Seminar on Stochastic Processes, 1987

296 Pages·1988·5.539 MB·English

by  Bruce W. Atkinson (auth.), E. Çinlar, K. L. Chung, R. K. Getoor, J. Glover (eds.)

Progress in Probability and Statistics
Volume 15
Series Editor
Murray Rosenblatt
Seminar on
Stochastic Processes,
1987
E. <;mlar,
K.L. Chung,
R.K. Getoor,
Editors
1. Glover,
Managing Editor
1988 Birkhauser
Boston . Basel
E. <;'mlar K.L. Chung
Civil Engineering Department Department of Mathematics
Princeton University Stanford University
Princeton. NJ 011544 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 University of Florida
La Jolla. CA 92093 Gainesville. FL 3261 I
U.S.A. U.S.A.
ISSN: OX92·063X
CIP·Titelaufnahme der Deuhchen Bibliothek
Seminar on Stochastic Processes:
Seminar on Stochastic Processes
Birkhiiuser.
Bis 6. 1986 mit d. Erscheinungsorten Boston. Basel. Stuttgart
WG 27:15 DBN 55.092348.9
5259/01 bg
7. 1987 ( 198X)
(Progress in probability and statistics: Vol. 15)
ISBN-13: 978-1-4684-0552-1 e-ISBN-13: 978-1-4684-0550-7
DOl: 10.1007/978-1-4684-0550-7
NE: GT
WG: 27 DBN 88.0\0688.3 87.12.21
5259/02" bg
~~. Birkhauser Boston. 1988
Softcover reprint of the hardcover 1st edition 1988
All rights reserved. No part of this publication may be reproduced. stored in a retrieval system. or
transmitted. in any form or by any means. electronic. mechanical. photocopying. recording or oth
erwise. without prior permission of the copyright owner.
Permission to photocopy for internal or personal use. or the internal or personal use of specific
clients. is granted hy Birkhiiuser Boston. Inc .. for libraries and other users registered with the Copyright
Clearance Center (CCC). provided that the base fee of $0.00 per copy. plus $0.20 per page is paid
directly to CCc. 21 Congress Street. Salem. MA 01970. U SA. Special re4uests should be addressed
directly to Birkhauser Boston. Inc .. 675 Massachusetts Avenue. Cambridge. MA 02139. U.S. A.
3381-2/88 $0.00 + .20
ISBN-13: 978-1-4684-0552-1
Text prepared by the authors in camera-ready form.
987654321
Dedication
Seven has always been an auspicious number in the Orient. As the ancients would
explain it, it is the five planets and the sun and the moon. It gives my oriental mind a
certain pleasure to note that this seminar, the seventh in the series, brought us together
to be with Professors Kai Lai Chung and Gilbert Agnew Hunt for a few days in the
seventieth year of their distinguished lives.
In fact, the idea of holding these seminars was hatched here, seven years ago, during
a visit by Professor Chung. I remember well his stories of the late forties in probability
in Princeton and of the events of those days presaging the roles the two classmates were
to play in the now well-known tale of probabilities and potentials.
Their fundamental contributions have inspired us all, pointed the way, and established
the standards. This series of seminars rings with the clearest expressions of our acknowl
edgement of our indebtedness to their pioneering spirits. This volume is dedicated to
them as a further token of our appreciation and in affectionate tribute to their leadership.
Princeton, 1987 Erhan (tnlar
FOREWORD
The 1987 Seminar on Stochastic Processes was held at Princeton
University, March 26 through March 28, 1987. It was the seventh
seminar in a continuing series of meetings which provide opportunities
for researchers to discuss current work in stochastic processes in an
informal and enjoyable atmosphere. Previous seminars were held at
Northwestern University, Evanston; University of Florida, Gainesville:
and University of Virginia, Charlottesville.
The success of these seminars has been due to the interest and
enthusiasm of probabilists in the United States and abroad. Many of
the participants have allowed us to pUblish the results of their re
search in this volume. The editors hope that the reader will be able
to sense some of the excitement present in the seminar by reading these
articles. This year's invited participants included M. Aizenman,
B. Atkinson, R. M. Blumenthal, C. Burdzy, D. Burkholder, R. Carmona,
K. L. Chung, M. Cranston, C. Dellacherie, J. D. Deuschel, N. Dinculeanu,
E. B. Dynkin, P. Fitzsimmons, R. K. Getoor, J. Glover, R. Gundy, P. Hsu,
G. A. Hunt, H. Kaspi, F. Knight, G. Lawler, P. March, P. A. Meyer,
J. Mitro, J. Neveu, E. Pardoux, M. Pinsky, L. Pitt, A. O. Pittenger,
Z. Pop-Stojanovic, P. Protter, M. Rao, T. Salisbury, M. J. Sharpe,
S. J. Taylor, E. Toby, S. R. S. Varadhan, R. Williams, M. Weber, and
Z. Zhao.
The seminar was made possible through the generous support of the
Office of Naval Research (Grant No. N00014-87-G-0138) and Princeton
University, and we are grateful for their support. Erhan Cinlar was
host during the meeting. The pleasant and stimulating ambience of this
meeting (and of several past meetings) was due to his efforts, and we
extend our thanks to him.
J. G.
Gainesville, 1987
TABLE OF CONTENTS
B. W. Atkinson Homogeneity for Two-Sided Discrete
Markov Processes
J. K. Brooks and Regularity and the Doob-~eyer 21
N. Dinculeanu Decomposition of Abstract Quasimartinqales
C. Dellacherie Autour des Ensembles Semi-Polaires 65
N. Dinculeanu Vector Valued Stochastic Processes III: 93
Projections and Dual Projections
P. J. Fitzsimmons On a Connection Between Kuznetsov 123
Processes and Ouasi-Processes
R. K. Getoor and More about Capacity and Excessive ~1easures 135
J. Steffens
J. Glover, W. Hansen Caoacities of Symmetric Markov Processes 159
and M. Rao
I. W. Herbst and Sobol ev Spaces, Kac-Requl ari ty and the 171
Z. Zhao Feynman-Kac Formula
F. B. Knight On Invertibility of Martinaale 193
Time Chanqes
J. Neveu Multiplicative Martingales for 223
Spatial Branchinq Processes
Z. R. Pop-Stojanovic Energy and Potentials 243
T. S. Salisbury Brownian Bitransforms 249
R. J. Williams On Time-Reversal of Reflected 265
Brownian Motions
R. Wu and M. Liao Remarks on Harmonic Functions and 277
Invariant Measures of Markov Processes
Z. Zhao Green Functions and Conditioned Gauqe 283
Theorem for a 2-Dimensional Domain
Z. R. Pop-Stojanovic Correction 295
HOMOGENEITY FOR TWO-SIDED DISCRETE MARKOV PROCESSES
by
BRUCE W. ATKINSON
1. INTRODUCTION.
We consider here three types of homogeneity for Markov processes
indexed by the integers with random times of birth and death, and
which take values in a countable state space S. The measures on the
path space corresponding to such processes are sigma-finite but not
necessarily probability measures. However, via the Radon-Nikodym
theorem, no difficulty arises in extending the definitions of cond
itional expectation and conditional independence.
Before describing the main results of this paper, it will be nec
essary to make some definitions.
(L1) DEFINITIONS Let a, b be distinct elements not in S.
0
(a) W ~functions w Z + S V ta,bj : there exists n G Z with wen) G
S, w(m) a ~ w(k) a ¥ k < m, and w(m) = b ~ w(k) b ¥ k> m}.
(b) For n G Z, define x(n) W ~ S U ta,bJ by x(n) (w) wen).
(c) For n G Z, define Qn W ~ W by Qn(w)(m) = w(m+n).
(d) A non-negative measure P on (W, O(x(n), n G Z)) is called a two
sided process if P(x(n) = i)<- ¥ i G Z. (Note: All processes in this
paper are two-sided and so we will drop the implicit prefix "two
sided" from now on.)
zJ
(e) A process P is called Markov if there exists a family ~p : n G
n
of substochastic matrices on S so that whenever m, n G Z, m ~ 1, and
iO' i1,o •• ,im G S, then
m-1
(1.2) P(x(n+k) = ik ' 0 ~ k~ m) P(x(n) io)J:foPn+k(ik,ik+1)·
I
2
(Note: (1.2) is equivalent to the standard definition stating that the
past and future are conditionally independent given the present.)
It is easy to show that such a Markov process P is also Markovian
in the following spatial sense:
Let I be a finite interval of at least 3 integers with endpoints m, n
where m < n. Let 10 = ~ k : m < k< nJ ( the interior of I), and let IC
= Z-I the exterior of I). Then, relative to P, c(x(k), k 6 10) and
c(x(k), k 6 IC) are conditionally independent given c(x(m),x(n)) on
~x(m),x(n) 6 S~o
The first type of homogeneity we consider relates to the spatial
Markov property described above.
(1.3) DEFINITIONo A Markov process is homogeneous if for every interval
of at least 3 integers, I, the distribution of the process evaluated
on 10 given the values (in S) at the 2 endpoints depends only on the
length of 1.
It turns out that (1 3) follows from the above definition restric
0
ted to intervals of just 3 integers. If I has just 3 integers then 10
simply represents the middle integer, and the conditional distribution
of the process evaluated at the middle integer given the values (in S)
at the endpoints we shall call the local characteristics in accord with
the terminology from Markov random field theory. Thus the technical
definition given in (2.1) will only deal with intervals of 3 integers,
£.
where we denote the local characteristics with
The other 2 definitions of homogeneity considered here are temp
oral in nature. A process is called homogeneous in the forward time
direction if for any integer n, the distribution of the process eval
uated on tn+1, n+2, ••• ) given the value (in S) at time n is independent
of n. A similar definition describes processes which are homogeneous
in the backward time directiono A Harkov process whic1.is homogeneous in
both time directions is called a two-sided Markov chain, and these
have been studied extensively in [1] We now briefly describe the main
0
results,
In section 2 it is shown that if P is Harkov and homogeneous and
3
P(x(n) i, x(n+1) = j) > 0 IJ n 6 Z, i, j 6 S, then there is a. positive
matrix p, not necessarily substochastic, a family of measures~n n ' n
6 ZJ, and a family of functions ~h ,n 6 zj, all strictly positive, so
n
that
0.4) P(x(n)
(1 5) P(x(n+k)
The argument is a modification of one found in 6 where Gibbs states
are studied. Indeed, if we also demand that P be a probability and that
x(n) 6 S IJ n 6 Z, P-a.s., then P is an example of a Gibbs state spec
ified EY p. With this as motivation we say that a process P is spec
1
ified EY p if there are non-negative families fn'"n' n6 Z~ and hn' n 6
z}
so that (1.4) and (1.5) hold. It is shown that such a P is Markov
and homogeneous; see (2.15). ( In [6J it is shown that extreme Gibbs
states are probability measures which satisfy (1.4) and (1.5) for
positive p, nn' and hn.)
In (2.16)we state that the local characteristics for a process P
specified by p are equal to p(i,j)p(j,k)[p2(i,k)]-1", which is quite
similar to Theorem of [6]. Indeed, in (6) it is mentioned that such
a form for the local characteristics follows from the equivalence of
homogeneous Markov random fields and nearest neighbor Gibbs states;
also see [5]. The proof given here is easier becausewe started with
the assumption that the process was Markovian in the traditional
sense, whereas a general Markov random field need not be a Markov
process.
Now let P be specified by p. Then IJ n 6 Z and j 6 S we have
1:i6S~n(i)p(i,j)hn+1 (j) = ~i6SP(x(n) = i, x(n+1) = j) ~
P(x(n+1) = j) = TIn+1 (j)hn+l (j). Thus
Similarly,