Probability Abstracts 7

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118. A NOTE ON SOME RATES OF CONVERGENCE IN FIRST-PASSAGE PERCOLATION

Kenneth S. Alexander

A variation is given of the van den Berg-Kesten inequality, enabling it to
apply to random variables, rather than just to events, associated with various
subsets of an index set. This is used to establish superadditivity of a
certain family of generating functions associated with first-passage
percolation.  This leads to improvements of some recent results of Kesten (see
abstract #116 of this series) on the rates of convergence of the expected
values of certain passage times.  Specifically, let mu be the time constant
and let a(0,n) be the passage time from (0,..,0) to (n,..,0).  Then Ea(0,n) -
n*mu = O(n^1/2 * log n).

alexandr@mtha.usc.edu

119. EXPONENTIAL WAITING TIME FOR A BIG GAP IN A ONE DIMENSIONAL ZERO RANGE PROCESS

The first time that the $N$ sites to the right of the
origin get empty in a one dimensional zero range process is shown to converge,
as $N\to\infty$ to the exponential distribution, when divided by its mean. The
initial distribution of the process is assumed to be one of the extremal
invariant measures $\nu_\rho$, $\rho\in (0,1)$ with density $\rho/(1-\rho)$. The
proof is based on the classical Burke's theorem.

pablo@ime.usp.br (TeX file available)
galves@ime.usp.br, landim@cmapx.polytechnique.fr 

120. A DOMAIN MONOTONICITY PROPERTY OF THE NEUMANN HEAT KERNEL

Elton P. Hsu

AMS Subject Classification : Primary 58G11, Secondary 60H15.

Let $p_\Omega (t,x,y)$ denote the heat kernel on an euclidean domain
$\Omega\in R^d$ with the Neumann (adiabatic) boundary condition.  We
prove the following monotonicity property of the Neumann heat kernel:
if $\Omega$ is a smooth convex domain and $D$ is a cell
(multidimensional rectangle) containing $\Omega$, then for every pair
of points $x,y$ in $\Omega$ and every $t>0$ we have
$$p_{\Omega}(t,x,y)\ge p_{D}(t,x,y).$$
The proof is probabilistic and involves studying the sample path
behavior of reflecting Brownian motion, the diffusion process
whose transition density function is the Neumann heat kernel.

elton@math.nwu.edu

121. EXPLOSION OF FIRST PASSAGE PERCOLATION ON A TREE.

Robin Pemantle and Yuval Peres

Let T be a tree in which each vertex at distance n from the root
has f(n) offspring and label the edges of T with i.i.d. random
variables.  Say that an explosion occurs if there exists an infinite
path along which the sum of these random variables is finite.  We give 
a necessary and sufficient condition for the almost sure occurrence
of an explosion; in the case where f is nondecreasing, the condition is

	SUM 1/f(n)  <  infinity .

In the general case, the proof uses a notion of stochastic domination
among trees which extends classical Hardy majorization.

peres@lom1.math.yale.edu (Latex file or hardcopy available)

122. SOLUTIONS OF STOCHASTIC DIFFERENTIAL-FUNCTIONAL EQUATIONS VIA BOUNDED STOCHASTIC INTEGRAL CONTRACTORS

Xuerong Mao

In this paper we shall use the bounded stochastic integral contractors
to investigate the existence and uniqueness of the solution of a 
general stochastic differential-functinal equation driven by a
non-linear integrator. Our results include the Lipschtiz continuous
condition as a special case.

xm@maths.warwick.ac.uk

123. LYAPUNOV FUNCTIONS AND ALMOST SURELY POLYNOMIAL STABILITY

Xuerong Mao

There exist a lot of stochastic differential equations which
are not exponentially stable almost surely, but the solutions
do tend to zero asymptotitcally almost surely. It is certainly
interesting and useful to know how fast the solutions tend to
zero now that they are not exponentially. We introduce a
new concept of almost surely polynomial stability in this paper
and establish several criteria for the stability with the help
of Lyapunov functions.

xm@maths.warwick.ac.uk

124. ALMOST SURE ASYMPTOTIC BOUNDS FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS

Xuerong Mao

Almost surely asymptotically upper bounds for solutions of Ito
equations are obtained under various hypotheses. The methods
introduced are then extended to treat much more general stochastic
differential equations driven by nonlinear integrators and the 
upper bounds for these solutions are also investigated with
the help of Lyapunov functions.

xm@maths.warwick.ac.uk

125. WAVE EQUATION WITH STOCHASTIC BOUNDARY VALUES

X. Mao and L. Markus

In this paper we investigate the stochastic vibrations
of a flexible string which is excited by a boundary
force of white noise, as modeled by the 1-dimensional
wave partial differential equation with stochastic
boundary values.  We present the appropriate existence,
uniqueness, and regularity theorems for such stochastic
solutions. We also obtain estimates for the asymptotic
growth of the stochastic amplitude, and formulate and 
resolve a barrier problem dealing with the prescription
of an amplitude limitation compatible with the linear
wave model.

xm@maths.warwick.ac.uk

126. LONG-TERM AVERAGE CONTROL OF A CONTINUOUS, MONOTONE PROCESS

Arthur C. Heinricher
Richard H. Stockbridge

We analyze optimal control problems for systems subject to
random deterioration and failure.  The system is replaced
at failure and our objective is to optimize the utilization
of the system between failures.  The problems
are new in that the payoff depends on the running maximum
of a diffusion.  This provides an intuitively
appealing model for naturally monotone phenomena such as wear.
The long-term average control problem is reduced to a family
of simpler, single-cycle problems and we give a policy improvement
algorithm based on this decomposition.

heinrich@ms.uky.edu
stockb@ms.uky.edu

127. AN INFINITE-DIMENSIONAL LP SOLUTION TO CONTROL OF A CONTINUOUS MONOTONE PROCESS

Arthur C. Heinricher
Richard H. Stockbridge

Consider the following optimal control problem: 
A process increases randomly in time until it reaches a specified level
at which time it is restarted at zero and the cycle is repeated.
A reward is earned while the process is running, but a cost is incurred
when the process is restarted at zero.
The controller is faced with the following conflict:  
A high drift rate improves the reward but also shortens the time before the
restart cost is incurred.  The performance of a strategy is measured by the
long-term average reward.

We take a different point of view in this paper than in the previous one.
The state process is characterized as a solution to a controlled version of 
Kurtz's  constrained martingale problem.
This leads directly to the stationary distributions for the
process and an infinite-dimensional linear program for the optimal
value.  The invariant measures for the replaced process
are identified and the solution to the LP problem 
is described.  A policy improvement algorithm is exhibited for computing
the solution.

heinrich@ms.uky.edu
stockb@ms.uky.edu

128. OPTIMAL CONTROL AND REPLACEMENT WITH STATE DEPENDENT FAILURE RATE: DYNAMIC PROGRAMMING

Arthur C. Heinricher
Richard H. Stockbridge

A class of stochastic control problems where the
payoff depends on the running maximum of a diffusion process
is described.  The controller must make two kinds of decision:
first, he must choose a work rate (this decision determines
the rate of profit as well as the proximity of failure), and 
second, he must decide when to replace a deteriorated system
with a new one.  
Preventive replacement is a realistic option if the cost for replacement
after failure is larger than the cost of a preventive replacement.

We focus on the profit and replacement cost for a single work cycle
and solve the problem in two stages.
First, the optimal feedback control (work rate) is determined
by maximizing the payoff during a single excursion of a controlled
diffusion away from the running maximum.  
This step involves the solution of the Hamilton-Jacobi-Bellman
partial differential equation.
The second step is to determine the optimal replacement set.
The assumption that failure occurs only on the set where the 
state is increasing implies that replacement is optimal only on this set.
This leads to a simple formula for the optimal replacement level
in terms of the value function.

heinrich@ms.uky.edu
stockb@ms.uky.edu

129. OPTIMAL CONTROL AND REPLACEMENT WITH STATE-DEPENDENT FAILURE RATE: AN INVARIANT MEASURE APPROACH

Stochastic control problems in which the payoff depends on the
running maximum of a diffusion process are considered.  Such
processes provide appealing models for physical processes that
evolve in a {\em continuous}\/ and {\em increasing}\/ manner and
fail at a random time.  The controller must make two decisions:
first, she must choose how fast to work (this decision determines
the rate of profit as well as the proximity of failure), and
second, she must decide when to replace a deteriorated system
with a new one.  Preventive replacement becomes an important
option when the cost for replacement after a failure is larger
than the cost of a preventive replacement.  Single--cycle and
long--term average criteria are used to evaluate the control and
replacement decisions.

We model the process via a martingale problem formulation. This 
enables the long--term average control problem to be rephrased as
an LP over the invariant measures of the process.  We identify the
invariant measures corresponding to each control and replacement
decision and determine the optimal solution using an iterative 
scheme.

heinrich@ms.uky.edu
stockb@ms.uky.edu

130. LARGE DEVIATIONS FOR MARKOV CHAINS WITH RANDOM TRANSITIONS

Timo Seppalainen

This paper presents almost sure uniform large deviation principles for
the empirical distributions and empirical processes of  Markov chains 
with random transitions. The results are derived under  assumptions  
which  generalize assumptions earlier used for time-homogeneous chains. 
The rate functions for the skew chain are expressed in terms of the 
Donsker--Varadhan functional and relative entropy. The sample chain 
rates are different, but they have natural  upper and lower bounds in 
terms of familiar rate functions.

timosepp@function.mps.ohio-state.edu

131. ON THE CRITICAL BEHAVIOR OF THE CONTACT PROCESS IN DETERMINISTIC INHOMOGENEOUS ENVIRONMENT

Neal Madras, Rinaldo Schinazi, Roberto Schonmann

We consider the contact process with inhomogeneous deterministic
death rates. We prove that
(i) such models may have discontinuous transitions, in the sense of surviving
at the critical point.
(ii) If the death rates are identically 1, except on a set which is small enough
in a proper sense and where the death rates take a fixed value smaller than 
1, then the critical point is identical to that of the homogeneous system.
   Extensions of the results to other $(d+1)$-dimensional systems with
$d$-dimensional deterministic inhomogeneities are also discussed.

schinazi@zeppo.colorado.edu (plain Tex file available)

132. A CHARACTERISATION OF CROSS-OVER MODELS THAT POSSESS MAP FUNCTIONS

Steven N. Evans, M.S. McPeek and T.P. Speed

This paper concerns genetic map functions and a particular 
probability model for crossovers.  Karlin and Liberman (1979) 
and Risch and Lange (1979) independently introduced what 
the former called the count-location (point) process model 
for crossovers, which leads to a probability model for 
multilocus recombination under the assumption of no chromatid 
interference.  Liberman and Karlin (1984) later explored the 
constraints on genetic map functions resulting from the 
requirement that they be realisable in terms of a probability 
model for multilocus recombination.  In this note we prove that 
under the assumption of no chromatid interference, the class 
of probability models for multilocus recombination that possess 
map functions is precisely the class of count-location processes.  
As a consequence, we give a complete analytic characterisation 
of the functions that can arise as map functions for some 
probability model of multilocus recombination under the 
assumption of no chromatid interference.

evans@stat.berkeley.edu

133. TWO REPRESENTATIONS OF A CONDITIONED SUPERPROCESS

Steven N. Evans

We consider a class of measure-valued Markov processes 
that are constructed by taking a superprocess over some 
underlying Markov process and conditioning it to stay 
alive forever.  We obtain two representations of such 
a process. The first representation is in terms of an 
``immortal particle'' that moves around according to 
the underlying Markov process and throws of pieces of 
mass, which then proceed to evolve in the same way that 
mass evolves for the unconditioned superprocess.  As a 
consequence of this representation, we show that the 
tail $\sigma$-field of the conditioned superprocess 
is trivial if the tail $\sigma$-field of the underlying 
process is trivial.   The second representation is 
analogous to one obtained by LeGall in the unconditioned 
case.  It represents the conditioned superprocess in 
terms of a certain process taking values in the path 
space of the underlying process.  This representation 
is useful for studying the ``transience'' and 
``recurrence'' properties of the closed support process.  
In particular, we find some evidence for the conjecture  
that the closed support of conditioned super Brownian 
motion is ``transient'' in more than $4$ dimensions and 
``recurrent'' otherwise.

evans@stat.berkeley.edu

134. ESTIMATION OF THE QUADRATIC VARIATION OF NEARLY OBSERVED SEMIMARTINGALES WITH APPLICATION TO FILTERING

Jean Picard

Consider a filtering problem in which the available
information is a noisy observation of a continuous semimartingale
$H_t$. In the case of a high signal-to-noise ratio, it is proved that
$H_t$ and its quadratic variation can be jointly estimated by means of
a finite-dimensional filter; moreover, for this result, the
observation noise and $H_t$ are not required to be independent. This
problem can be viewed as a linear filtering problem with randomly
time-varying parameters, and our filter is auto-adaptive with respect to
changes of the parameters. These results are then applied to the non
linear filtering of Markov diffusion processes when the observation
function is not injective but satisfies a weaker detectability
assumption. It appears that filtering such a system involves two time
scales. The study is based on time discretization; the main tools are
an averaging principle, and an application of the asymptotic ordinary
differential equation method for the study of stochastic algorithms.

picard@ucfma.univ-bpclermont.fr

135. A NECESSARY AND SUFFICIENT CONDITION FOR THE MARKOV PROPERTY OF THE LOCAL TIME PROCESS

Nathalie Eisenbaum and Haia Kaspi

Let $X$ be a Markov process on an interval $E$ of the real line, with
lifetime $\zeta$, admitting a local time at each point and such that
$P_x(X hits y) > 0$ for all $x,y$ in $E$. We prove here that the local time 
process $(L_\zeta^x, x\in E)$ is a Markov process if and only if $X$
has fixed birth and death points and $X$ has continuous paths.

The sufficience of this condition has been established by Ray,
Knight and Walsh. The necessity is proved using arguments based
on excursion theory. We have proved this result before for
symmetric processes using the existence of a zero mean Gausssian
process with the Green function as covariance.

IERKH01@TECHNION.bitnet

136. ON CONSERVATIVENESS AND RECURRENCE CRITERIA FOR MARKOV PROCESSES

Y.Oshima

We give general criteria which guarantee the conservativeness and
recurrence of Markov processes associated with (not necessarily 
symmetric) Dirichlet spaces. Using the criteria we discuss comparison
theorem as well as some sufficient conditions for recurrence or
conservativeness. 

ta020@higo.kumamoto-u.ac.jp

137. ON A CONSTRUCTION OF MARKOV PROCESSES ASSOCIATED WITH TIME DEPENDENT DIRICHLET SPACES

Y.Oshima

For a given family of time dependent Dirichlet spaces with common 
domain, we construct an associated space-time Hunt process. The 
construction is carried out using the similar argument to the case
of symmetric Dirichlet spaces by using the notion of parabolic 
capacity. 

ta020@higo.kumamoto-u.ac.jp

138. SOME PROPERTIES OF MARKOV PROCESSES ASSOCIATED WITH TIME DEPENDENT DIRICHLET FORMS

Y.Oshima

For the space-time Markov processes associated with time dependent 
family of Dirichlet spaces, we show that the stochastic calculus 
similar to that given by Fukushima in symmetric case remains to hold,
that is, the correspondence between smooth measures and positive
natural additive functionals, decompositions of additive functionals 
into martingale part and energy zero part .... can be discussed
under our settings. 

ta020@higo.kumamoto-u.ac.jp