Seminar on Stochastic Processes, 1992

277 Pages·1993·21.28 MB·English

by  Robert J. Adler (auth.)

Progress in Probability
Volume 33
Series Editors
Thomas Liggett
Charles Newman
Loren Pitt
Seminar on
Stochastic Processes, 1992
E.Qnlar
K. L.Chung
J.
M. Sharpe
Editors
R. EBass
K. Burdzy
Managing Editors
Springer Science+Business
Media, LLC
E. ClDlar R.F. Bass
Dept. of Civil Engineering K. Burdzy
and Operations Research (Managing Editors)
Princeton University Dept. of Mathematics
Princeton, NJ 08544 University of Washington
Seattle, W A 98195
K. L. Chung
Dept. of Mathematics
Stanford University
Stanford, CA 94305
M.J. Sharpe
Dept. of Mathematics
University of California-San Diego
La Jolla, CA 92093
Library of Congress CataIoging-in-Publication Data
Seminar on Stochastic Processes (12th : University of
Washington)
Seminar on Stochastic Processes, 1992 I E. ~mlar, K. L. Chung,
M. J. Sharpe, editors ; R. F. Bass, K. Burdzy, managing editors.
p. cm. - (Progress in probability; v. 33)
Seminar held at the University of Washington Mar. 26-28, 1992.
Includes bibliographical references.
ISBN 978-1-4612-6714-0 ISBN 978-1-4612-0339-1 (eBook)
DOI 10.1007/978-1-4612-0339-1
acid-free)
1. Stochastic processes--Congresses. I. ~1D1ar, E. (Erhan),
1941- . II. Chung, Kai Lai, 1917- . III. Sharpe, M. J., 1941-
IV. Title. V. Series: Progress in probability : 33.
QA274.A1S44 1992 93-22016
519.2--dc2O CIP
Printed on acid-free paper.
© Springer Science+Business Media New York 1993
Originally published by Birkhlluser Boston in 1993
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ISBN 978-1-4612-6714-0
Camera-ready copy prepared by the Authors in TEX.
987 6 5 432 1
Contents
Superprocess local and intersection local times
and their corresponding particle pictures
Robert J. Adler .......... . 1
Strong approximations to Brownian local time
Richard Bass and Davar Khoshnevisan . . . . . . . . 43
Some path properties of iterated Brownian motion
Krzysztof Burdzy . . . . . . . . . . . . . . . . . 67
Conditioned superprocesses and a semilinear
heat equation
Alison M. Etheridge . . . . . . . . ....... 89
Markovian bridges: construction, Palm interpretation,
and splicing
Pat Fitzsimmons, Jim Pitman, and Marc Yor . 101
The martingale problem for a differential operator
with piecewise continuous coefficients
Ping Gao .............. . 135
Generalized Schrodinger semigroups
J. Glover, M. Rao, and R. Song . 143
On a limit theorem for non-stationary branching processes
Tetsuya Hattori and Hiroshi Watanabe ..... 173
On the covering time of a disc by simple
random walk in two dimensions
Gregory F. Lawler ..... . 189
4>-Variation of the local times of symmetric
Levy processes and stationary Gaussian processes
Michael B. Marcus and Jay Rosen .... 209
Critical random walk in random environment on trees
of exponential growth
Robin Pemantle ............ . 221
Uniform invariance principles for intersection local times
Jay Rosen ................. . 241
vi CONTENTS
On a two-armed bandit problem with both continuous
and impulse actions and discounted rewards
A. A. Yushkevich . . . . . . . . . . . . . 249
Existence and uniqueness of the solution of a semilinear
stochastic evolution equation on the whole real line
Bijan Z. Zangeneh ............ . 267
FOREWORD
The 1992 Seminar on Stochastic Processes was held at the Univer
sity of Washington from March 26 to March 28, 1992. This was
the twelfth in a series of annual meetings which provide researchers
with the opportunity to discuss current work on stochastic processes
in an informal and enjoyable atmosphere. Previous seminars were
held at Northwestern University, Princeton University, University of
Florida, University of Virginia, University of California, San Diego,
University of British Columbia and University of California, Los An
geles. Following the successful format of previous years, there were
five invited lectures, delivered by R. Adler, R. Banuelos, J. Pitman,
S. J. Taylor and R. Williams, with the remainder of the time being
devoted to informal communications and workshops on current work
and problems. The enthusiasm and interest of the participants cre
ated a lively and stimulating atmosphere for the seminar. A sample
of the research discussed there is contained in this volume.
The 1992 Seminar was made possible through the support of
the National Science Foundation, the National Security Agency, the
Institute of Mathematical Statistics and the University of Washing
ton. We extend our thanks to them and to the publisher Birkhauser
Boston for their support and encouragement.
Richard F. Bass
Krzysztof Burdzy
Seattle, 1992
SUPERPROCESS LOCAL AND
INTERSECTION LOCAL TIMES
AND THEIR CORRESPONDING
PARTICLE PICTURES
Robert J. Adler
Faculty of Industrial Engineering & Management
Technion - Israel Institute of Technology
Abstract
The super Brownian motion has a local time up to three di
mensions, an intersection (over disjoint time sets) local time up to
dimension seven, and a renormalised self-intersection local time up
to dimension five.
The branching Brownian motions that, in the infinite density
limit, provide a particle picture for the super Brownian motion
have a local time in only one dimension, an intersection local time
up to three dimensions, and a renormalised self-intersection local
time only in dimensions one and two.
This paper is primarily concerned with the "dimension gap"
between the particle picture and the superprocess, and shows how
to explain this gap in terms of weak convergence of functionals of
the finite system to functionals on the superprocess. The first sec
tion gives a detailed review of the recent literature on local and
intersection local times for super Brownian and super stable pro
cesses.
AMS 1991 subject classifications: Primary, 60J55, 60H15; Secondary,
60F17,60F25.
Key words and phrases: Superprocesses, local time, intersection local time.
1. Research supported in part by US-Israel Binational Science Foundation
(89-298), Israel Academy of Sciences (702-90), and Office of Naval Research
(NOOOI4/89/J /1870).
2 ROBERT J. ADLER
O. INTRODUCTION
This paper is concerned with the local time and intersection local time
of superprocesses. In particular, we are interested in how these relate to
certain functionals of the branching particle system whose high density limit
is the superprocess.
The first section is a mini-review of the recent literature on the local
times of superprocesses. It contains no new results (although one or two
"well known" facts may appear here in print for the first time), and so can
be skipped by a reader already familiar with this material.
Section 2 contains the new results, as well as summarising briefly some
previous, qualitatively different, results that arise when the questions raised
in this paper are applied to the Brownian density process.
Section 3 contains some moment formulae for the finite systems of
branching particles mentioned above. These are closely related to Dynkin's
(1988c) diagrams for calculating the mixed moments of the superprocesses,
and should be of independent interest. In the same vein that we believe the
particle picture adds probabilistic insight into the structure of superprocess
local time, we believe moment formulae for the particle picture add further
understanding to Dynkin's analytically derived formulae.
Sections 4 and 5 are devoted to proofs.
1. LOCAL TIMES
(a) Superprocesses and particle pictures. We start with a parameter
J.l > 0 that will eventually become large, and a finite measure m on ~d,
d ~ 1, of total mass Iml = m(~d). With these we generate a particle picture
based on J( = [J.L·lml1 initial particles which, at time zero, are independently
distributed in ~d, according to the probability measure me) I Im i. Later on
we shall occasionally require that m is absolutely continuous with respect
to Lebesgue measure. We define a branching rate p > O. Each of these
J( particles follows the path of independent copies of a Markov process B,
until time t = pi J.l.
At time pi J.l each particle, independently of the others, either dies or
splits into two, with probability ~ for each event. The individual particles
in the new population then follow independent copies of B, starting at their
place of birth, in the interval [pIJ.l, 2plJ.l) , and the pattern of alternating
critical branching and spatial spreading continues until, with probability
one, there are no particles left alive.
The process of interest to us is the measure valued Markov process
{Number of particles in A at time t}
(1.1) xt(A)
J.l
SUPERPROCESS LOCAL TIMES 3
where A E Bd = Borel sets in !Rd. Note that, for fixed t and J.l, Xi is an
atomic measure. Note also that if p = 00 there is no branching occuring.
It is now well known that under very mild conditions on B the se
quence {XI'h,>l converges weakly, on an appropriate Skorohod space, to
a measure valued process which is called the superprocess for B. A precise
formulation of this convergence is given in §2. We shall be interested pri
marily in the cases in which B is either an !Rd valued Brownian motion with
infinitesimal generator a, (so that the components of B have variance 2 at
time t = 1) or an !Rd valued symmetric stable process whose generator is
the fractional Laplacian ao< := -( _a)0</2, a E (0,2). (cf. Yoshida (1980).)
When a = 2 we define a2 =: a. Set p = 1. Then in either of these cases
the corresponding superprocess, known, respectively, as a super Brownian
motion or super stable process, can be defined as the unique solution of the
following martingale problem:
For all ¢ E Cl(!Rd), the space of all bounded, continuous, !R valued
functions on !Rd with continuous first and second order partial derivatives,
(1.2a)
is a continuous square integrable martingale such that Zo = 0 and
(1.2b)
(We have taken the obvious liberty here of denoting integration via
J =
¢(x) v(dx) v(¢) for a measure v. Later, without further comment,
we shall also write this as (¢, v}.)
It can be shown that the solution of this martingale problem is also the
unique solution of the stochastic partial differential equation (SPDE)
(1.3)
We shall call the system of branching particles described in the first
two paragraphs the "particle picture" for the measure valued solution of
(1.2). A more precise description of this system is given in the following
section.
When p = 00 the situation is a little different, and Xi, as defined above,
converges with probability one to the absolutely continuous measure whose
density Xt(u) satisfies the deterministic PDE 8xt(u)/8t = aaXt(u), with
initial condition Xo = m. (This is just the strong law of large numbers.) In
this case we look at the processes
(1.4) TJr<¢) = J.ll/2 {Xf(¢) - EXi(¢)},