zI g, where z is a complex number,
such that Re(z) > 0, is dense in Co(M+), since Exp(M+, D(A)) is the core for g.

By 4.6 the same is true with respect to the operator zI (9" + 9f). It follows that
the closure

of G9 + G is the infinitesimal generator of a contraction Co semigroup, which is given
by limit in 4.7. By 4.6 and 2.109 we obtain that

D(a) C n(Gd + C)

and

(4.8)

However, both g and g* + ge are infinitesimal generators for Co semigroups of con-

tractions. By 4.8 this is possible only if Q = G + Qg.

Q.E.D.

We have explicit formulas for (Qf~) for several cases (see section 3.4), and for (Qf)

always. In particular, (Qf) has a very simple description (see 4.3). One may hope to

find explicit formulas for (Qe), by using Theorem above. Unfortunately, it does not

seem to be so. The following example illustrates how complex these formulas can be

in general. In some special cases we will obtain nice explicit formulas in section 4.3.

Example 4.1.1 Let (t) be a two-state Markov chain, which stays for exponential

time in state 1, then jumps to state 2 and stays there forever. More precisely, its

semigrouop is
e-' 1 e-t
T = 0 1 (4.9)

and its infinitesimal generator is

A= -' (4.10)

Of course, in this case E = E = {1, 2}, and C(EA)= R2, and M' = R_.

Let the branching mechanism be T(x, A) = -A2, i.e., it has constant coefficients.

Then the (Tt, P)-superprocess (Xt) is an R -valued stochastic process, whose in-

finitesimal generator is given by (see 2.115),

82 02 0 0
x jl- + x2 -- X1 + X1 (4.11)
1 OX2 1X 1 O2

In this case we have explicit formulas for (Qf) and (Qf) (we do not need "tilde"

notation here, since E = Ea). Moreover, we can find also an explicit formula for

a nonlinear semigroup (Vt), which is the solution of 2.61. However, even in this

(relatively simple) case we were not able to extract explicit formula for (Qt).

Let us compute the explicit formula for (Vt). For every f = (fl, f2) in C(EE)+ =
R+, Vtf = ((Vf), (Vtf)2) is also in R_ We claim that Vtf is given by
(Vtf) f2 + (fi f2)e-t (4.
f2t + 1 (f2t + 1)2 [ft -Id. f 1
(>,)()1 = + 1((4)12)
f2 (f2+1)2 (fl
and

(f)2 + (4.13)
f2t + 1
Using results from section 2.1, it is enough to check that Vtf is well-defined, positive,
and that it satisfies 2.61. There are no problems with the second coordinate 4.13,
since this problem is solved already in section 3.1 (see formula 3.8). Thus, we will
concentrate on 4.12. If fi f2 > 0, then (Vtf)x is obviously well-defined and positive.
Consider the case when fi f2 < 0. We define a real-valued function h : R+ -- R
by
ft e-'ds
h(t)= (feds + f2) + 1
fi (f2S + 1)2
Since fi f2 < 0 we obtain, for every t > 0,

ds 1 1
h(t) > (s (f- f2) +1=- 1 i- (f f2)+1
(f2s + 1)2 f2 f2t + I
t fit + 1
=^ (-12)+1= >o .
f2t +1- f2t +- 1
Therefore, (Vtf)i is well-defined. Let us prove that it is positive. By the proof above
we obtain

et(f2t + 1)2. h(t) 2 et(f2t + )2 1 >
f2t+ 1-
> (f2t + 1)(ft + 1) > 0,

which implies, since fi f2 < 0,

(Vtf > 2 f f2
f2t + 1 (f2t + 1)(fit + 1)

fi + ff2t
= >0
(ft + 1)(fit + 1)
It remains to be proved that (Vt) satisfies 2.61. Let us define a real-valued function
H : R+ Rby
1
H(t) =
(f2t + 1)2 h(t)
By differentiating H(t), and using that H(0) = 1, we obtain

H(t) = 1 H(s) e-(f, f2)H(s) +2 f2 ds
JO f2S + 1

Multiplying both sides by e-t(f f2), and then adding f2/(f2t + 1) to both sides,
and using the fact that

(Ttf)i ft f2 ds f2 + (f f2)
.o if2s + 1 f2t + 1
we obtain

(Vtf)i = (Ttf)i t f- 2 ds-
0 f2s + 1

Ie` e [e-'(fi f2)H(s)]2 + 2 e-'(f, f2)H(s) ds .

By changing the variable of integration, and by writing 1 = (1 e-') + e-', we get

(Vf)' = (Tf) [T,(V_,f)2] ,

which is exactly 2.61 for the first coordinate in this case.
As we have just seen it was lengthy, but elementary, to check that 4.12 and 4.13
give the solution of 2.61 for our (Tt) and l. However, it was not so easy to find 4.12.
To do so we were trying to solve the corresponding differential equation. Combination
of several useful suggestions from [38], and many guesses led us to 4.12.
Using results from section 3.4 and formula 4.3 we obtain

(4.14)

Qf((Al,2); (d 2))= A(A- .,-, (1_e-c)+,)(dvi, dV2) ,

and

t '((l, 2); (dvi, dV2)) = (dvi, d2) e (4.15)
(0 ) n / k ] n-k
1 n vt) l k e-v/t ,( 2(n-k)- le-(/t
n=O n! =0(k) (n k)
where r is the well-known gamma-function.
These results imply the following formula for the m-th term in Trotter-Kato ap-

proximation formula 4.7

[ e(dvi dv t/e m
[Q,/mQ/m] ((pl, #2); (dv, dv2)) = (dv, dv2) .e t/
V1 '2

Ae nt/1m) = [kle-t/m]' 0[1(1 e-t/') + I2 ]-'-

n [1e ^/ nk -t /m 2
ne=0 n2!(t/m)2 n, k= 0 (
(t/2m)njn +n ( t2 ) k2 1 n2-k2.
f=2O fl2!(t/m)2n2 k2kO
nz-kz
.22 ( n2 k2 1
12=0 12 (1 e-t/m)n2-k-1

nim=0 km=0
nm-k nm, kIm 1
*==o Im (1 e-'/")"=-km-"
F(k1 + k2 + 12) P(ni + n2 (kI + k2 + 12))
r(ki)Pr(n- ki)Fr(k2)r(n2 k2)
00 (t/2m )n _l+n (- ) \2(1 -
E ni!r(t/m)2nm 2 [" Eietl 2(

r(k^_ + km + m) r(nni + nm (kin- + k + m))
r(km )r(nm km)
Despite the fact that the formula above seems to show some sort of "regularity", and

that we have explicit formula for (Vt), we were not able to recognize if there is an

explicit formula for (Qt). This question remains open.O

4.2 Superprocesses over Markov Chains

Let ((e) be a Markov chain with state space {1,2,...,n}. It is determined by
its semigroup (Te), i.e., by the infinitesimal generator A = (aij)". As we have seen

already in section 2.4 and 2.5 the superprocesses over (t) are branching Markov pro-
cesses on R+ whose infinitesimal generators are given in Theorem 2.4.3. The class

of branching Markov processes on R' has been studied by Watanabe in [57] (the
paper actually deals with the case n = 2, but only to make notation simpler) and
by Rhyzhov and Skorokhod in [51]. Of course, at that time (1969) they did not con-
sider the terminology of superprocesses. However, in their papers they characterized
completely the infinitesimal generators of branching Markov processes on R" (see,

for example, [57], Theorem 2, pp.449-450). Comparing these formulas to 2.115 we
can see that the class of superprocesses over Markov chains is smaller than the class
of branching Markov processes (there are no termination coefficients in 2.115 and
drift coefficients are restricted by b and A). Even more general class, of branching
processes on R+ with immigration, has been studied in [40].

Following the notation of the previous section, we obtain from 2.115 that (Id,, ')-
superprocess (X') has the infinitesimal generator 9', given by

r 2 F F o00
('F)(p) = 2E c, b-b + [F( + eu) F(I)]n,(du) (4.16)

In particular, it shows that (Xt') is equal to the process (see 3.51 and 3.52)

( t t ,..., .) (4.17)

where (d"') are independent one-dimensional superprocesses with respect to ',(A) =

%P(i, A); i = 1,... ,n. Recall (chapter 3) that each (T') is the time-changed process
of the L6vy process (LP') stopped at zero (see 3.20). In particular, it shows that, for