Probability Abstracts 14

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233. RATES OF CLUSTERING IN STRASSEN'S LIL FOR PARTIAL SUM PROCESSES

Uwe Einmahl and David M. Mason

We provide a detailed description of the rate of clustering in Srassen's
functional law of the iterated logarithm for partial sum processes.
Necessary and sufficient conditions are given for certain clustering rates
in terms of moment-type conditions. In the process, we also derive a new
strong approximation of sums of iid random variables by a Wiener process.

ueinmahl@ucs.indiana.edu 

234. LAWS OF THE ITERATED LOGARITHM FOR LOCAL TIMES OF THE EMPIRICAL PROCESS

Richard F. Bass and Davar Khoshnevisan

We give exact expansions for the upper and lower tails of the distribution 
of the maximum of local time of standard Brownian bridge on the interval 
[0,1]. We use the above expansions to prove upper and lower laws of the 
iterated logarithm for the maximum of the local time of the uniform empirical 
process. This solves two open problems cited in the book of Shorack and Wellner.

davar@math.washington.edu

235. SINGULAR STOCHASTIC CONTROL FOR DIFFUSIONS AND SDE WITH DISCONTINUOUS PATHS AND REFLECTING BOUNDARY CONDITIONS

Jin Ma

In this paper we continue to study a diffusion-type, finite-fuel singular 
stochastic control problem and the related stochastic differential equations 
with discontinuous paths and reflecting boundary conditions as defined in 
the previous work of the author. The measurable dependence of the solution 
with respect to the initial state and the underlying probability measures 
(with Prohorov metric) is derived. Also, the approximation of certain 
complete class of controls by those with continuous paths is proved to be 
possible in a weak sense. With the help of these results, we prove the 
Dynamic Programming Principle (Bellman principle) rigorously and show that 
the value function is the viscosity solution of certain Hamilton-Jacobi-
Bellman equation.

majin@math.purdue.edu

236. SOLVABILITY OF FORWARD-BACKWARD SDE'S AND THE NODAL SET OF HAMILTON- JACOBI-BELLMAN EQUATIONS

Jin Ma and Jiongmin Yong

In this paper, the solvability of a class of forward-backward stochastic 
differential equations over an arbitrary prescribed time duration is studied. 
We design a stochastic relaxed control problem, with both drift and diffusion 
all being controlled, so that the solvability problem is converted to a 
problem of finding the nodal set of the viscosity solution to a certain HJB 
equation. Our method overcomes the fatal difficulty encountered in the
traditional contraction mapping theorem approach to the existence theorem 
of such SDEs. For a certain class of forward-backward SDEs, the results 
concerning the solvability, non-solvability and the maximum solvable interval 
are derived using our new approach.

majin@math.purdue.edu

237. A MARTINGALE CHARACTERIZATION OF THE SET-INDEXED POISSON PROCESS

B.G. Ivanoff and E. Merzbach
     
      A Martingale characterization of the set-indexed Poisson process
is proved: a set-indexed point process is a Poisson process if and
only if there exists a deterministic and increasing process such that
the difference is a strong martingale.

merzbach@bimacs.cs.biu.ac.il

238. MARTIN BOUNDARIES OF SECTORIAL DOMAINS

M. Cranston, Thomas S. Salisbury

Let $D$ be a domain in $R^2$ whose
complement is contained in a pair of rays leaving the origin. That is, $D$
contains two sectors whose base angles sum to $2\pi$. We use balayage to
give an integral test that determines if the origin splits into exactly
two minimal Martin boundary points, one approached through each sector.
This test is related to other integral tests due to Benedicks and
Chevallier, the former in the special case of a Denjoy domain. We then
generalise our test, replacing the pair of rays by an arbitrary number.

salt@nexus.yorku.ca (AMSTeX file available, to appear in Ark. Mat.)

239. SWITCHED DIFFUSION PROCESSES AND SYSTEMS OF ELLIPTIC EQUATIONS ----A DIRICHLET SPACE APPROACH

Z.Q. Chen and Z. Zhao

	The switched diffusion process associated with a weakly
coupled system of elliptic equations is studied via a Dirichlet
space approach and is applied to prove an existence theorem of
the Cauchy initial value problem. A probabilistic representation
theorem for solutions of the Dirichlet boundary value problem
and a generalized Skorokhod decomposition for the reflecting
switched diffusion process are obtained.

zchen@euclid.ucsd.edu  (Hard copy available)

240. DUFFUSION PROCESSES AND SECOND ORDER ELLIPTIC OPERATORS WITH SINGULAR COEFFICIENTS FOR LOWER ORDER TERMS

Z.Q Chen and Z. Zhao

	The connection between diffusion processes and second order
elliptic operators of the form ${1\over 2}\nabla\cdot (a\nabla)
+b\cdot\nabla +q$ with singular coefficients $b, q$ and measurable
coefficient $a$ is studied. Among other things, we prove a probabilistic
representation theorem for solutions of Dirichlet boundary value
problem on bounded domains.

zchen@euclid.ucsd.edu  (AmsLaTeX file available)

241. POTENTIAL THEORY FOR ELLIPTIC SYSTEMS

Z.Q. Chen and Z. Zhao

	The existence and uniqueness theorem is proved for solutions
of the Dirichlet boundary value problem for weakly coupled elliptic
systems on bounded domains. The elliptic systems are only assumed 
to have measurable coefficients and having singular coefficients
for the lower order terms. Probabilistic representation for solutions
of Dirichlet problem is obtained by using the switched diffusion
process associated with the system. A strong positivity result for
Dirichlet solutions is proved. Formulas expressing resolvents and
kernel functions for the system by those of the component elliptic
operators are also obtained.

zchen@euclid.ucsd.edu  (AmsLaTeX file available)

242. REFLECTING BROWNIAN MOTIONS ON UNBOUNDED EUCLIDEAN DOMAINS AND A DELETION RESULT FOR SOBOLEV SPACES OF ORDER (1,2)

Z.Q. Chen

	Reflecting Brownian motions on general unbounded Euclidean
domains $D$ are studied. When there exists an increasing sequence of
smooth subdomains with union $D$ such that their surface measures
on each compact set are uniformly bounded, we show that the reflecting
Brownian motion on the Euclidean closure of $D$ is a semimartingale
and has a generalized Skorokhod decomposition. As an application of
our method of proof, a deletion result for Sobolev spaces of order
(1,2) is derived. More precisely, suppose that $D$ is a Euclidean
domain and $E$ a relatively closed subset of $D$ having zero Lebesgue
measure, a necessary and sufficient condition is obtained for 
$W^{1,2}(D\setminus E)=W^{1,2}(D)$. An extendibility result for
harmonic functions is also presented.

zchen$euclid.ucsd.edu  (TeX file available)

243. INVARIANT DIFFUSION PROCESSES IN LIE GROUPS

Ming Liao

Let g_t be a left invariant diffusion process in a noncompact semisimple 
Lie group G, let  G = NAK be an Iwasawa decomposition, and let ition. 
g_t = n_t a_t k_t be the corresponding decomposition of the process. 
We show that k_t is a diffusion process in the compact subgroup K and the 
Haar measure on K is a stationary measure for k_t.  Write  a_t = exp(H_t). 
As our main result, we show that if k_t is ergodic, then almost surely, 
as t tends to infinity, n_t converges in N and  H_t/t  tends to  -cH, 
where c>0 is a constant and H is a vector independent of g_t. 
Our results can be restated in Cartan (polar) decomposition. It means 
that almost surely, as t tends to infinity, the projected process  g_tK 
in the symmetric space G/K has a limiting "angle" and its "radial" 
component tends to infinity according to a limiting rate. 

mliao@auducvax.bitnet

244. THE NET OUTPUT PROCESS OF A SYSTEM WITH INFINITELY MANY QUEUES

P.A. Ferrari, L. R. G. Fontes

We study a system of infinitely many queues with Poisson arrivals and
exponential service times. Let the net output process be the difference
between the departure process and the arrival process. We impose certain
ergodicicity conditions on the underlying Markov chain governing the customer
path. These conditions imply the existence of an invariant measure under which
the average net output process is positive and proportional to the time.
Starting the system with that measure we prove that the net output process is
a Poisson process plus a perturbation of order 1. This generalizes a classical
theorem (Burke (1956), Kelly (1979)) asserting that the departure process is a
Poisson process.

pablo@ime.usp.br     Latex file available 

245. HARNACK PRINCIPLE FOR ELLIPTIC SYSTEMS

Zhen-Qing Chen and Zhongxin Zhao

We prove the Harnack inequality for weakly coupled elliptic systems
$S$ which have Holder continuous coefficients for first and second order
terms and singular coefficients for the zero order terms. When the
system $S$ is irreducible, a full Harnack principle is obtained. These
results fulfill the general form of the Harnack principle for the weakly
coupled elliptic systems and improve significantly the previous result 
on Harnack inequality for the systems given by F. Mandras.

zchen@euclid.ucsd.edu (AmsLaTeX file available)

246. STOCHASTIC DIFFERENTIAL GEOMETRY WITH JUMPS 1

Serge Cohen

The intrinsic second-order calculus for manifold-valued semimartingales of
Meyer-Schwartz is extended to the non-continuous
case . The main idea is to substitute 2-jets at  x  (only local behavior is
needed in the continuous case) by functions that are twicely differentiable at
x  (to deal with macroscopic jumps). We first introduce an intrinsic 
integral; when the manifolds are vector spaces, this is but an integral with 
respect to a random measure (Jacod). It is then used to define 
quadratic variations of semimartingales and SDE's. Last we get existence and 
uniqueness of strong solutions for such SDE's with locally Lipschitz 
coefficients. 

sc@newaphro.enpc.fr (Latex file available, in French)