Stochastic processes and point processes of excursions

POINT PROCESSES OF

EXCURSIONS

J.A.M. VAN DER WEIDE

TR diss

1545

POINT PROCESSES OF

EXCURSIONS

POINT PROCESSES OF

EXCURSIONS

PROEFSCHRIFT

Ter verkrijging van de graad van doctor aan de

Technische Universiteit Delft, op gezag van de

Rector Magnificus, Prof. Dr. J.M. Dirken, in het

openbaar te verdedigen op dinsdag 26 mei 1987 te

16.00 uur ten overstaan van een commissie door het

College van Dekanen daartoe aangewezen,

door

JOHANNES ANTHONIUS MARIA VAN DER WEIDE

geboren te Leeuwarden

Doctorandus in de Wiskunde

TR diss

1545

STOCHASTIC PROCESSES AND POINT PROCESSES OF EXCURSIONS van

J.A.M.van der Weide

Zij P een Markov-kern op een meetbare ruimte (E,E). Definieer de Markov-kern P op de meetbare ruimte (E x[0,l], f 8 B(E0,1])) door

1

Pf(x,a) = J P(x,dy.) ƒ f(y.,yo)dy

E 0 ' l

waarbij f : E x [0,1] •* K een begrensde, meetbare functie is en (x,a) e Ex [0,1]. De in Cl] voor meetbare functies h : E ■+ [0,1] gedefinieerde potentiaalkern U heeft de volgende stochastische interpretatie. Voor x e E en A e E is

V

'

"

(x,«o[

\

Ax[0,l]

V]

L n= 1

waarbij (Y ) „ d e Markovketen op E x [0,1] is met overgangskern P en waarbij n n £ U

T de eerste terugkeertijd is van A, = {(x,y) e E x [0,1] : y < h(x)}. [ 1] Revuz, D. : Markov Chains. North-Holland, Publ.Comp. Amsterdam 1975. Zij X = (X ) .. standaard Brownse beweging startend vanuit 0 en zij A = (A ) het additief functionaal gedefinieerd door

A

t

J

\oM

s

-Zij T = (x ) . de van rechts-continue inverse van A. Het stochastisch proces X = (X_). ^ „ gegeven door X_ = X • t. kan direct geconstrueerd worden uit het

t t ^ O t t

Itö-Poisson puntproces van excursies vanuit 0 van de Brownse beweging X. Uit een eenvoudige berekening volgt dat X gereflecteerde Brownse beweging is.

Williams,D : Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. ch. Ill, section 38, Wiley, New York 1979.

In [ 1 ] wordt een Suslin ruimte gedefinieerd als een Hausdorff topologische ruimte E waarvoor een Suslin-metriseerbare ruimte P bestaat en een continue, surjectieve afbeelding van P op E. De opmerking dat dit Bourbaki's definitie van een Suslin ruimte is als aangenomen wordt dat P een Poolse ruimte is, is onjuist.

[1 ] Dellacherie, C. and Meyer, P.A. : Probabilities and Potential. North-Holland, Amsterdam 1978.

stelling (4.2.4) zwak naar de kansmaat P .

5. Zij Y = (Y (t)) . het Markov proces geconstrueerd uit het Ito-Poisson punt-proces van excursies vanuit 0 van standaard Brownse beweging waarbij de term YT, Y > 0, is opgeteld bij de som ( T + A ( T ) ) van de lengtes van de excursies tot en met tijd T(zie sectie (4.3) van dit proefschrift). Zij verder L de Blumenthal-Getoor locale tijd in 0 van proces Y . Dan is voor X,u > 0 en 0 < a < £X2

ƒ e"a t En(exp[-XY(t) - pL(t)])dt = ^ - ^ . '.■

0 ' (2a-X^)(y(v'2"+Y)+aY+\/2a')

6. Laat het Markov proces Y gedefinieerd zijn als in stelling 5 en laat m = min(t : Y (t) = a) het eerste treffen van toestand a e IR zijn. Dan

is voor X > 0

u i "X m a^ >/TT -av^T ,-, ^rrr . -2a\/2T.-l

E (e ) = V2X e (X-y+vZX -Xye )

7. Het verdient aanbeveling om ook bij het onderwijs aan de TU Delft in de vakkei Analyse, Lineaire Algebra, Kansrekening en Statistiek meer gebruik te maken van computers.

8. In de Breitkopf uitgave van Johann Sebastian Bach's Matthaeus Passion is in de aria "Gebt mir meinen Jesum wieder" de tweede tel van de vierde maat niet in overeenstemming met het handschrift van Bach. Bij uitvoeringen van deze aria verdient de oorspronkelijke,door Bach aangeduide stokvoering de voorkeur

Prof.dr. C.L. Scheffer drew my attention to the subject of this thesis, I am most grateful for many inspiring discussions which often clarified the issues considerably. I thank Prof.dr. E.G.F. Thomas for his kind help during the preparation of the manuscript and prof.dr. W. Vervaat who drew my attention to a paper of Whitt. I sincerely thank Mrs. Netty Zuidervaart-Murray for the excellent typing of the manuscript, ir. S.J. de Lange for a careful proof-reading and ir. H.J.L. van Oorschot for his assistance in mastering the software package T3. Finally I want to acknowledge the "Vakgroep SSOR" for giving me the opportunity to write

I n t r o d u c t i o n and Summary 1

1. Point processes 15

1.1. Topological spaces of Borél measures 18

1.2. Poisson point processes 31

1.3. Itö-Poisson point processes. 35

2. Excursion theory 51

2.1. Ray processes 52

2.2. Point processes of excursions of a Ray process from a given

state 54

2.3. Construction of stochastic processes from Itö-Poisson point

processes 67

3. Applications 89

3.1. The Itö-Poisson point process attached to Brownian motion 90

3.2. The Itö-Poisson point process attached to Brownian motion

with constant drift 93

3.3. Feller's Brownian motions 95

3.4. Brownian motion on an n-pod 99

3.5. A remark on Blumenthal's construction of the Markov process

attached to an Itö-Poisson point process 103

4. Random walk approximations 107

4.1. Approximation by discrete semigroups 108

4.2. Skew Brownian motion 112

4.3 Stickiness 125

Appendix 135

References 147

Samenvatting (summary in Dutch) 153

In his studies [35] and [36] of the sample paths of Brownian motion,

Levy developed the idea to decompose the time set [0,ro[ in a part Z at which the process is in state 0 and intervals of time spent in K\{0}.

Throughout the years this has proved to be a very fruitful idea. On

one hand the study of the set of zeros Z led Levy to the description of

local time as an occupation density (Levy used in [35] the term "mesure

du voisinage" . See for occupation densities the survey article of

Geman and Horowitz [14], who discuss connections between the behaviour

of a (non-random) real-valued Borel function and the behaviour of its

occupation density. Local times for general Markov processes were

introduced by Blumenthal and Getoor in [3]). On the other hand Levy's

study of the behaviour of Brownian motion on zero-free intervals was

the starting point of excursion theory. Levy's theory was extended in

Itö-McKean [27], (2.9) and (2.10). See also Chung's article [6], in

which elementary derivations are given of a number of Levy's results.

This research led to many deep theorems about the behaviour of the

paths of diffusions, see for instance Williams [58] and Walsh's

discussion of Williams' results in [53]. Another important application

of excursion theory can be found in the construction of those strong

Markov processes, which behave outside a fixed state (or more generally

outside a set D) as a given Markov process X. In this area the works of

Dynkin [9], [10] and Watanabe [54], [55] are important. For excursions

from a subset S, see the works of Maisonneuve [38], [39] and Getoor

[16]. Getoor gives also an application to invariant measures, see also

Kaspi [30] and [31]. Unlike occupation densities, which are also useful

in the study of non-random functions, excursion theory takes its use

from the Markov character of the random process. To make clear the

chain with state space E. Let a € E be a given state. Denote for

k=l,2,... by u^ the time at which the Markov chain X visits the state a

for the k time; iC = w if there are less than k visits to a. Suppose that a is a recurrent state, i.e. IP„[ui < °°] = 1. Then the k

excursion Vk = (Vk(n))n>0 ^rom a °f t n e Markov Chain X is defined as follows N U N f X- . .K ulSt-n Vk(n) X , ... for 0 < n < ur'-u: ua+ n for n > uk + 1- ^ Let V0 = ( Vo( n ) )nN0 be defined by

f

K for n < „1

V

>

It follows from the strong Markov property that the sequence of

excursions (Vj.)kNi *s independent and identically distributed. It is clear that the process X can be reconstructed pathwise from the

sequence (Vk)kNQ.

For Markov processes with continuous time parameter the situation is

more complicated. As an example take standard Brownian motion

B = (B(.)txQ and consider the excursions from state 0. Let Z be the set of zeros of B. The component intervals of [0.00[\Z are called excursion intervals. Since Z is a topological Cantor set of Lebesgue measure 0

(see Ito-McKean [27], problem 5, p.29), with probability 1 there is no

first excursion interval. Let I=]a,|3[ be an excursion interval. The map

Vj : [0,o°[ -» R defined by

■ Ba + t for 0 < t < p-a

Vj(t) = ■

[ 0

t >

is the excursion made by B from 0 corresponding to the excursion

interval I; f=/3-a is called the length of the excursion. Put Tj=<p(a),

where <p is the local time of B at zero. Itó proved in [25] (see also Meyer [41]) that the random distribution of points (TT - V T ) in [0,<D[ x U, I running through the excursion intervals and U being the

space of excursions from 0, is a Poisson point process on [0,<x>[ x U whose intensity measure is the product of Lebesgue measure X on [0,o°[ and some a-finite measure u on U. This means that the number of points

( X T . V T ) in a subset [u,v[ x U of [0,<x)[ x U is Poisson distributed with expectation (v-u)u(U ) whilst the numbers of points ( TT. V J ) in disjoint subsets of [0,°°[ x U are independent. Ito proved this result actually for excursions of a standard Markov process X from a regular point a, and he gave a characterization of the excursion law v of a recurrent extension of X, i.e. a strong Markov process, which behaves as X until the first hitting of state a.

It is interesting to look at Itö's definition of a point process. Let (S,y) be a measurable space. A point function p : ]0,<°[ -» U is defined to be a map from a countable set D C, ]0,«>[ into U. Meyer in [41] considers a point function p as a map defined for all points in ]0,<°[ by putting p(x)=3 for x € ]0,<°[\D where 9 is an extra point added to U. Let now IT be the space of all point functions: ]0,ro[ -» U. Denote for p € ! I and for E € S8(]0,°°[) ® if by N(E.p) the number of the time points t € D for which (t,p(t)) € E. The Borel a-algebra S6(!T) on ÏÏ is defined as the cr-algebra generated by the sets {p € IT : N(E,p) = k } , E € <3J(]0,«>[) ® y, k=0,l,2,... I to defined a point process as a

(IT, 36(17))-valued random variable. For instance the point process of excursions from 0 of Brownian motion is the (stochastic) point function p defined by

D = {TJ-: I an excursion interval} and p(t) = V j . t = Tj € Dp.

This definition gives a clear picture of point processes such as they appear in excursion theory and that is presumably the reason why in studies about excursion theory this definition is always used, see for instance Watanabe [54] and Greenwood and Pitman [17]. Beside this definition of a point process as a stochastic point function, there

exists a fairly general theory of point processes which views a point

process as a discrete random measure. See Neveu [44], Jagers [29] and

Krickeberg [34] for point processes on a locally compact space and

Matthes, Kerstan and Mecke [40] for point processes on a complete,

separable metric space. This measure-theoretical approach to excursions

makes it possible to use some important results from this theory, such

as e.g. the Palm-formula, which were up to now not used in the

literature about excursion theory. An example of the use of the

Palm-formula can be found in the construction of a Markov process from

a Poisson point process of excursions. Itö only remarks in [25] that

this can be done by reversing the procedure of deriving the excursion

process from a Markov process. In Ikeda & Watanabe [22] Brownian motion

is constructed from its excursion process using the general theory of

stochastic processes (compensators and stochastic integrals). And in

[2] Blumenthal gives a construction of which he claims that it is the

construction Itö had in mind; this contruction consists of a pathwise

approximation of the Markov process. The most recent and complete work

along these lines can be found in Salisbury [47] and [48]. The

construction that we will give is based on an application of the Palm

formula and on the so-called renewal property of a Poisson point

process of excursions. This construction has in our opinion the

advantage that it makes clear why the constructed process has the

Markov property and it displays the role of local time in the

construction. The same method can be used to write down a formula for

the resolvent of the constructed process.

We continue with the definition of a point process as a discrete random

measure. Let X be a topological space with Borel a-algebra 55(X).

Roughly stated, a point process on X is a probability measure on the

space of locally finite point measures on (X, 26(X)), or a random

(X,2S(X)) by which we identify a random variable with its distribution. From now on we will use the word point process only in this sense. In excursion theory the topological space X is the (topological) product of the set of nonnegative reals [0,co[ with the usual topology and the space of excursions U endowed with the Skorohod topology. For example the point process of excursions from 0 of Brownian motion is the random measure S{6/ y ■>: I an excursion interval} where 6X is the notation for the Dirac measure in x. Note that X = [0,00[ x U is a polish space. The main reference on point processes on polish spaces is the book [40] of Matthes, Kerstan and Mecke. The theory which they develop depends essentially on a fixed metric d on X, chosen in advance, such that the metric topology coincides with the topology of X and (X,d) is a complete, separable metric space. A nonnegative Borel measure on X is locally finite if it is finite on the sets in 28(X), which are bounded in the sense of the metric d. This theory is not directly applicable to excursion theory. The point measures which arise in excursion theory are finite on the sets [a,b[ x [f > I ] , I > 0, (remember that f is the

length of the excursion) and the most interesting cases are those where the set [a,b[ x U has infinite mass. Note that the set [f > I] is dense in U. Thus it is not clear how to choose a metric d on X for which the set of locally finite measures contains this family of point measures. Instead of trying to find such a metric, it seems more natural to define local finiteness directly in terms of the sets [a,b[ x [f > I ] . More general, let V be a family of Borel subsets of X. A nonnegative Borel measure jx on X is called y-finite if fx(A) < °° for every A e V and a point process P is an y-finite point process if the probability measure P is concentrated on the space of y-finite measures. The set of locally finite measures'in the sense of Matthes, Kerstan and Mecke coincides then with the y-finite measures, y being the family of all open balls with finite radius. Point processes on locally compact spaces are probability measures on the set of Radon

measures, which is the same as the set of y-finite measures with y

consisting of the compact subsets.

So far we did not discuss a measurable structure on the set jt{V) of y-finite measures, which is of course necessary for the definition of

probability measures on A+(y). A a-algebra on M+(if) should at least

measure the maps \x € jt{V) -* ]x(k), A € 26(X). In Matthes et al. [40] the a-algebra on M (y) (y being the family of open balls of finite radius) is defined in an abstract way as the o—algebra d generated by these maps. In the literature about point processes on locally compact

spaces, on the other hand a a-algebra on the set of Radon measures is

introduced in a topological way as the Borel a-algebra $(M+)

corresponding to the vague topology on M+(if). It turns out that 3)(M+)=si

in this case, so we have a definition of si as a Borel a-algebra corresponding to a nice topology on Jl (if), which makes it possible to use the apparatus of topological measure theory. In section (1.1) we

will define a topology on the set A (y) of y-finite measures on an arbitrary polish space X. Let *(y) = {f e C ^ X ) •' 3A e y : supp(f) C A}

and let T(y) be the topology o(M+(y), 3f(y)) of pointwise convergence on

3f(y). If y is a family of open subsets of X filtering to the right such

that y covers X and such that y contains a countable, cofinal subset,

then it will turn out that (A (y), T(y)) is a Suslin space whilst the Borel a-algebra on M (y) coincides with si. At the end of the section we compare our results with the results of Harris in [19] and [20], who

also defines a topology on some family of nonnegative Borel measures on

a complete, separable metric space.

Section (1.2) contains standard results for y-finite point processes,

in particular the Palm-formula which is now a direct consequence of a

general theorem on disintegrations of measures from topological measure

theory. Further y-finite Poisson point processes and Cox processes are

^-finite Poisson point processes on X=[0,°>[ x U, if being the family of subsets I x U of X where I is a bounded, open sub-interval of [0,»[

and (Un)n>i is a sequence of open subsets of U, increasing to U. It is clear that V is a filtering family of open subsets of X which covers X and has a countable, cofinal subsequence. Denote by -dj(y) the set of

^-finite point measures fi for which n({t} x U) £ 1, t 2 0. An Itö-Poisson point process is a Poisson point process P on X with

intensity measure X ® u, X denoting Lebesgue measure on [0,°°[ and v a a-finite measure on U satisfying u(Un) < », n > 1. We choose the name Itö-Poisson point process, because the point process of excursions, as

constructed by Ito, is of this type. Following Ito. the measure u is

called the characteristic measure of P. Further P(J(J(y)) = 1 for an

Itö-Poisson point process P. The first important property of

Itö-Poisson point processes is the renewal property which is treated

here as a generalization of the property that a Poisson process is free

from after-effects. The renewal property was already mentioned in Itö

[25], but without a proof. We continue with Itö's characterization of

Itö-Poisson point processes with a proof using "point process

techniques". We end section (1.3) with a beautiful theorem of Greenwood

and Pitman [17], which states that an Itö-Poisson point process P has

an intrinsic time clock in the following sense: if fi € jj(y), denote by

ffcld-1). Et&iU-) • ■ ■ • t n e Uk-sequence of p., i.e.

supp(u) D ([0,°°[ x Uk) = (Tk i(u), f k i C " ) ) ^ ! where the enumeration is such that the sequence (Tjci(fi))i>i l s increasing in the order of K. The sequence fk = (firi ^i>1 *s a n i-i-d. sequence on the probability space (j«J(y) ,P). The theorem of Greenwood and Pitman states that the time

coordinates Ti^ can be reconstructed from the sequence fifi-fkQ---- if

u(U)=+a>. We give a complete proof of a slightly more general version of this theorem, which was formulated in [17] as a theorem on stochastic

In chapter 2 excursion theory is treated for Ray processes. We have

chosen to treat excursion theory for Ray processes, since this class is

in some sense the most general class of strong Markov processes, see

Getoor [15] and Williams [59]. After a brief survey of Ray processes in

section (2.1), we construct in section (2.2) the Itö-Poisson point

process of excursions from a given state a of a Ray process Y. Since we

want to include branchpoints in our discussion, we use a definition for

excursions which differs a bit from Itö's definition, see also Rogers

[45] who uses the same definition. We call excursion intervals the

connected components of the complement in [0,"»[ of the closed set of

time points where the process hits or approaches the state a. Let

(rk)k>l ^e a decreasing sequence of positive real numbers and let Uj={u € U : C > r^}. Denote by Vi^ the n excursion of Y with length

exceeding r^. The strong Markov property implies that the sequence

(^kn)n>l i s a n independent, identically distributed sequence. Let T =inf{t > 0 : Yt=a or Yt_=a}. An application of the theorem of Greenwood and Pitman yields:

- If Pa[Ta=0]=l there exists an y-finite Itö-Poisson point process N defined on (fl,?,P ) whose [f > l]-subsequence is the sequence

of excursions of Y of length greater than I. The characteristic

measure u of N is the unique (modulo a multiplicative constant)

measure on U of which the conditional distribution u|y is the

probability distribution of V.j; u is a a-finite measure with

total mass U(U)=-H». The Markovian properties which u inherits

from the process Y are described in theorems (2.2.3) and

(2.2.4).

- In the remaining case where P „ [ T =0] = 0 there exists an i.i.d.

[C > l]-subsequence is the sequence of excursions of Y of length greater than I.

Note that it was not necessary for this construction to introduce explicitly a local time at state a. Local time at state a will be discussed in section (2.3), in which we construct Markov processes from an y-finite Itö-Poisson point process. The basic idea is the following. If (i € jKJ(Sf), then supp(n) can be considered as a countable, ordered subset (uC7)CTgifu'\ °f U where J(p) denotes the projection on [0,°>[ of supp(^) and where u =u iff (a,u) 6 supp(n). Note that (u ) e T(„\ is not necessarily a totally ordered subset of U. Let L : U -» [0,CD[ be a given, measurable function on U. Define for a € [0,°°[

B(er.n) = 2{L(uT) : T € J(^) 0 [O.o-]}

= Jn(dTdu)l

-,(T)L(u)

C ( n ) = U [ B ( a - . ^ ) . B ( o . j i ) [ . a € J ( n )

If T=C((j) then denote, by |x the c o n c a t e n a t i o n of the functions

ua | [ O . L ( uC T) [ ' ° € JM- t h a t i s

p. ■ [0.°°[ -» E

JI(s) = ua( s - B ( a - , M ) ) = JM( d T d v ) ( v l | -0 L ( v ) [) ( s - B ( T -f ti ) )

where a € J(p) such that s € [B(o--,fi), B(CT,(X)[. In general we do not have that [0,ro[=C(|i). If B(a,pi) is strictly increasing as a function of a. then [0,°°[ is the disjoint union of C(u) and the range R of B(.,u). Let now P be an ^-finite Itö-Poisson point process with characteristic measure v. We want to construct Markov processes, so we have to assume

that u satisfies the properties of the characteristic measures which arose by the construction of the Itö-Poisson point processes of excursions in section (2.2). But it is not necessary to assume that u(U)=+a>. In this context it is more natural to consider a family (P ) er of point processes, where Px is the ^-finite Itö-Poisson point

process P to which is added a first excursion corresponding to a start

from x, taking in account the transition mechanism which is contained

in the measure v. For our construction we will follow the above described basic idea with the lifetime f in the role of L. Considered

as a function of p. € J,(y), B(T,fi) is a random variable on the probability space (■^i(y), ?x) • The Poisson-property of the point process P implies that the stochastic process ( B ( T ) ) ^ is a

subordinator (i.e. the process ( B ( T ) ) >Q has nondecreasing cadlag

realizations and stationary independent increments). In our

construction we add a linear term stationary -IT to B(t), with i a nonnegative real parameter, which gives us the general form of a

subordinator with the same Levy measure as B ( T ) . An interpretation of

the parameter i will be given in chapter 4. The simple Markov property for the constructed process is proved in theorem (2.3.6). In theorem

(2.3.8) we give an expression for the resolvent and in theorem (2.3.9)

the strong Markov property is proved under a weak extra condition. In

theorem (2.3.10) we give an explicit formula for the Blumenthal-Getoor

local time at state a. We end this section with an example of the

construction of a stochastic process from a more general point process

than an Itö-Poisson point process. This construction is based on a Cox

process and leads to a strong Markov process which is killed

exponentially in the local time at a.

In chapter 3 we give some applications of excursion theory. In the

first two sections we derive explicit expressions for the

characteristic measures of the Itö-Poisson point processes of

excursions from 0 attached to standard Brownian motion and Brownian

motion with constant drift. A natural problem is to describe all strong

Markov processes which behave like a given Ray process Y until the

first hitting or approach of a given state a. As far as we know the

only complete solution for this problem is given in Itö and McKean [26]

[0,°°[. In section (3.3) we given an interpretation in terms of

excursion theory of the parameters which appear in its description. In

section (3.4) we will construct a model for random motion on an n-pod

E , that is a tree with one single vertex 0 and with n legs having

infinite length. This is the most simple example of random motion on a

graph. In defining the process on EL we should like it to be Markovian

with stationary transition probabilities. We should also like to have

the process to behave like standard Brownian motion restricted to a

half line, when restricted to a single leg. Using the results for

reflecting Brownian motion from section (3.3) we are able to

characterize all strong Markov processes which satisfy this

description. Frank and Durham present in [12] for the first time an

intuitive description of such a process for the case n=3. They

considered the case of continuous entering from 0 in a leg, which was

chosen according to some given probability distribution. The difficulty

which arises in the construction of this process is that the process,

when starting from 0, will visit 0 infinitely many times in a finite

time interval. It is therefore not possible to indicate the leg which

is visited first starting from 0. In section (4.2) we will explain what

is meant with choosing a leg according to some given probability

distribution with the help of a random walk approximation. The

construction that we will give is based on section (2.3); our model

allows also jumping in a leg, stickiness at 0 and killing with a rate

proportional to local time at 0. In section (3.5) we show how theory of

section (2.3) can be applied to the construction of certain Markov

processes which Blumenthal uses in [2] and for the construction of

which he refers to Meyer [42]. As already mentioned above, chapter 4

contains random walk approximations. Let S =(S^)k£|jj. n=l,2,... be a

sequence of Markov chains on Z with transition matrices P and initial

continuous time parameter by

X ^ t ) = n-* Sn [ n t ] (n)

and let P be the distribution of 5C where u is the distribution of

X (0). In section (4.2) we consider the case where Pn=P does not depend on n where P 'is given by

1

P(m.m-l) = P(m.m+1) = - for m * 0 P(0,k) = pj^ for k e 2

in which {p^ : k € Z} is a probability distribution on TL. Harrison and Shepp proved in [21] that for the special case p1=a, p_1=P, a+/J=l the

(n)

sequence of probability measures (PQ ) converges weakly to the distribution of skew Brownian motion starting from 0. Skew Brownian

motion was introduced in Itö-McKean [27] as an example of a diffusion

process. In the terminology of section (3.4) skew Brownian motion can

be considered as a random motion on a 2-pod which behaves like standard

Brownian motion outside 0 and which enters continuously in a leg chosen

with probability distribution (a, 1-a). We will prove a part of a more

general result which was stated without proof in Harrison and Shepp. If

the probability distribution (pi,) has a finite first moment and if the

sequence of probability measures (u ) %i on K with supp(i> ) C n TL converges weakly to a probability measure m on \R. then the finite

(n)

dimensional distributions of the sequence (P ) converge weakly to the n

finite dimensional distributions of skew Brownian motion with initial — +

cP k distribution m and with parameter a

S M P , /

Section (4.3) gives a random approximation for the stochastic process

Y constructed from the It6-Poisson point process of excursions from 0

of standard Brownian motion, where we have added to the total excursion

length B ( T ) up to time T the term TT. Let the transition matrix P of

the Markov chain Sn be defined by

Pn(0.0) = %

Pn(0.1) = Pn( 0 , - l ) = ^ ( 1 - ^ ) -r-n1*

where a_ = . It turns out that the distributions of the processes 1+T-n14

)C defined as above converge weakly to the distribution of Y^.

In this thesis only excursions from a single state a are treated. It looks as if it is not too difficult to generalize this approach to the description of excursions from a finite set of states. It seems that one will need Cox processes to describe the excursions from a finite set of states. These Cox processes will not satisfy the renewal property, which will be replaced by some kind of Markov property. See also Itö [25], who proposes to call the excursion point process Markov in this case. However these Markov excursion point processes are not discussed by Itö.

POINT PROCESSES

A point process is a random distribution of points in some space X. The case where X is the real line, more generally a locally compact, second countable Hausdorff space or a separable metric space, has been studied extensively. One always assumes that there is a family y of subsets of X, each of which can contain only.a finite number of points, y is the family of compact subsets if X is locally compact and if X has a metric structure then V is the family of bounded subsets of X.

Mathematically the concept of a point process is formalized as follows. Let X be a topological space and let if be a family of open subsets of X. To a distribution Z of points in X we assign the point measure

2 ö„, where 6„ is the Dirac measure in z. The description with

z€Z z z

measures on X has greater flexibility than the description with subsets of X and is mathematically more convenient because of the richer structure of the linear topological nature of the space of measures. Moreover in the case of point processes with multiple points the approach via measures is more natural. So let M = M (if) be the set of all nonnegative Borel measures on X which are finite on the elements of V. Denote by d the smallest a-algebra on Ji which measures the maps lx€M+ -> ^ ( A ) , A € #(X). Let A" = M"{V) be the subset of 1+ consisting of

the point measures on X. An ^-finite point process on X is a probability measure on (M M) which is concentrated on Ji", or an ./«"-valued random variable where we identify a random variable with its distribution. However, the measure-theoretic introduction of the CT-algebra d is not quite satisfactory. There are several reasons to

prefer a definition of sA as the Borel a-algebra corresponding to some topological structure on Ji '■ a topology on M , which induces the corresponding narrow topology on the space of measures on M+, makes it possible to discuss weak convergence of point processes. Further, measurabi 1 ity properties of subsets of M (for instance M") can be derived from topological properties and there is a powerful disintegration theorem for measures on topological spaces.

As an example, consider briefly the set of nonnegative Borel measures on a locally compact, second countable Hausdorff space X. Then M = JM (y) is the set of all Radon measures on X (if is the family of compact subsets of X ) . Let 3f = 3f(y) be the set of all continuous functions on X with compact support. Endow M with the vague topology T = o(M ,3f) of pointwise convergence on the elements of 'X. A net (u„) in M* converges vaguely to u € A iff f^a(f) -» f-t(f) for each f € 3f, where n(f) is the functional-analytic notation for the integral of f with respect to p.. The vague topology renders M a polish space, i.e. M is metrizable with a complete metric. The Borel a-algebra 28 on (,/M , T ) coincides with the a-algebra si generated by the maps p. € Jk -* u(A), A € 28(X). The basic result on weak convergence is Prohorov's theorem, which gives a characterization of the relative compact subsets of (M , T ) . The set of point measures M" is a vaguely closed subset of M+. See for proofs Bourbaki [5] and Krickeberg [34].

In the literature about point processes on complete, separable metric spaces (X,p) one studies always y-finite point processes, where if is the family of bounded Borel subsets of X. The point processes which arise in excursion theory turn out to be ^-finite point processes on a polish space U, where y is some family of open subsets of U.. The theory of point processes on complete, separable metric spaces is not applicable in this case, since it is not clear whether there exists a complete metric d for U such that if coincides with the family of

d-bounded subsets of U. So, before we can study excursion theory we

have to study the set M (y) of y-finite Borel measures on a polish space U for some family if of open subsets of U.

Let X be a completely regular Suslin space (for example a polish space)

and let if be a family of open subsets of X such that

(i) y is filtering to the right with respect to inclusion,

(ii) y has a countable cofinal subset, and

(iii) y covers X.

We will construct in section (1.1) a topology on the set M+ = A+(y) of

nonnegative, y-finite Borel measures on X. Let % = 3f(y) be the set of all bounded, continuous functions on X with support contained in an

element of y. Equipped with the topology T = a{M ,3f) of pointwise convergence on the elements of 3f, M turns to be a Suslin space whilst the Borel a-algebra on (A ,T) is identical to the a-algebra si generated by the maps |x € i -> n(A) , A € 28(X). For polish spaces X we have the

stronger result that (J ,T) is a Lusin space. Our treatment is based on

the results for polish, Lusin and Suslin topological spaces in Schwartz

[49]. The set of y-finite point measures M"(y) turns out to be a closed subset of M , as it is for the vague topology on the set of Radon measures on a locally compact, second countable Hausdorff space. In

section (1.2) we will discuss y-finite point processes on a polish

space X. The existence of Palm measures, which are, loosely stated, the

conditional distributions of the point process if it is known that a

certain element x € X occurs (see Jagers [28] for the case that X is

locally compact), follows now from a general disintegration theorem

from topological measure theory. Further Poisson point processes and

Cox processes (so called doubly stochastic point processes) are

discussed. It will be shown that for every y-finite measure v on X there exists an y-finite Poisson point process with intensity measure

Poisson point processes on a polish space X, which is the topological

product of the half line [0,°°[ and a polish space U, whilst 9 consists of the sets I x 11 where I is an open, bounded sub-interval of [0,co[ and (U ) vi is an increasing sequence of open subsets of U which covers

U; the intensity measure is the product of the Lebesgue measure X and a

Borel measure u on U which is finite on the sequence (Un)n>x- It° w a s the first who studied these processes as stochastic point functions and

that is the reason why we call them Itö-Poisson point processes. In

section (1.3) we discuss the renewal property for Itö-Poisson point

processes and the characterizations of these processes which were given

in I to [25] and in Greenwood & Pitman [17]. We will give full proofs of

slightly more general versions of these theorems which were originally

formulated in terms of stochastic point functions; the renewal property

was stated in Ito [25] without proof.

1.1 Topological spaces of Borel measures.

Let X be a Suslin space with Borel a-algebra S8(X). A Suslin space is a

Hausdorff topological space for which there exists a polish space Y and

a continuous surjection from Y to X, see Schwartz [49], p. 96. To have

enough continuous functions on X we will assume that X is a completely

regular space, i.e. for each x € X and each open neighbourhood U of x

there is a continuous function f on X to the closed unit interval such

that f(x)=l and f is identically zero on X\U.

Let G be an open subset of X. Equipped with the relative topology, G is

a completely regular Suslin space (Schwartz [49] theorem 3, p.96).

Denote by (^(G) the space of bounded continuous functions on G and by

3f(G) the subspace of CV(G) consisting of restrictions to G of bounded

continuous functions on X with support contained in G

«(G) = {f |G : f € C y X ) , supp(f) C G}.

by ^u(G). Endowed with the narrow topology, that is the topology

TjfG) = a(j«£(G), C^G)) of pointwise convergence on C ^ G ) . ^ ( G ) I s a Suslin space, see Bourbaki [5], p.6. Denote by T Q ( G ) the topology

CT(^(G), S(G)) on JH^(G). J t i s clear that T2( G ) C T ^ G ) .

Note that T^CG) £ Ti(G). Indeed if (x^) is a sequence in G converging for n-*» to a point x in the boundary of G, then the sequence of Dirac

measures (ö ) converges in the space (^(G), T^fG)) and diverges in

the space (^(G) , T ^ G ) ) .

1.1.1 Proposition. Let X be a completely regular Suslin space. If G is an

open subset of X, then (A.(G), To(G)) is a Suslin space.

Proof. Since To(G) C T,(G) and (JL (G), T , ( G ) ) is a Suslin space, it is sufficient to prove that (A,(G). To(G)) is a Hausdorff space. Let 0 C G

be an open subset of G and let x € 0. X is a completely regular Haus­

dorff topological space, so there is an open neighbourhood V of x such

that V C 0 and there is a continuous function f on X to the closed

unit interval such that f (x)=l and f is zero on X\V. It is clear that

supp(fx) C V C O C G a n d l0= sup{fx : x € 0}.

Any Suslin space is a Lindelof space, so that the family {f : x 6 0}

has a countable subfamily (fn)n\i with the same upper envelope 1Q, see Schwartz [49], p. 103 and 104. Define for n > 1 the function g^ as the

pointwise supremum of f j up to and including fR. The sequence (gj,) is an increasing sequence of functions on X with support contained in G

and with supremum 1Q. Hence for v,p. € ^ ( G ) we have V f € «(G) : u(f) = u(f) => u(0) = u(0)

for any open 0 in G. Let

V = {A : A € «8(G), u(A) = *i(A)}.

if is a d-system containing the open subsets of G. By the monotone class theorem it follows that 9 = SS(G), see Williams [59], p.40. So «(G)

separates the points of ^D(G) and this implies that TgfG) is a Haus­ dorff topology on ^ ( G ) . □

1.1.2 Remark. A Hausdorff topological space is said to be a Lusin space if

there exists a polish space Y and a continuous bijective mapping from Y

to X, see Schwartz [49], p.94. It is clear that any Lusin space is a

Suslin space. If we assume that X is a polish space, then the topologi­

cal space (JCU(G) , T , ( G ) ) is also a polish space (see Bourbaki [5],

p.62) and we may conclude that (A.(G),To(G)) is a Lusin space.

Let 9 be a family of open subsets of X, which is filtering to the right with respect to inclusion, i.e. V A,B e V 3 C € y> : A C C and B C C. For all pairs A,B € SP, A C B , we define the map Tr.R by

*AB : < W "» ^b(A) ■ " A B W = A"

where «fi denotes the restriction of jx to A: A^(G) = n(G), G € 26(A). It

is clear that ((JKH(A), Tn{^)). ^ A R ) is a projective system of Suslin spaces. Note that ((^K(A), T , ( A ) ) , ■"'AR) is not a projective system of

topological spaces, as the TTAD are not Ti-continuous. The projective

limit M = M(9) = lim Tr*g -*u(B) is the subspace of the product IT A,(A) whose elements fi = (u.^) satisfy the relation ^A^ARCI-'B) whenever A C B . The projective topology on M is the coarsest topology which makes the

projections

TT

: M -».J£(B). ir

((n

)) = u

continuous and is therefore the trace on M of the product topology on

II ^u(A). Let A+ - <K+(y) be the space of nonnegative Borel measures on (X, 28(X)), which are finite on y. Elements of M are called ^-finite measures.

1.1.3 Proposition. If if covers X, then the map <p : n € jf -» (Afx) € M

is a bijection of JT onto M.

Proof. It is clear that ^ maps M into M.

Let (fxA) € M and G € «6(A) D 28(B) for some A.B € if. Since if is filter­

ing, there exists a C € if such that A,B C C.

Then HA( G ) = ('"AC ^ t ) ^ ) = ^ ( ^ l"l G ) = Mc(G) and in the same way

fig(G) = M<-.(G) .Therefore

V ■ U «8(A) -» R , n(G) = nA(G) if G e «8(A)

is an unambiguously defined setfunction on the ring LHS(A) .

If (G_)_xi is a pairwise disjoint sequence in USS(A) with union 2G

contained in 1128(A), then G . S G ^ e 28(C) for some C € if.

It follows that

n(2Gn) = fic(2Gn) = 2fic(Gn) = 2n(Gn).

So u. is a finite, a-additive measure on (X, U B(A)). AGf

Being an open cover of a Lindelof space, if has a countable subcover. It

follows that 28(X) is the a-ring generated by the ring LE8(A) and p. has a

unique extension to a measure fi € M . See Halmos [18], p.54. It is

clear that >p(p.) = (u^).

This proves that f is a bijection of Ji onto M. □

Assume that if covers X. Denote by T = r(if) the coarsest topology on M

which makes the bijection <p : M -» M continuous.

Defining

3f(y) = {f € (^(X) : 3A € if : supp(f) C A } ,

T is equal to the topology a(M , X{if)) of pointwise convergence on

%(if). If 3 is a cofinal subset of if (i.e. for each A € if there is a

D e SB such that A C D ) then X(if) = 3f(2J) and o(jt. 3f(y)) = a(J«+, S(ffl)).

Denote the Borel a-algebra on (jt,T) by 28 = 28(jM+(y)) and define the

a-algebras s^ and s^ on ^ by

and

si2 = a(u £ J+^ u(f), f € X(if)).

1.1.4 Theorem. Let X be a completely regular Suslin space and let if be a family of open subsets of X filtering to the right, if is a cover of X and if if contains a countable cofinal subset, then

(i) (A+{if), r(if)) is a Suslin space and

(ii) J^1 = s$2 = 28.

Proof. Let 2) be a countable cofinal subset of if. Being a countable projective limit of Suslin spaces, M(2)) is a Suslin space, see Schwartz

[49],p. 111. Hence (J«+(2)), T(2!)) is a Susl in space. It is clear that

M+(if) = J?(*b) and T(2!) = r(i?). which proves (i).

The family of continuous maps u € M -* fi(f), f € #(y) , separates the points of jt(if). Indeed, let p.,v € M+ so that n(f) = u(f) for every f € #(y) and let A e if. Every f € #(A) being the restriction to A of a function g € %(if) with support contained in A,

Au(f) = u(g) = «(g) = Au(f).

So M = iu by the proof of proposition (1.1.1) and it follows that p. = v since A was arbitrarily chosen in if. Since (A+(if), r(if)) is a

Suslin space, there is a countable subfamily (fn)n\i of X(if) such that the points of A {if) are separated by the maps ^ : fi € M+(if) -» n(f ).

By Fernique's lemma, the sequence (>/'_) generates 28. See Schwartz [49],

p. 104, p.105 and p. 108.

So 28 C

s&2-Let now f € %(if) and let A € if be such that supp f C A. Since f is a continuous function, there exists a sequence of 28(X)-stepf unctions,

zero outside A, converging uniformly to f. It follows that the map

p. € A (if) -» u(f) is s0,-measurable. So s^9 c ^ i •

Let A € if. Let O C A be an open subset of G. As in the proof of prop­ osition (1.1.1) we can construct an increasing sequence (fn)n>i in *{^) with supp(f ) C A and with supremum IQ.

It follows that the map fi € M (if) -» fi(0) is SS-measurable. A monotone class argument gives the S8-measurabi 1 ity of the maps p. € jt{if) -» uifi), G € (J 2Sf A"). Since if has a countable cofinal subset, every Borel set

KGf K '

in X can be written as a countable union of elements of U 3i(A) .

So tiy C 28.

It follows that slj = sk, = 38 • D

From now on we will assume that the space X is a polish space. Let d be

a metric on X such that the metric topology is the topology of X and

(X,d) is a complete metric space, if will be a fixed family of open subsets of X satisfying the conditions of theorem (1.1.4). By if' we will denote the family of all Borel subsets of X contained in some

element of if. Note that the space (U(+,T) of y-finite measures is a Lusin space in this case, see remark (1.1.2).

1.1.5 Remark. Even for polish spaces it need not be true that a filtering

family of open subsets, which covers the space, has a countable cofinal

subset. For example, let X be the space of all pairs of non-negative

integers with the discrete topology. X is a polish space. A set A is

member of the family if iff for all except a finite number of integers m the set {n : (m,n) e A} is finite, if is a filtering family of open subsets of X, which covers X. But if does not have a countable cofinal subset. Indeed, let (Ai^^M be a sequence of subsets of X contained in

if. For every k > 1 we can choose an element x^ = (m,n) € X such that n > k and x^ t A^. The set B = {x^ : i>l} is an element of if and there is no Aj^ such that B C Ak.

The following proposition provides a condition equivalent to

convergence of a net in (M , T ) .

1.1.6 Proposition. Let X be a polish space and let 9 be a family of open subsets of X satisfying the conditions of theorem (1.1.4) and let

((i ) be a net in M+ and (i € M .

Then the following statements are equivalent:

(i) va -» (J. in (jlf4", T ) .

(ii) limsup Ha(F) < n(F) for all closed F € if' and liminf Ha(0) > u(0) for all open 0 e if . Proof.

(i) * ( Ü )

Let F be a closed subset of X, F C A for some A € y and let 2) be a

countable dense subset of X. Define

I = {(x.q) : x € 2), q € Q+, Bx(q) R F = 0}

where B (q) = {y : y € X, d(x,y) < q } . I is a countable set. For i = (x,q) € I, the sets F and A U B (q) are disjoint closed sets,

where A denotes the complement of A. Since X is a normal topological

space, there are disjoint open sets U and V such that F C U and

A U B (q) C V. By Urysohn's lemma there is a continuous function {■ on X to the interval [0,1] such that f, is zero on U and one on F. It is

clear that supp (fj) C U 0 V* C A, so f; e 1 If y € F, then there is an element (x,q) € I such that y € Bx( q ) .

It follows that

1F = inf {ft : i € I}.

Define g^ = inf{fi fj }, n > 1, where (in)n>i is a n enumeration of I. It is clear that (gn) is a sequence in 3f converging pointwise to

lim Uai&n) = n(gn) and

limsup u ( F ) < limsup ^ ( g j = ^(gn)• a a It follows that

limsup ua(F) < u(F).

Let 0 be an open subset of X, O C A for some A € if. Let (gn) be an increasing sequence of bounded continuous functions such that

supp(g ) C A and 1 Q = sup g^. (See the proof of proposition (1.1.1).)

For each n > 1

lim ua(gn) = ^(gn) and

liminf Ua(0) > liminf ^a(gn) = n(Sn)• It follows that

liminf na(0) > H(0). This completes the proof of (i) ^

(ii)-( ü ) =>(ii)-(i)

Let f be a bounded nonnegative continuous function on X with

supp(f) C A for some A € if.

Define f o r k > 1 the functions UK, vk : X -» K by 1 u . = 2 - 1 K ,•_{i>l k} M I, _ 1

[f < -] n A

k

k =

W f

j

\

1 •

the summations being finite summations since f is bounded.

It is clear that uk < f < vk, for all k>l and that uk | f and v^ 1 f. Hence 1 1 l i m i n f u„(u. ) > 2 - liminf yin ( [ f < - ] PI A) a a K i > l k a. k > 2 - u ( [ f < - ] 0 A) by ( i i ) i > l k k = n ( uk) and a n a l o g o u s l y limsup ^a( vk) < f i ( vk) .

It follows that

u(uk) < liminf Ha(f) < limsup Ma(f) < Viv^)• By taking limits for k -* °° we get

lim uQ(f) = u(f) which completes the proof of (ii) => (i).n

A measure |i£ J is called an ^-finite point measure if

V G € a" : u(G) € IN.

An ^-finite point measure is called simple if

Vx 6 X. ^ = u({x}) € {0.1}.

The set of y-finite point measures will be denoted by M" = M"(V) and the set of simple ^-finite point measures by Ji' = M'(y).

i

1.1.7 Proposition. Let X be a polish space and let if be a family of open subsets of X satisfying the conditions of theorem (1.1.4).

Then M." is a closed subset of (M , T ) .

Proof. Let (na) be a net in M" converging to \i € M . Take x € supp(jj.) and let U be an open neighbourhood of x, U € V .

Then by proposition (1.1.6)

0 < u(U) i liminf ua(U). Since Ha(U) 6 IN, it follows that

liminf uQ(U) > 1.

Consider now a decreasing sequence (U ) vj of open neighbourhoods of x

in y'such that U 1 {x} and U+^ C U for every n 2 1.

From Urysohn's lemma follows the existence of a sequence (1\.) in 3f(y)

such that ly < 1^ < ly for every n > 1. n+1 n

ufhj = H m ^(hjj) > limsup Jia(Un+1)

> liminf H

( V

) *

and

u = H m fi(lu) > 1.

It follows that supp(^) is a discrete set and therefore for n suf­

ficiently large

^x = ^Un ) = ^n ). . Proposition (1.1.6) implies that

»i(Ün) * limsup p. (0n) a

> limsup na( Un) > liminf ua( Un) > n ( Un) . Hence f o r n s u f f i c i e n t l y l a r g e

and i t f o l l o w s t h a t fi € M". D

L e t Uj Un be a f i n i t e sequence of open s u b s e t s of X such that Uj Un € if and l e t k j 1^ € IN.

Define

Vu. U -.k, k = ^ € ^ " : ^Ui ) = ^Ui ) = k i . i = l . . . T i } . 1 n 1 n

I t f o l l o w s from p r o p o s i t i o n ( 1 . 1 . 6 ) t h a t the map

n e / ^ u(G)

is lower semicontinuous (resp. upper semicontinuous) for each open

(resp. closed) subset G C X in if'. Hence

\ Un;k1 1^=-*" n & : ^ Ui) > ki- i a n d ^ ( Üi) < ki +i 1=1....n} is open in M".

Let 11 be a countable base for the topology of X consisting of open subsets with closure in if' (see appendix Al) and let A,, AQ, . . . be an increasing, countable cofinal subfamily of if.

Define for k, n > 1

°k.n = U VU1 Un;l 1

elements are contained in Ak. It follows that (V is open in A" and that the set

{u£T : u(Ak)=n} = {Ue*" : n(Ak)>n-l} \ {p£A" : u(Ak)>n} is a Borel subset of A".

So

A' = 0 U {u € A" : u(Ak) = n} fl <\ k=l n=l

is a Borel subset of A".

So we have derived the following proposition:

1.1.8 Proposition. Let X be a polish space and let if be a family of open subsets of X satisfying the conditions of theorem (1.1.4). Then M' is a Borel subset of (A , T ) .

In chapter 2 we will be interested in a special class of point meas­

ures on a product space.

Let X be the product T x U of the halfline T = [0,°°[ with the usual

topology and a polish space U. The space X with the product topology is

a polish space. Let (UJ,)I,\I be an increasing sequence of open subsets

of U, Uk| U. Define

y = { I x G : I C T open and bounded, G C U open

and G C l l for some k > 1}.

y is a filtering family of open subsets of X which satisfies the con­

ditions of theorem (1.1.4). Denote by A the set of ^-finite measures on (X,S8(X)) and by A\ the set of simple y-finite point measures p. sat­ isfying the condition

Vt € T : u({t} x U) < 1.

1.1.9 Proposition. Let X and V be defined as above. Then A\ is a Borel subset of {A , T ) .

Proof. The proof is analogous to the proof of proposition (1.1.7) and

is therefore omitted. D

1.1.10 Remarks.

(i) If X is a locally compact, second countable Hausdorff space and

y the family of compact subsets of X, then A' is a dense Gg set in A".

(ii) Let (X.d) be a complete, separable metric space and let if be the family of all bounded open subsets of X. The family y satisfies the conditions of theorem (1.1.4); a countable cofinal subset of

y is the sequence of open balls (B (z)) •.■• with radius n € IN and center a fixed point z€X. Matthes, Kerstan and Mecke define in

[40], section (1.15) a metric p on A". It turns out that (X',p) is a complete, separable metric space and the metric topology on

A" coincides with the relative topology on A" as a subspace of

(A

.r).

(iii) Let (X,d) be a complete, separable metric space and x^ be a

fixed point of X. Harris calls in [19] a (nonnegative Borel)

measure fi on X x^-finite if

(a) n(X\V) < » for each open set V containing Xo, and

(b) M({x„}) = 0.

Let M be the class of x^-finite measures and let 1

Ef = {x : x € X, dfx.x..) > - } , t > 0. The sets Ef are closed and

c t

have disjoint boundaries.lt is clear that u 6 M <=* ^(Et) < m for all t > 0.

Harris introduced in [19] a topology on M, which we will de­

scribe now. Denote for t > 0 by Lt the Levy-Prohorov distance on ^ ( Et) , that is a metric on ^ (Et ^ s u c h t h a t (•*b(Et^ L^ i s a complete, separable metric space and the Lt-topology on ^ ( Et)

If fi, i) € M put

00 -t

Lfu.u) = ^ - d t . J 1+Lt(n.i>)

0 t

where Lt(ji,u) denotes the Lt distance of the restrictions of pi and u to E£. L is well-defined and is a metric for M such that (M,L) is a complete, separable metric space.

Consider now the polish space XXjx,,,}. Let if be the family of 1

open subsets A£ = {x € XXjXo,} : d(x,xu) > — } , t > 0.

if satisfies the conditions of theorem (1.1.4). Denote by jl the restriction of the measure p. € M to XNfx^}, jl is a tf-finite measure on X^x,»}.

Proposition. The map \-]i€.}i-*p,€.M is a continuous b i ­ section from (M,L) onto (M*~ , T ) .

Proof. It is clear that \ is a bijection. To see that \ is con­ tinuous, let f € Sf and let (iK.) be a sequence in M converging to jx, i.e. lim L(u._,fi)=0. Then supp(f) C A,, for all t sufficiently

n-*» n *•

small. It follows that there exists t > 0 such that supp(f) C At 1

and fi({x : x € X, d(x,xK>) = —}) = 0. From Harris [19], theorem t

(2.2) we conclude that

l i m ^ f f ) = l i £A t^ ( f |A t)

= At^fl At) = ^f) .

So the maps n € M -» (\(v))(f), f £ Jf are continuous, which im­ plies the continuity of )(.G

If |i,u £ M+ put

Uli.v) = L (X - 1( n ) . X- 1( " ) )

d-topology on A . From the foregoing proposition it follows that T C Td.

Proposition. Let (nn) be a sequence in M and p € A . Then T-lim (i = |i <=> d-lim p = p.

n-*» n n-i«> '

Proof. The implication (<=) holds since T C TJ. S O assume that 1

T-lim u = u. If fx({x : x€X, dfx.x^) = -}) = 0. then u(6E,.) = 0,

n-*° " t L

where öEt denotes the boundary of Et- By proposition (1.1.6) we have that lim u (Er) = fi(Et). Identifying u. and T(/J.) , it follows

that the restrictions of (p. ) to Et converge in (^b(Et), Lt) to tx, see Topsoe [50], p.40. From Harris [19], theorem 2.2 we may

conclude that d-lim u_ = p. D

So for the topologies T and T , on M we have: ("" ,Td^ *s a P°lisn space,

T Td

"n -* V iff "j, -» (■*•

One cannot conclude from this that r = TJ. Take for instance ( X . T ) as in example E of Kelley [33], p.77 and take for TJ the

discrete topology on X. It is clear that T and T , satisfy the

above conditions and that T ^ Ti.

1.2 Poisson point processes.

Let X be a polish space and let y be a family of open subsets of X

which is filtering to the right with respect to inclusion. Assume that

V has a countable cofinal subset and that y filters to X. Denote by jt the Lusin space of nonnegative Borel measures on X which are finite on

For a finite sequence B, B in SS(X) the finite-dimensional

distribution Pp g is defined as the image of P under the map 1' m

i* e i* ■♦ W B j ) fi(BJ] e ([0.<»])m.

Note that it is a consequence of theorem (1.1.4) that probability

measures on A with the same finite-dimensional distributions are identical.

The Laplace transform P of P is defined by

P(f) = J P(dn) exp [- J f(xMdx)]

M+ X

where f runs through the cone S8(X)+ of nonnegative measurable functions. The moment generating functions of the finite-dimensional

distributions Pg 3 are determined by P as follows from

1 ■ ' ■ m

ƒ«?•■■

m™

B

.--

m

---

f ^ C

V

-where 0 < Uj < 1, i=l m. So P is uniquely determined by its Laplace

transform. The intensity measure i=ip of P is the Borel measure on X

defined by

i(B) = J P(d*i)n(B). B 6 28(X).

We say that P has y-finite intensity if i e M+. Denote by p the

Campbell measure of P, that is the measure on J x X defined by

J" F(fx,x)p(d^.dx) = J P(dn) |fx(dx) F(u.x).

jtxX *+ X

It is clear that p is a a-finite measure if the intensity measure i of

P is ^-finite. The projection p(M x .) of p on X is the intensity measure i of P. If the intensity measure ip of P is ^-finite, then a

general theorem on disintegrations of measures (see Bourbaki [5],

section 2.7) implies the existence of a measurable family of probability measures (Px)xgx o n ^+ s u c n that

J F dp

"

r r f r

= J P(dji) J n(dx) F(jx.x) = J i(dx) J Px(d^)F(tx.x) ( K )

M+ X X ,«+

probability measures P on M are called Palm measures of P and the formula (*) is the so-called Palm formula for P. Note that P is

completely determined by the intensity measure ip and the Palm measures

(P ) ev- A straightforward calculation gives a formula for the Laplace transforms of the Palm measures P . Let f ,g € 28(X)+, then

| ip(dx) Px(f)-g(x) = - — P(f + t g ) |t = 0.

A probability measure P on M will be called an y-finite point process with phase space X or an ^-finite point process on X if ?(M"(iP))=l. If the phase space and the family if are clear from the context we will speak of a point process. A point process P will be called a simple

point process if P(JM*)=1. A S usual in probability theory, an M -valued random variable N will also be called a (simple) point process if its

distribution on Jk is so. A point process is said to be free from after-effects if its finite-dimensional distributions satisfy the

relation

\...Bm = ?B® •■• 8 PBm

1 m 1 m

where Pg ®. . .8 Pg is the product of the measures Pg on IN. A point

process P will be called a Poisson point process with intensity measure

i) if P is free from after-effects and if the one-dimensional distribu­

tions Pg. B € 28(x), are Poisson distributions with expectation u(B),

l>(B)]*

k!

PB({<*>}) = 1 if v(B) = co. PB({1<}) = ±-±-^- e " W , k=0,l,2.... if u(B) < «>,

1.2.1 Proposition. Let P be a Poisson point process with ^-finite intensity

measure u. Then the Laplace transform P and the Palm-measures Px are given by

P

= ö

*

. x e x.

where 6 denotes the Dirac measure on M in the point 6X. Proof. Follows from standard calculations.D

1.2.2 Proposition. For every v € M+(y) there exists a unique ^-finite Poisson

point process on X with intensity measure v.

Proof. Let A € y. The restriction i» of v to A is a finite measure on (A,3S(A)). So there is a unique {A}-finite Poisson point process »P on A

with intensity measure .p, see Matthes et al [40], section 1.7. iP is a

probability measure on

(ib(

). #(•*£(

)))'

9S(^b(

))

CT-algebra on (A,(A), To(A)), see for To(A) the definitions preceeding

proposition (1.1.1). Let A,B 6 y, A C B and let ir^g be the projection

of JM^(B) on ^ ( A ) as defined in section (1.1). The image T ^ C R P ) of gP is a probability measure on (J«Ï(A) , S6(J<K(A))). A straightforward

calculation gives (T^QI-QP)) = (^P) . So ^^B^P) = ^P and it follows that (^(A), 3KA,(A)), ^P, TTAD) is a projective system of probability spaces. Since if has a countable cofinal subset, it is a consequence of Bochner's theorem (see Bochner [4], p. 120) that there exists a

projective limit P, which is a probability measure on (M+, 26(JM+ )). An easy calculation yields that P is the ^-finite Poisson point process on

X with intensity measure P.D

Denote by P the ^-finite Poisson point-process on X with intensity

measure u € JH. Note that P is a simple point process iff the intensity measure v is a diffuse measure (i.e. i>({x}) = 0 for every x € X). The family of point processes {P^: u € M } is a measurable family, i.e. for every G € S5(J+) the map u € J+ -» P (G) is measurable. Let V be a probability measure on {M. , 2S(Ü )) and let Q be the probability measure

on (A+. $(M+)) defined by

Q = J V(dp)P

.

It is clear that Q is a point process on X, which is simple iff V is

concentrated on the diffuse measures in j t . Such a process Q is called a Cox process.

1.2.3 Proposition. Let Q be a Cox process as defined above. The intensity

measure iQ, the Laplace transform Q and the Palm measures (Ovivcv

of Q are given by

iQ(B) = iv(B) , B € «(X). Q(f) = V(l-e"f) . f € Jg(X)+. Ox = «x * Jvx(d«)Pn • x e X.

where 6 denotes the Dirac measure on M in the point 6 .

Proof. The formulas for iq and Q follow directly from the definitions.

To prove the formula for C^, let F : M+ x X -» K be a measurable,

non-negative function. Then

JQ(dn) Jii(dx)F(n.x) = Jv(du) |Pu(dn) Jn(dx)F(n.x)

= Jv(du) |u(dx) |(6X * PB)(dn)F(n.x)

= Ji

(dx) Jv

(d

)Jp

(d^)F(^+ö

,x)

= Jiq(dx) |(6

* Jv

(d«)P„)(d^)F(^.x)

from which the result follows. D

1.3 Ito-Poisson point processes.

Let X be the product T x U of the halfline T = [0,<°[ with the usual

topology and a polish space U. The Borel a-algebras on T and U will be

denoted by SU. and "U. Endowed with the product topology X is a polish space and its Borel cr-algebra 3S(X) is identical to the product

cr-al-gebra SU- ® "11. Let (Uk)kvi be an increasing sequence of open subsets of U which covers U. The family if of open subsets of X defined by

y = { A : A = I x G , I C T open and bounded,

G C U open and G C Uk for some k > 1}

is filtering with respect to inclusion and contains a countable,

cofinal subset. The topological space of y-finite measures on X will be

denoted by (A+ ,T) , see section (1.1). The Borel a-algebra 'S on A is

identical to the a-algebra generated by the family of maps

{p»: A€S8(X)}, where p^ is defined by p^ : v € A+ -» u(A). The family CSt)t>0 of sub CT-algebras of 'S defined by

<St = CT(PA, A € 28(X), A C [O.t] x U)

is a filtration on (A+,"£) . A measurable map -ty ■ A -» T is called

($,.)-adapted if [^ < t] € ^L for every t € T. An Itö-Poisson point process

on U is an ^-finite Poisson point process P with phase space X whose

intensity measure p. is the product of the Lebesgue measure X on T and a nonnegative Borel measure v on U, which is finite on the sequence

(Ui.)j.vi. Following Itó [25], u is called the characteristic measure of

the Ito-Poisson point process P.

1.3.1 Proposition. Let P be an y-finite Itö-Poisson process on U, then

P(Jj)=l.

Proof. Since the intensity measure fi = A®i> of P is diffuse, P is a

simple point process on X. Define the mappings TT^ (k>l) by

irk : A' -> Aj , irk(u) = [B e Aj. -♦ n(B x Uk)],

where Aj is the space of point measures on T which are finite on all bounded subintervals of T. The map Fu is a P-a.e. defined, measurable

map on A . The measure Pk = wk(P) is the Poisson point process on T with intensity measure ik = u(Uk)*A. Since the intensity measure i, is diffuse, the point process Pk is a simple point process. It follows

that

and

P U k1 (-»!■)) = PkC-*r) = 1

-i, k>l

which completes the proof of the proposition.D

Let <p ■ M -» T be a measurable map. Define the transformation R by

R : jt -» J<+. 9

|R^(u)(dadu)f(a.u) = J^(dadu}f(a.u)l[ 0 i V ( ( l ) ](a). f€S(x)., Let for CT € T the map t be defined by

ta ■■ ( T . V ) € ]a.»[ x U -♦ ( T - O . V ) C X.

Define the transformation T by

T : M+ -» J<+,

JT^fuJCdaduJfCa.u) = Jfi(dadu)fotv,(>1j(a.u}l^(|x)ia,[(a) = JfiCdaduJffa-^CfiJ.uJl-j^^j^for).

We will write simply R and T if f is the constant map p. € JH -* s. The following picture illustrates these definitions. The picture shows supp(u). supp(R u ) and supp(Tj^) for a simple point measure p..

supp(RpH) «

supp(u)

s u p p ( T ^ )

1.3.2 Lemma. Let f '■ M -» T be a measurable map. The above defined transformations R and T are measurable.

Proof. Define for k,n = 1,2,...

Akn = & € M+ : k # 2 _ n * *M < (k+l)-2_n} and

*n = 2k t ^1) *2"1 1^

-The sequence of measurable stepfunctions (vn)nsi is a strictly de­ creasing sequence, which converges pointwise to <p. It is clear that for every bounded continuous function f : X -» IR with support contained in

some element of V and for every |i £ i lim (T ,x)(f) = (Tu)(f)

n-*° rn ^

and

lim (R u)(f) = (Rji)(f). n-*» *n ^

It follows that the sequences (T ) •.. and (R ) v1 converge pointwise to T and R . So it is sufficient to prove the measurability of T and

R . Let A € SS(X) and fi € M+. Since

MI (R n)(A) = 2 lAl M n(A n [0.(k+l)-2-n[ x U) MI k kn and (T H)(A) = 2 1A (n) n((t 0-n)_1(A)). MI ',ik "kn (k+l)'2

it is clear that the maps |i C / -» (R n)(A) and )i € / -» (T JLI)(A) are

*n Mi measurable maps which implies measurability of the transformations R

Ml

and T„ .D

MI

1.3.3 Theorem (Renewal property). Let f '■ M -» T be a measurable map and let P be an Itö-Poisson point process. If </> is CSt)-adapted, then R and T are independent M -valued random variables on {JT.'S.?) and

VP)=P.