Seminar on Stochastic Processes, 1986

209 Pages·1987·12.58 MB·English

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

Progress in Probability and Statistics
Volume 13
Series Editor
Murray Rosenblatt
Seminar on
Stochastic Processes,
1986
E. <;mlar,
K.L. Chung,
R.K. Getoor,
Editors
J. Glover,
Managing Editor
1987 Birkhauser
Boston . Basel . Stuttgart
E. C;mlar 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 University of Florida
La Jolla, CA 92093 Gainesville, FL 32611
U.S.A. U.S.A.
ISSN: 0892-063X
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Seminar on Stochastic Processes:
Seminar on Stochastic Processes ...- Boston ; Basel;
Stuttgart : Birkhauser
6. 1986 (1987).
(Progress in probability and statistics ; Vol. 13)
ISBN-13: 978-1-4684-6753-6 e-ISBN-13: 978-1-4684-6751-2
DOl: 10.1007/978-1-4684-6751-2
NE: GT
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Softcover reprint of the hardcover 1st edition 1987
ISBN-13: 978-1-4684-6753-6
9 8 7 6 5 4 3 2 I
FOREWORD
The 1986 Seminar on Stochastic Processes was held at the University
of Virginia, Charlottesville, in March. It was the sixth 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 support of the
Office of Naval Research (Contract # A86-4633-P) and the University of
Virginia. We are grateful for their support. The participants were
welcomed to Virginia by S. J. Taylor, whose store of energy and
organizing talent resulted in a wonderful reunion of researchers. We
extend to him our warmest appreciation for his efforts; his hospitality
makes us hope that we can someday return to Virginia for another
conference.
J. ~.
ISBn
~aineauille,
TABLE OF CONTENTS
K. L. CHUNG Green's Function for a Ball 1
P. J. FITZSIMMONS On the Identification of Markov Processes
by the Distribution of Hitting Times 15
P. J. FITZSIMMONS On Two Results in the Potential Theory of
Excessive Measures 21
R. K. GETOOR Measures that are Translation Invariant
in One Coordinate 31
R. K. GETOOR and Constructing Markov Processes with
J. GLOVER Random Times of Birth and Death 35
P. HSU Branching Brownian Motion and the Dirichlet
Problem of a Nonlinear Equation 71
K. JANSSEN Representation of Excessive Measures 85
J. F. LE GALL The Exact Hausdorff Measure of Brownian
Multiple Points 107
J. F. LE GALL and The Packing Measure of Planar Brownian
S. J. TAYLOR Motion 139
Z. MA and Truncated Gauge and Schrodinger Operator
Z. ZHAO with Both Sign Eigenvalues 149
B. MAISONNEUVE Subordinators Regenerated 155
D. MONRAD and Local Nondeterminism and Hausdorff
L. D. PITT Dimension 163
Z. POP-STOJANOVIC Last Exit Time and Harmonic Measure for
Brownian Motion in Rd 191
J. STEFFENS Some Remarks on Capacities 195
F. B. KNIGHT Correction 215
Green's Function for a Ball
K. L. Chung*
=
Let B B(a,r) be the open ball with center a and
*
radius r in Rd, d ) 3~ aB its boundary sphere. For x a,
its inversion with respect to B is defined to be
2
(1) x * a + r 2(x - a).
Ix - al
We have
(2 ) Ix - allx* - al
from which it follows that the mapping is involutary:
(x*)* = x. Also we have from (1):
2
la - Yl2 + 2r (a - y,x - a)
Ix - al2
4 I 12
+ r 4 x - a ~
Ix - al
(3 )
* Research supported by AFOSR Grant 85-0330.
1
2
The right-hand member of (3) being symmetric in (x,y), we
see that
x*
Figure
It follows from (2) and the Figure that if z € aB, we have
by similar triangles:
=
Ix - al _,.,.....:r:..-_
(5)
r Ix· - al
*
namely for any x € B, x a:
= Iz - xwl
(6) aB { z € Rd : -
z - x
It is clear that we may put a = 0 by the mapping
x,...:;rx - a. Next, we put
(7) f(x,y)
and compute the key formula:
(8) f(x,y)2
3
We now introduce
(9) U(x,y) I Id-2'
x - Y
with U(x,x) +"', where
r("d2 - 1)
(10)
d!2 •
41t
The function u is known as the Green's function for Rd.
The Green's function for B is the function G defined on
B U (aB) as follows:
(11) G(x,y) (_---:r....-._) d-2 } •
IXllx* YI
Since Ixl Ix* - YI > rlx - yl by (8), it follows that
(12) o ( G(x,y) ( U(x,y)
in B x B: while
(13 ) G(x,z) o
on B x aB by (5). For each y E B, it can be verified that
U(o,y) - G(o,y) is harmonic in B - {y} and takes on the
boundary value of U(o,y) on aB. The last two properties
uniquely determine G, and is its raison d'etre in classic
potential theory. The constant Ad has its significance,
but since it plays no role in what follows it is sometimes
omitted in the difference of U.
The role of the radius r is not so clear. However, a
4
straight forward computation shows that if we denote
temporarily the G in (11) by Gr , we have the reduction
formula:
(14)
=
This permits us to concentrate on B B(O,l) and the G in
=
(11) with r 1. It follows from (4) that G is symmetric
in (x, y) :
G(x,y) G(y,x).
We shall denote the distance of x in B to ~B by
o(x) = 1 - Ixl.
Proposition 1. We have
(15) %min (
Ix
( 1 4(d - 2)o(x)o(y»)
.. min
Ix - yld-2' Ix _ yld
Proof. The inequality on the right with the first term
under min is just (12). Now we write
G(x,y)
(16)
Ad
Since f(x,y) > Ix - YI by (8) with r 1, the numerator in
(16) is less than