Seminar on Stochastic Processes, 1985

331 Pages·1986·4.261 MB·English

by  R. M. Blumenthal (auth.), E. Çinlar, K. L. Chung, R. K. Getoor, J. Glover (eds.)

Progress in Probability and Statistics
Volume 12
Peter Huber
Murray Rosenblatt
series editors
Seminar on
Stochastic Processes,
1985
E. <;mlar
K.L. Chung
R.K. Getoor
editors
J. Glover
managing editor
1986 Birkhauser
Boston . Basel . Stuttgart
E. Cmlar K.L. Chung
Civil Engineering Department Department of Mathematics
Princeton University Stanford University
Princeton, NJ 08544 Stanford, CA 94305
U.S.A. U.S.A.
R.K. Getoor J. Glover (managing editor)
Department of Mathematics Department of Mathematics
University of California-San Diego University of Florida
La Jolla, CA 92093 Gainesville, FL 32611
U.S.A. U.S.A.
Library of Congress Cataloging in Publication Data
Seminar on Stochastic Processes (5th: 1985: University of Florida)
Seminar on Stochastic Processes, 1985.
(Progress in probability and statistics; vol. 12)
Includes bibliographies.
I. Stochastic processes-Congresses. I. <;mlar,
E. (Erhan), 1941- . II. Chung, Kai Lai, 1917- .
III. Getoor, R.K. (Ronald Kay), 1929- .
IV. Title. V. Series: Progress in probability and statistics; v.12.
QA274.AIS45 1985 519.2 86-11285
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Seminar on Stochastic Processes:
Seminar on Stochastic Processes ... -Boston; Basel
Stuttgart : Birkhauser
5, 1985 (1986).
(Progress in probability and statistics; Vol. 12)
ISBN-13: 978-1-4684-6750-5 e-ISBN-13: 978-1-4684-6748-2
DOl: 10.1007/978-1-4684-6748-2
NE: GT
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© Birkhauser Boston, Inc., 1986
Softcover reprint of the hardcover 1st edition 1986
FOREWORD
The 1985 Seminar on Stochastic Processes was held at
the University of Florida, Gainesville, in March. It was
the fifth seminar in a continuing series of meetings which
provide opportunities for researchers to discuss current
work in stochastic processes in an informal atmosphere.
Previous seminars were held at Northwestern University,
Evanston and the University of Florida, Gainesville.
The participants' enthusiasm and interest have
resulted in stimulating and successful seminars. We thank
them for it, and we also thank those participants who have
permitted us to publish their research here.
The seminar was made possible through the generous
supports of the Division of Sponsored Research and the
Department of Mathematics of the university of Florida, and
the Air Force Office of Scientific Research, Grant No. 82-
0189. We are grateful for their support. Finally, the
comfort and hospitality we enjoyed in Gainesville were due
to the splendid efforts of Professor Zoran Pop-Stojanovic.
J. G.
Gainesville, 1986.
Table of Contents
R.M. BLUMENTHAL. A Decomposition of Excessive Measures 1
J. BROOKS and N. DINCULEANU. HI and BMO Spaces of
Abstract Martingales 9
K. BURDZY. Brownian Excursions and Minimal Thinness
Part II Applications to Boundary Behavior
of the Green Function 35
K.L. CHUNG. Doubly-Feller Process with Multiplicative
Functional 63
P.J. FITZSIMMONS. Another Look at Williams'
Decomposition Theorem 79
R.K. GETOOR. Some Remarks on a Theorem of Dynkin 86
R.K. GETOOR. Some Remarks on Measures Associated with
Homogeneous Random Measures 94
P. HSU. Brownian Exit Distribution of a Ball 108
F.B. KNIGHT. On the Duration of the Longest Excursion 117
B. MAISONNEUVE. Strict Past Conditioning at Arbitrary
Times 148
J.B. MITRO. Discontinuous Time Changes and Duality
For Markov Processes 155
E. PERKINS. The Cereteli-Davis Solution to the
HI-Embedding Problem and an Optimal
Embedding in Brownian Motion 172
Z.R. POP-STOJANOVIC. Thinness and Hyperthinness 224
J. STEFFENS. Note on the Generator of a Ray Resolvent 233
M.I. TAKSAR. Infinite Excessive and Invariant Measures 243
G.A. BROSAMLER. Brownian Occupation Measures on
Compact Manifolds 290
J. GLOVER. Correction to: Topics in Energy and
Potential Theory 323
Seminar on Stochastic Processes, 1985
Birkhauser, Boston, 1986
A DECOMPOSITION OF EXCESSIVE MEASURES
by
R. M. Blumenthal
Let {Pt;t ) O} denote the transition semigroup for a
Borel right Markov process on a state space (E,8). We set
U(x,D) = J Pt(x,D)dt and denote by ~U and Uf the potential
o
of a measure ~ and a function f respectively. Given a
measure m on 8 and a set D in 8 let mD be the measure
mD(B) = m(D n B). A measure m on 8 is called excessive if
m is a-finite and m ) mPt for all t. If ~ is a measure on
8 the formula ~U(A) = J~(dx)U(X,A) defines a measure, the
potential of ~, on 8. ft will be excessive if and only if
it is a-finite. An excessive measure m is called invariant
if mPt = m for all t.
In [1] Fitzsimmons and Maisonneuve call an excessive
measure m dissipative if m is the set-wise supremum of the
potentials ~U which are set-wise less than m, and they call
m conservative if the only potential less than m is the
zero measure. Then they prove that every excessive m has a
unique decomposition into a sum, mC + md, of a conservative
and a dissipative measure. In fact they prove that if q is
any strictly positive Borel function on E with m(q) finite
1
2
then mC is m restricted to {U ~} and md is m restricted
q
to {U <~} so that up to m-nuII sets these sets are
q
independent of q.
The purpose of this note is to prove that for every
=
excessive function $ and positive t we have $ Pt $ almost
everywhere relative to mC. It turns out that at the
expense of a few extra lines one can obtain this invariance
result and the mC + md decomposition all at once, and
without the intervention of the Kuznecov theory of
stationary measures that formed the basis for [l]~ and so
we will give this slightly expanded presentation. The
invariance result verifies a conjecture of Getoor and
Steffens, who came upon it while preparing their paper [3]
on capacity. I appreciate their calling the problem to my
attention.
We will start off with some definitions and simple
observations.
Call a set A dissipative if A is in B, A is finely
open, and there is a function h in B+ with h strictly
positive on A and Uh ( I on all of E. One argues easily
that any countable union of dissipative sets is itself
dissipative. For an example of such a set take a bounded
Borel excessive $ such that Pt $ decreases to zero as t
increases to infinity, and set A = {$ > Pr$} where r is a
r
positive number. Then U($ - P $) = f Pt$dt so we can take
r 0
=
h ($ - Pr$)/rn$n. The requirement that Pt $ decreases to
zero is unnecessary: we can replace any bounded $ by
~ = $ - a, where a is the limit as t approaches ~ of Pt $.
Then ~ is excessive, Pt~ decreases to zero and {~ > Pr~}
=
{$ > Pr$}. Also $ need not be bounded or even finite
3
because we can replace ~ by min(~,n) and use the fact that
a countable union of dissipative sets is again one. Any
set of the form {O < Uq < m} with q in 8+ is a countable
union of sets {Uq > Pruq} and so is dissipative.
Now given an excessive measure m, take a finite
measure e equivalent to m and a sequence {An} of
dissipative sets such that lim e(A) sup{e(D)ID
n n
A
dissipative} and let be the union of the An' Then A is
dissipative, and obviously if D is any dissipative set then
A) A)
e(D - is zero so that m(D - is zero also. In
particular if ~ is any Borel excessive function and t is
positive we have ~(x) = Pt~(x) for almost all (m)x in
A. A
E - We want to replace by a slightly larger set.
Specifically, take a function h in 8+ which is strictly
A
positive on and with Uh ( 1 on E and set A = {Uh > O}.
A
Then A is in 8, A is dissipative and is contained in A,
so that m(A - A) O. Let B be the complement of A. Since
A is the set where an excessive function is strictly
positive, B must be absorbing~ that is pX(Xt is in A for
some t ) 0) o for all x in B. The sets A and B are the
ones referred to in the next statement.
THEOREM 1. rnA is dissipative. mB is conservative. If ~
=
is excessive then for every t, ~ Pt~ almost everywhere
PROOF. Since A and A differ by an m-null set we have
established already, in the previous paragraph, the
assertion about ~, at least if ~ is a Borel function. This
implies the conclusion for a general excessive ~ because
4
according to (6.11) of Getoor and Sharpe [2] for any
excessive $ there is a Borel excessive ~ with the
equalities $ = ~ and Pt $ = Pt~ holding almost surely (m).
Before continuing the proof we will make some remarks
about measures.
(1.1) if M and N are a-finite measures with M(D) ) N(D)
for all D in 8 then there is a unique measure 9 on 8 with
N + 9 = M. Of course 9(f) = M(f) - N(f) if f is in 8+ and
M(f) is finite. We will just write M - N when we mean 9~
(1.2) if Ml,M2' ••• is a sequence of a-finite measures with
Ml (C) ) M2(C) ) ••• for all C in 8 then there is a unique
=
measure ~ such that ~(C) limnMn(C) whenever Mn(C) is
finite for some n. We write simply ~ = limnMn.
(1.3) suppose m is excessive, tn is a sequence of numbers
increasing to = and we set ~ = limnmPt in the
n
interpretation from (1.2). Then ~ is independent of the
sequence tn' ~ is invariant and 9 = m - ~ is excessive with
o whenever m(D) < =.
Assertions (1.1) through (1.3) are trivial to verify.
Now let 9 be excessive and suppose there is an
increasing sequence {en} of sets whose union is E and such
=.
that for each n, 9P (C ) ~ 0 as t ~ Then
t n
(1. 4)
for all F in 8. To see this note that (1.4) follows from