Probability Abstracts 1

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1. STOCHASTIC CALCULUS AND THE MODULUS OF CONTINUITY OF LOCAL TIMES OF LEVY PROCESSES

R. Bass and D. Khoshnevisan 

60J

Barlow and Barlow and Hawkes gave a necessary and sufficient condition
for the local times of a Levy process to be jointly
continuous. A stochastic calculus proof is given of the sufficiency
part.

bass@math.washington.edu  (plain Tex file available)

2. HITTING A BOUNDARY POINT WITH REFLECTED BROWNIAN MOTION

Krzysztof Burdzy and Donald Marshall 

60J

Suppose that $X$ is a reflected Brownian motion in the upper half-plane 
with the variable angle of reflection $\theta (x)$.  
Then $X$ hits $0$ with positive probability if and only if
$$\int_0^1 {1\over y} \exp \left [\int_{-1}^1 {{\theta (x) x dx}
\over {\pi (x^2 + y^2)}}\right ] \cos \left [\int_{-1}^1 
{{\theta (x) y dx}\over {\pi (x^2 +y^2)}}\right] dy < \infty . $$

burdzy@math.washington.edu

3. APPLICATIONS OF SYMMETRY GROUPS IN MARKOV PROCESSES

J. Glover

60G

Algebraic structures have enjoyed a special role in 
Markov processes from the very beginning of the subject.  Probabilists
focussed much of their attention initially on independent increment 
processes in the group R^d.  Since these processes have convolution semigroups,
they are especially amenable to concrete analysis by Fourier techniques.
As Markov processes matured, probabilists began to study them in more general
settings, and there is often no algebraic structure in evidence on the state
space these days.  We will explore methods of introducing algebraic structures
on the state space which are naturally associated with the process.  In 
several instances, after an invertible probabilistic transformation, they will
become independent increment processes in the new groups structure.

glover@math.ufl.edu

4. AN UMBRELLA METHOD FOR UNIFORM APPROXIMATIONS OF STABLE LOCAL TIMES

D. Khoshnevisan

60G, 60J

We show that there is an underlying order governing uniform approximation 
theorems for the local time of a stable process. Even in the case of Brownian 
motion, this method yields new results.

davar@math.washington.edu  (Latex file available)

5. LOCAL ASYMPTOTIC LAWS FOR BROWNIAN CONVEX HULL

D. Khoshnevisan

60J

Using an equivalence theorem for a class of variational problems, we prove 
theorems for the upper and lower rates at which Brownian convex hull gets 
created in d-dimensional Euclidean space. This links large deviations 
to results of Paul Levy. Several examples, as well applications to 
random walks are presented.

davar@math.washington.edu   (Latex file available)

6. ARCSINE LAWS AND INTERVAL PARTITIONS DERIVED FROM A STABLE SUBORDINATOR

Jim Pitman and Marc Yor

60J

Levy discovered that the fraction of time a standard one-dimensional
Brownian motion B spends positive before time T has arcsine distribution,
both for T a fixed time when B(T) is not 0 almost surely, and for T an inverse
local time, when B(T) = 0 almost surely.  This identity in distribution
is extended from the fraction of time spent positive to a large collection of 
functionals derived from the lengths and signs of excursions of B away from 0.
Similar identities in distribution are associated with any process
whose zero set is the range of a stable subordinator, for instance a 
Bessel process of dimension d for  0 < d < 2.

pitman@stat.berkeley.edu

7. TRANSITION DENSITIES FOR BROWNIAN MOTION ON THE SIERPINSKI CARPET

M.T.Barlow and R.F. Bass

60J

In previous work the authors constructed "Brownian motion" on the
Sierpinski carpet, a continuous, nondegenerate strong Markov process
whose state space is the Sierpinski carpet, and which is invariant
under local isometries. Let p(t,x,y) be the transition densities of X(t) 
with respect to the Hausdorff-Besicovitch measure on the carpet. Good 
upper and lower bounds are found for p(t,x,y), both on and off the 
diagonal.

mtb3@phx.cam.ac.uk or bass@math.washington.edu (plain Tex file available)

8. BROWNIAN MODELS OF FEEDFORWARD QUEUEING NETWORKS: QUASIREVERSIBILITY AND PRODUCT FORM SOLUTIONS

J. M. Harrison and R. J. Williams 

60J

We consider a very general type of d-station open queueing network, with 
multiple customer classes and a more or less arbitrary service discipline at
each  station, but restricted by the requirement that customers always flow
from lowered  numbered stations to higher numbered ones.  To approximate the
behavior of such  a queueing network under heavy traffic conditions, a
corresponding Brownian network model is proposed, and it is shown that the
approximating Brownian model reduces to a d-dimensional reflected Brownian
motion W whose state space is the non-negative orthant.  A necessary and
sufficient condition for W to have a product form stationary
distribution (that is, a stationary distribution with independent components),
and a probabilistic interpretation  for that condition, are  given. 
Our interpretation involves an analog of the notion of quasireversibility
introduced by F. P. Kelly and elaborated by J. Walrand in their
brilliant  analysis of product form solutions for conventional queueing network
models.


rjwilliams@ucsd.bitnet

9. REFLECTED BROWNIAN MOTION IN A CONE WITH RADIALLY HOMOGENEOUS REFLECTION FIELD

Y. Kwon and R. J. Williams. 

60J

This work is concerned with the existence and uniqueness of a strong Markov
process that has continuous sample paths and the following additional
properties.
(i)  The state space is a cone in d-dimensions  (d>2),  and the
process behaves in the interior of the cone like ordinary Brownian motion.
(ii) The process reflects instantaneously at the boundary of the cone,
the direction of reflection being fixed on each radial line emanating from the
vertex of the cone.
(iii) The amount of time that the process spends at the vertex of the
cone is zero (i.e., the set of times for which the process is at the vertex has
zero Lebesgue measure).

The question of existence and uniqueness is cast in precise mathematical terms
as a submartingale problem in the style used by Stroock and Varadhan for
diffusions on smooth domains with smooth boundary conditions.  The question is
resolved in terms of a real parameter a which in general depends in a
rather complicated way on the geometric data of the problem, i.e., on the cone
and the directions of reflection.  However, a criterion is given for
determining whether  a>0.  It is shown that there is a unique
continuous strong Markov process satisfying (i)-(iii) above if and only if 
a<2,  and that starting away from the vertex, this process does not
reach the vertex if a is less than or equal to 0,  and does reach the vertex 
almost surely
if  0<a<2.  If a is greater than or equal to 2,  there is a unique continuous 
strong
Markov process satisfying (i) and (ii) above; it reaches the vertex of the cone
almost surely and remains there.  These results are illustrated in concrete
terms for some special cases.

The process considered here serves as a model for comparison with a 
reflected Brownian motion in a cone having a non-radially homogeneous reflection
field. This is  discussed in a subsequent work by Kwon.

rjwilliams@ucsd.bitnet

10. LIMIT THEOREMS AND VARIATION PROPERTIES FOR FRACTIONAL DERIVATIVES OF THE LOCAL TIME OF A STABLE PROCESS

P. J. Fitzsimmons and R. K. Getoor 
 
60J

We obtain limits theorems for the occupation times of 1-dimensional stable Markov processes.  These results are refinements of the classical limit theorems
of Darling and Kac, and they generalize results obtained by  Yamada for
Brownian motion.  The resulting limit processes  are fractional derivatives and
Hilbert transforms of the stable local time.  We also study the  $p$-variation
properties of these limit processes.

pfitzsim@ucsd.bitnet

11. ON THE DISTRIBUTION OF THE HILBERT TRANSFORM OF THE LOCAL TIME OF A SYMMETRIC LEVY PROCESS

P. J. Fitzsimmons and R. K. Getoor

60J

We derive simple explicit formulas for the Fourier-Laplace transforms of the 
Hilbert transform and related functionals of the local time of a symmetric 
Levy process. These formulas generalize results of Biane and Yor for Brownian 
motion. The method of proof provides an explanation (of sorts) for the presence
of the hyperbolic functions in such formulas.

pfitzsim@ucsd.bitnet