polar sets and palm measures in - American Mathematical Society

TRANSACTIONSOF THE AMERICANMATHEMATICALSOCIETY Volume 208, 1975

POLAR SETS ANDPALMMEASURESIN THE THEORY OF FLOWSt1) BY

DONALD GEMANAND JOSEPH HOROWITZ ABSTRACT.

Given

prove that certain

decompose

uniquely

and a measure discrete

parts

those

which neglect

actly

those

functional

into a measure

by one. sets

neglected

properties,

functional

which

by supermartingales

set"

These

results

in the theory

and a

are exactly

and polar

Finally,

sets

are ex-

we characterize

and continuity,

measure.

set" and

to 0, we find that

no "semipolar

the flow neglects,

such as predictability

no "polar

Palm measures

by every Palm measure.

ß, we

of the flow)

the continuous

corresponding

charging

space

space

0 which charges Considering

As a consequence,

in terms of its Palm

the role played

set.

over a probability as the state

measure

by a polar

of the additive

supported

various

on 12 (viewed

into a Palm

supported

Q further decomposes measure

a flow (0 ), t real,

measures

of an additive further

of flows,

illuminate

as pointed

by

J. de Sam Lázaro and P. A. Meyer.

0. Introduction. Let (0, ?°, dynamical

that

system (all terminology

a finite

tinuous" where

measure

uniquely

is the restriction

able additive

composing

will be explained

Q on J„

decomposes P~

P, 9 ), t £ R (the real line), be a filtered

functional

of the Palm measure

a, and p is supported

a into its continuous

and discrete

polar"

which is carried

set and a measure set.

PJ

P~

Q = P~

con+ p,

Pa of a predict-

by a "polar"

parts,

into a measure

In §1 we prove

absolutely

into the sum of two measures to 3q_

will see (in §2) that P~ splits polar,

below).

which is "progressively

setin

JQ_. De-

say &c and a¿, we which charges

no "semi-

by a semipolar, but charges no

Thus we have a decomposition

(1)

Q = P~c+Pd + d

analogous

to that of a measure

on the state

space of a Markov process

[l,

Received by the editors April 11, 1974.

AMS(MOS)subject classifications (1970). Primary 28A65, 60G10. 7 of all continuous

tions having left limits)

in R. The basic

which is constant

to add that many of the results

arising

and (ii) for every reaK-valued

be a Blackwell

/ £ (ë)

though they are more complicated tion spaces

is separable, ¿j(A) is analytic

all sets of the form Í0] x A, A £

and (t, oo) x A, t £ R+, A £ ÍF°_). This is similar

able cr-field [2], but more appropriate

of one another.

by all sets of the form

in the present

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

to the usual

context.

predict-

POLAR SETS AND PALM MEASURESIN THE THEORY OF FLOWS

145

(6) Lemma. 9° = 0+(iB+® 3*}_),i.e. u £ (9°) iff 9~u £ (S+ ® ?£_). First note that (R+ x O, $+ ®? ) is a Blackwell space, and 58+ ® ÍFq_ is a separable

sub cr-field of 58+ ® j

[t, oo) x A, with A £ J?

9~u is constant contained

, and let u(s, a>) = 7r, Âs)!^^*

Then, for every

of 9 , say

We wish to show

(s, co), (s1, co') are

B £ ÍB+, C £ JQ_,

we have

= ¡As ')Ic(co '), so s = s ' and co, co ' lie in the same atom of 3"0_.

Hence 9~u(s, both vanish

ïj_.

a generator

on the atoms of 58+ ® 3~0_> Suppose

in such an atom.

IB(s)Ic(a)

. Consider

co) = I[ftB0^)IA(e_jo)

and 0~«(s ', t, because

then 9sA £ J,.s\_

We have shown ?° C 0+(53+® S*_), Next, let v £ (S+ ® 3* J

the form i 0.

\Nt+s+Yt+s>

di)

zi+s \e-tNso9t+e-tYso6l.

It is tedious,

but straightforward,

s > 0, are local class

(D) potentials,

uniqueness, e~lN

martingales

° 9

fications,

all relative

the two expressions for all s, a.s.,

zV, Y are "almost

By the Doob-Meyer

that both Y,

correctly, for Y

and e~lMs

J

° 6(,

o 0 , s >0, , s > 0.

are

By

and we conclude

. Putting

and may be replaced by N and

N

and e~lYs

to the cr-fields

"match"

and similarly

homogeneous"

which we again denote

M - A, where

to check

and that both

zV

=

s = 0, we find

by homogeneous

modi-

Y.

decomposition

theorem

M = (AL) is a uniformly

integrable

[13, p. 119] we may write martingale,


and A = (A )

Y=

147

POLAR SETS AND PALM MEASURES IN THE THEORY OF FLOWS is a predictable geneous,

(= natural)

integrable

increasing

we have two decompositions

of Y,

process.

Since

Y is homo-

for s > 0 (t fixed) analogous

to (11):

M,t+s -A, t+s = AL -A,-(A, t+s t

t+s

-A),/"

Yt+s

e~'M St o0 -e-'A Noting that

e~lMs

and e~lAs

° 91, A

tive to Íj

° 9

and Al

- A

St

o0#.

ate uniformly

- At are predictable

î, s > 0, t fixed),

integrable

increasing

martingales,

processes

we have by the uniqueness

(all rela-

of the decomposi-

tion,

(12)

At+s-At

Now let follows

at = flQes dAs. easily

Maisonneuve

that

(By /*

= e~tAs°ef

we will always

a is a predictable

mean f,a &i.) From (12) it

AF. (This

argument

was inspired

by

[10].)

Now for any Palm measure,

say

Pa,

we have (see

[5])

( 13)EJ" u(s, co)ß(ds,co)= f J* z/s, 9_sco)dsPß(dco), Hence for any A £ S

u £ (9>® S°) +.

,

EiYt-A)a Efie"daa-,

a) -Jê-Pjd^ds.

So we may write

( 14) J* e~sQo 9siA) ds = EiNt; A) +f" e~sPao 9 siA) ds, Now define a measure Q - P~,

P~

ft is positive

fit on 9

by equation

being the restriction

A e 5%.

(8) with Q replaced

of Pa to 3q_.

by p =

Using ( 14), we see that

on sets of the form (t, oo) x A, A e J,

, and hence

on all of

9 ; moreover

(15)

E(Nt;A)=p[(t,~)xA]afê-spo9s(A)ds,

(The existence

of a measure

( 15) is established Having chosen

function

j£ on R+ x 0 satisfying

by FoTlmer [3] in a different

that p > 0 so it only reamins

AeS%.

situation.)

to prove that p lives

zV homogeneous,

i.e.

the first equality

finite left limits

Moreover,

for all f € R a.s.

since

It is easy to see

on a polar set.

N s e~'ñ

ñ e (S q ) + , note first that we may extend

still have a supermartingale.

° 9

for an excessive

N( to all t £ R and

N, will be right-continuous t —ñ

in

° 0


with

has these properties.

148

D. GEMANAND J. HOROWITZ

Define,

for each

n > I, Rn = inf \r> 0, rational:

family of stopping

incides

with inf \r> 0: TV > «i. Starting

shows that the stochastic

interval

K = ÍI ]]7? , oo[[ £ 9°.

and that

Nr> n]. Each

7? £ S°, the

times of I**I+l» ' e R+> and on right-continuous

The argument

K is evanescent.

with discrete

]]T, 0. Then (r, 0_fco) £ K for some r > 0, i.e.

R (0_rco) < r for all « > 1, and this puts

co in Gc. But {£> 0] C Gc implies

{f > 0] is polar

null set.

since

Gc is an invariant

This

completes

the proof

of (9). We now sketch

the proof of another

decomposition

Q on SQ_, which is valid under the additional standard

for an arbitrary

assumption

is ff-isomorphic

(b) for any increasing

to the Borel cr-field of a Polish

sequence

/ , and decreasing

sequence

such that A n is an atom of J, t' , we have M A 4 0. ' 'n n ^ n Unfortunately, the usual filtrations on the standard

such as E and 52 previously We will indicate For each

jj

i is a

system [3], [15], which means

(a) each J

Let

that

(finite)

mentioned,

later how to circumvent

O be a finite measure

t e R+, the measure

spaces

are not standard this difficulty

cm

on J n and define u —

of sets A ,

of flow theory,

in the above sense.

for those two cases.

Q as before (see (8)).

f\,

Q has a Lebesgue

space;

decomposition

on J~t+,

namelyQf(A)= Q'fA)+ Q'¡(A),with Q¡ « P on JJ+, and Q" 1 P on 3^ (1 means

"singular").

An easy argument using

besgue decomposition

shows that Q = e~lQ^

Zj = dQt/dP

. We may choose

Z = (Z()

on J(

° 0

the uniqueness

a homogeneous

just as in the proof of (9), and this splits

plus a class the Föllmer

(D) potential measure

such that the first equality

in ( 15) holds.

N, i.e. the unique

We thus have,

proof of (9),

(16)

version

Q = Qa+p+V


° 0 . Let

of the potential

into a local martingale

Y, both of which are homogeneous.

of the local martingale

of the Le-

and Q = e-íO"

N

Now let pt be measure

on 9

a being as in the

POLAR SETS AND PALM MEASURESIN THE THEORY OF FLOWS where

Qa is defined

by the right-hand

side of (8) with Pa in place

149

of 0, "v =

Q - Qa- p. In this way, v[(t, oo)x A] = 0¡(A), A e S°t+. Equation tinuous

(16) exhibits

measure

Q as the sum of the progressively

M = Qa + p

One establishes

easily

and the "progressively

that such a decomposition

Define measures p, v on S\_ (17) Lemma.

For every

(18)

Proof.

Define

a transformation

(o)dsp(dco)

T

on S+ ® J class

fot each

argument

t € R+ by

shows

that

T : (f

)+

From (8) we find that

QiTâe'Qil^^ttu),

By looking

at the generators

gressively

absolutely

holds

is unique.

for v" and v.

(19)

singular

v'.

u £ (9 )+,

T u(s, co) a u(s + /, 9_ co). A monotone

->(?°)+.

con-

measure

by p(A) = p7(lA o 0), „(A) = V(IA ° 9).

p(u) =iaf™e~S°~v(s,

and similarly

absolutely

singular"

(t fixed).

r£R+J

of J

continuous

, we also find that

measure,

The uniqueness

for M (resp.

v)

in place

while

A1(T zz) defines

v(T

a pro-

u) is progressively

of the decomposition

of Q; since

In view of (6), it will suffice

u£(9°)+.

(16) shows

Qa also satisfies

that (19)

(19), so does "p.

to verify ( 18) for zz = 0 (7r go.^), t>0,¿¡£

(J0_)+, in which case the right-hand side of (18) reduces to e~lp(¿J ° 9). The left-hand side is

M('[[i,oo[[0+a = e-$iTti{ = *"£(£ The fact that p is carried

can state,

given that

(20) Theorem. p + v, where

\j

measure

functions

just as before

and we

as Q = P~

+

in (9), and v is such that

co)dsv(dco),

u£(9°)

+,

singular.

As we indicated, how to overcome

set is proven

0 on AQ_ may be written

and p are as described

v(u) ^JJe-r^, is progressively

0 0).

\ is standard:

A finite

P~

by a polar

o 9)) (by (19))

the spaces

this difficulty,

/: R —»R U JA], where

that / is continuous

E, 3 and S are not standard. introduce

the space

E

A é R is an adjoined

on R for all t < C aud fit)

To illustrate

consisting "death

of all

point"

such

= A for all t > £. The "life-

time" £< oo depends on /. Clearly .®S)^,

s

process,

such that

T

we note that well-measura-

to being adapted.

For an integrable measurable

latter

from our discussion.

of the process

(26)

while, in

the dual pre-

and A = (A{((o)) an increasing

and with each J,

fot the well-measurable

These

from the general

\ an increasing

and note that 9 C (L. The accessible be omitted

for example,

of some of the work of Papangelou

Before going on, we recall

ÍJ j.

the first type classifies

of Pa under projection,

under which,

of a. is a.s. absolutely

in (9 )

etc. refer to the family

will be of two kinds:

type, we give conditions

with ¿j and is unique

¿j ° 0 is actually

of well-measurability,

in this section

an AF a in accordance the second dictable

¿j ) is bounded or nonnegative We note that the process

RAF a, we now denote by a

(resp. predictable)

projection

(resp.

as defined above.


a*) the dual well-

154

D. GEMANAND J. HOROWITZ (27) Theorem.

measures

The increasing

processes

a", a* are AF's

whose Palm

are

(28)

BjB-Btf'),

Proof.

It suffices

analogous,

EjO=Ea(t),

to treat the predictable

even somewhat

easier.

Suppose,

f«