DONALD GEMAN AND JOSEPH HOROWITZ. ABSTRACT. Given a .... if (E, л) is a measurable space, we write (ambiguously). / £ (©) to mean that / is an .... We hasten to add that many of the results below do not depend on (I), though they are ...
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
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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, ^As)!^^*
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,
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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) -J^e-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]af^e-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
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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
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
° 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
QiT^ae'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.
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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«