Probability Abstracts 55

1430. LAYERED MULTISHIFT COUPLING FOR USE IN PERFECT SAMPLING ALGORITHMS (WITH A PRIMER ON CFTP)

David Bruce Wilson

In this article we describe a new coupling technique which is useful in a
variety of perfect sampling algorithms. A multishift coupler generates a random
function f() so that for each real x, f(x)-x is governed by the same fixed
probability distribution, such as a normal distribution. We develop the class
of layered multishift couplers, which are simple and have several useful
properties. For the standard normal distribution, for instance, the layered
multishift coupler generates an f() which (surprisingly) maps an interval of
length L to fewer than 2+L/2.35 points --- useful in applications which perform
computations on each such image point. The layered multishift coupler improves
and simplifies algorithms for generating perfectly random samples from several
distributions, including the autogamma distribution, posterior distributions
for Bayesian inference, and the steady state distribution for certain storage
systems. We also use the layered multishift coupler to develop a Markov-chain
based perfect sampling algorithm for the autonormal distribution.
  At the request of the organizers, we begin by giving a primer on CFTP
(coupling from the past); CFTP and Fill's algorithm are the two predominant
techniques for generating perfectly random samples using coupled Markov chains.

dbwilson@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/9912225.

1431. PERCOLATION IN THE HYPERBOLIC PLANE

Itai Benjamini and Oded Schramm

This is a study of percolation in the hyperbolic plane and on regular tilings
in the hyperbolic plane. The processes discussed include Bernoulli site and
bond percolation on planar hyperbolic graphs, invariant dependent percolations
on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the
existence of three distinct nonempty phases for the Bernoulli processes. In the
first phase, $p\in(0, p_c]$, there are no unbounded clusters, but there is a
unique infinite cluster for the dual process. In the second phase,
$p\in(p_c,p_u)$, there are infinitely many unbounded clusters for the process
and for the dual process. In the third phase, $p\in [p_u,1)$, there is a unique
unbounded cluster, and all the clusters of the dual process are bounded. We
also study the dependence of $p_c$ in the Poisson-Voronoi-Bernoulli percolation
process on the intensity of the underlying Poisson process.

schramm@wisdom.weizmann.ac.il

• This article is available from the xxx mathematics archive as math.PR/9912233.

1432. DIFFEOMORPHIC FLOWS DRIVEN BY LEVY PROCESSES

David R.E. Williams

We prove that the stochastic differential equation $$ Y_{s,t}(x) = Y_{s,s}(x)
+ \int_0^{t-s} f(Y_{s,s+u}(x)) dX_{s+u},
  Y_{s,s}(x)=x\in\R^d. $$ driven by a L\'evy process whose paths have finite
p-variation almost surely for some $p\in[1,2)$ defines a flow of locally
C^1-diffeomorphisms provided the vector field f is $\alpha$-Lipschitz for some
$\alpha>p$. Using a path- wise approach we relax the smoothness condition
normally required for a class of discontinuous semi-martingales.

williams@stats.ox.ac.uk

• This article is available from the xxx mathematics archive as math.PR/0001016.

1433. PATH-WISE SOLUTIONS OF SDE'S DRIVEN BY LEVY PROCESSES

David R.E. Williams

In this paper we show that a path-wise solution to the following integral
equation $$ Y_t = \int_0^t f(Y_t) dX_t \qquad Y_0=a \in \R^d $$ exists under
the assumption that X_t is a L\'evy process of finite p-variation for some $p
\geq1$ and that f is an $\alpha$-Lipschitz function for some alpha>p. There are
two types of solution, determined by the solution's behaviour at jump times of
the process X, one we call geometric the other forward. The geometric solution
is obtained by adding fictitious time and solving an associated integral
equation. The forward solution is derived from the geometric solution by
correcting the solution's jump behaviour. L\'evy processes, generally, have
unbounded variation. So we must use a pathwise integral different from the
Lebesgue-Stieltjes integral. When X has finite p-variation almost surely for
p<2 we use Young's integral. This is defined whenever f and g have finite p and
q-variation for 1/p+1/q>1 (and they have no common discontinuities). When p>2
we use the integral of Lyons. In order to use this integral we construct the
L\'evy area of the L\'evy process and show that it has finite (p/2)-variation
almost surely.

williams@stats.ox.ac.uk

• This article is available from the xxx mathematics archive as math.PR/0001018.

1434. RANDOM VICIOUS WALKS AND RANDOM MATRICES

Jinho Baik

Lock step walker model is a one-dimensional integer lattice walker model in
discrete time. Suppose that initially there are infinitely many walkers on the
non-negative even integer sites. At each tick of time, each walker moves either
to its left or to its right with equal probability. The only constraint is that
no two walkers can occupy the same site at the same time. It is proved that in
the large time limit, a certain conditional probability of the displacement of
the leftmost walker is identical to the limiting distribution of the properly
scaled largest eigenvalue of a random GOE matrix (GOE Tracy-Widom
distribution). The proof is based on the bijection between path configurations
and semistandard Young tableaux established recently by Guttmann, Owczarek and
Viennot. Statistics of semistandard Young tableaux is analyzed using the Hankel
determinant expression for the probability from the work of Rains and the
author. The asymptotics of the Hankel determinant is obtained by applying the
Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the
related orthogonal polynomials.

jbaik@math.princeton.edu

• This article is available from the xxx mathematics archive as math.PR/0001022.

1435. RANDOM WALKS AND ELECTRIC NETWORKS

Peter G. Doyle and J. Laurie Snell 

A popular account of the connection between random walks and electric
networks.

doyle@math.dartmouth.edu

• This article is available from the xxx mathematics archive as math.PR/0001057.

1436. ON THE HYDRODYNAMIC EQUILIBRIUM OF A ROD IN A LATTICE FLUID

Pablo. A. Ferrari, Christian Maes, Laura Ramos, Frank Redig

We model the behavior of a big (Brazil) nut in a medium of smaller nuts with
a stochastic asymmetric simple exclusion dynamics of a polymer-monomer lattice
system. The polymer or `rod' can move up or down in an external negative field,
occupying N horizontal lattice sites where the monomers cannot enter. The
monomers (at most one per site) or `fluid particles' are moving symmetrically
in the horizontal plane and asymmetrically in the vertical direction, also with
a negative field. For a fixed position of the rod, this lattice fluid is in
equilibrium with a vertical height profile reversible for the monomers' motion.
Upon `shaking' (speeding up the monomers) the motion of the `rod' dynamically
decouples from that of the monomers resulting in a reversible random walk for
the rod around an average height proportional to log N.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/0001073.

1437. MARKOV TRANSITIONS AND THE PROPAGATION OF CHAOS

Alexander David Gottlieb

The propagation of chaos is a central concept of kinetic theory that serves
to relate the equations of Boltzmann and Vlasov to the dynamics of
many-particle systems. Propagation of chaos means that molecular chaos, i.e.,
the stochastic independence of two random particles in a many-particle system,
persists in time, as the number of particles tends to infinity.
  We establish a necessary and sufficient condition for a family of general
n-particle Markov processes to propagate chaos. This condition is expressed in
terms of the Markov transition functions associated to the n-particle
processes, and it amounts to saying that chaos of random initial states
propagates if it propagates for pure initial states.
  Our proof of this result relies on the weak convergence approach to the study
of chaos due to Sznitman and Tanaka. We assume that the space in which the
particles live is homeomorphic to a complete and separable metric space so that
we may invoke Prohorov's theorem in our proof.
  We also show that, if the particles can be in only finitely many states, then
molecular chaos implies that the specific entropies in the n-particle
distributions converge to the entropy of the limiting single-particle
distribution.

yeager@math.berkeley.edu

• This article is available from the xxx mathematics archive as math.PR/0001076.

1438. THE SUPREMUM OF BROWNIAN LOCAL TIMES ON HOLDER CURVES

Richard Bass and Krzysztof Burdzy

For $f: [0,1]\to \R$, we consider $L^f_t$, the local time of space-time
Brownian motion on the curve $f$. Let $\sS_\al$ be the class of all functions
whose H\"older norm of order $\al$ is less than or equal to 1. We show that the
supremum of $L^f_1$ over $f$ in $\sS_\al$ is finite is $\al>\frac12$ and
infinite if $\al<\frac12$.

burdzy@math.washington.edu

• This article is available from the xxx mathematics archive as math.PR/0002012.

1439. ON THE COVER TIME OF PLANAR GRAPHS

Johan Jonasson and Oded Schramm

The cover time of a finite connected graph is the expected number of steps
needed for a simple random walk on the graph to visit all the vertices. It is
known that the cover time on any n-vertex, connected graph is at least
(1+o(1))nlogn and at most (1+o(1))(4/27)n^3. This paper proves that for
bounded-degree planar graphs the cover time is at least c n(logn)^2, and at
most 6n^2, where c is a positive constant depending only on the maximal degree
of the graph.

schramm@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/0002034.

1440. A MIXTURE OF THE EXCLUSION PROCESS AND THE VOTER MODEL

Vladimir Belitsky, Pablo A. Ferrari, Mikhail V. Menshikov, 
Serguei Yu. Popov

We consider a one-dimensional nearest-neighbor interacting particle system,
which is a mixture of the simple exclusion process and the voter model. The
state space is taken to be the countable set of the configurations that have a
finite number of particles to the right of the origin and a finite number of
empty sites to the left of it. We obtain criteria for the ergodicity and some
other properties of this system using the method of Lyapunov functions.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/0002051.

1441. IS BOOTSTRAP REALLY HELPFUL IN POINT PROCESS STATISTICS?

Martin Snethlage 

There are some papers which describe the use of bootstrap techniques in point
process statistics. The aim of the present paper is to show that the form in
which bootstrap is used there is dubious. In case of variance estimation of
pair correlation function estimators the used bootstrap techniques lead to
results which can be obtained simpler without simulation; furthermore, they
differ from the desired results. The problem to obtain confidence regions for
the intensity function of inhomogeneous Poisson processes can be easily solved
without bootstrap techniques.

kerscher@theorie.physik.uni-muenchen.de

• This article is available from the xxx mathematics archive as math.PR/0002061.

1442. DETERMINANTAL RANDOM POINT FIELDS

Alexander Soshnikov

The paper contains an exposition of recent as well as old enough results on
determinantal random point fields. We start with some general theorems
including the proofs of the necessary and sufficient condition for the
existence of the determinantal random point field and a criterion for the weak
convergence of its distribution. In the second section we proceed with the
examples of the determinantal random point fields from Quantum Mechanics,
Statistical Mechanics, Random Matrix Theory, Probability Theory, Representation
Theory and Ergodic Theory. In connection with the Theory of Renewal Processes
we characterize all determinantal random point fields in R^1 and Z^1 with
independent identically distributed spacings. In the third section we study the
translation invariant determinantal random point fields and prove the mixing
property of any multiplicity and the absolute continuity of the spectra. In the
fourth (and the last) section we discuss the proofs of the Central Limit
Theorem for the number of particles in the growing box and the Functional
Central Limit Theorem for the empirical distribution function of spacings.

soshniko@math.ucdavis.edu

• This article is available from the xxx mathematics archive as math.PR/0002099.

1443. BLOCKING MEASURES FOR ASYMMETRIC EXCLUSION PROCESSES VIA COUPLING

P. A. Ferrari, J. L. Lebowitz, E. Speer

We give sufficient conditions on the rates of two asymmetric exclusion
processes such that the existence of a blocking invariant measure for the first
implies the existence of such a measure for the second. The main tool is a
coupling between the two processes under which the first dominates the second
in an appropriate sense. In an appendix we construct a class of processes for
which the existence of a blocking measure can be proven directly; these are
candidates for comparison processes in applications of the main result.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/0002193.

1444. SOME MEASURE-PRESERVING POINT TRANSFORMATIONS ON THE WIENER SPACE AND THEIR ERGODICITY

A.S. Ustunel and M. Zakai

Suppose that T is a map of the Wiener space into itself, of the following
type: T=I+u where u takes its values in the Cameron-Martin space. Assume also
that u is a finite sum of H-valued multiple Ito-Wiener integrals. In this work
we prove that if T preserves the Wiener measure, then necessarily u is in the
first Wiener chaos and the transformation corresponding to it is a rotation in
the sense of [9]. Afterwards the ergodicity and mixing of such transformations,
which are second quantizations of the unitary operators on the Cameron-Martin
space, are characterized.

zakai@ee.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0002198.

1445. ENTROPY-DRIVEN PHASE TRANSITIONS IN MULTITYPE LATTICE GAS MODELS

H.-O. Georgii, V. Zagrebnov

In multitype lattice gas models with hard-core interaction of
Widom--Rowlinson type, there is a competition between the entropy due to the
large number of types, and the positional energy and geometry resulting from
the exclusion rule and the activity of particles. We investigate this
phenomenon in four different models on the square lattice: the multitype
Widom-Rowlinson model with diamond-shaped resp. square-shaped exclusion between
unlike particles, a Widom-Rowlinson model with additional molecular exclusion,
and a continuous-spin Widom-Rowlinson model. In each case we show that this
competition leads to a first-order phase transition at some critical value of
the activity, but the number and character of phases depend on the geometry of
the model. Our technique is based on reflection positivity and the chessboard
estimate.

georgii@rz.mathematik.uni-muenchen.de

• This article is available from the xxx mathematics archive as math.PR/0002233.

1446. VALUATION OF EXOTIC OPTIONS UNDER SHORTSELLING CONSTRAINTS

Uwe Schmock, Steven Shreve and Uwe Wystup

Options with discontinuous payoffs are generally traded
above their theoretical Black-Scholes prices
because of the hedging difficulties created by their
large delta and gamma values. A theoretical
method for pricing these options is to constrain
the hedging portfolio and incorporate this constraint
into the pricing by computing the smallest
initial capital which permits super-replication of 
the option.
We develop this idea for exotic options, in which
case the pricing problem becomes one of stochastic
control. Our motivating example is a call which
knocks out in the money, and explicit formulas
for this and other instruments are provided.

shreve@cmu.edu

• To see a preprint or other information provided by the author click here.
• Or here.
• Or here.

1447. OPTIONS ON A TRADED ACCOUNT

Steven Shreve and Jan Vecer

In this article we study options on a traded
account.  In terms of the actions available
to the buyer, the options we study
are more general than a class of options known
as passport options; in terms of
the model of the underlying asset
they are more restrictive.
Using probabilistic techniques, we find the value
of these options, the optimal strategy of the
buyer, and the hedging strategy the seller should
use in response to a (not necessarily optimal)
strategy by the buyer.

shreve@cmu.edu vecer@andrew.cmu.edu

• To see a preprint or other information provided by the author click here.
• Or here.

1448. REAL-TIME QUEUES IN HEAVY TRAFFIC WITH EARLIEST-DEADLINE-FIRST QUEUE DISCIPLINE

Bogdan Doytchinov, John Lehoczky and Steven Shreve

This paper introduces a new aspect of queueing 
theory, the study of systems that service 
customers with specific timing requirements
(e.g. due dates or deadlines).   Unlike 
standard queueing theory in which common 
performance measures are customer delay, 
queue length and server utilization, 
real-time queueing theory focuses on the
ability of a queue discipline to meet 
customer timing requirements, e.g., the 
fraction of customers who meet their 
requirements and the distribution of 
customer lateness.  It also focuses on queue
control policies to reduce or minimize 
lateness, although these control aspects 
are not explicitly addressed in this paper.  

To study these measures, one must keep
track of the lead-times (deadline minus 
current time) of each customer, hence the 
system state is of unbounded dimension.
A heavy traffic analysis is presented 
for the earliest deadline first
(EDF) scheduling policy.  This analysis 
decomposes the behavior of the real-time 
queue into two parts: the number in the 
system (which converges weakly to
a reflected Brownian motion with drift) 
and the set of lead-times given the queue 
length.  The lead-time profile has a 
limit which is a non-random function
of the limit of the scaled queue length 
process.  Hence, in heavy traffic, 
one can characterize the system as a
diffusion evolving on a one-dimensional 
manifold of lead-time profiles. Simulation 
results are presented which indicate that this
characterization is surprisingly 
accurate. A discussion of open
research questions is also presented. 

bodgand@wpi.edu jpl@stat.cmu.edu shreve@cmu.edu

• To see a preprint or other information provided by the author click here.

1449. THE AVERAGE DENSITY OF SUPER-BROWNIAN MOTION

Peter Moerters

We prove the existence of average densities for the support
of a super-Brownian motion at a fixed time. Our result 
establishes a dimension-dependent fractal parameter for 
super-Brownian motion, which enables us to compare the 
local mass density of the super-Brownian motion at a fixed 
time with the local mass density of the occupation measure 
of a standard Brownian motion.

peter@mathematik.uni-kl.de

1450. A UNIQUENESS RESULT FOR HARMONIC FUNCTIONS

Richard Bass

Let d>2, D the upper half space, and suppose u is
C^2 on the closure of D. If the gradient of u vanishes
continuously on a subset of the boundary of D of positive
d-dimensional Lebesgue measure and u satisfies certain
regularity conditions, then u must be identically constant.
This is related to a problem of L. Bers.

bass@math.uconn.edu

• To see a preprint or other information provided by the author click here.

1451. SUPPORT FRAGMENTATION FOR MULTIPLICATIVE CASCADE MEASURES

Mina Ossiander

The family of h-cascades arises naturally in the 
study of multiplicative cascades.  We show that
the limiting h-cascades deriving from a single
collection of cascade generators have disjoint
supports with probability one and are consequently
a.s. mutually singular.  This result relies on the
derivation of a fragmentation level for the individual
multiplicative h-cascade measures using large
deviation type techniques.

ossiand@math.orst.edu

1452. MARTINGALES AND LARGE DEVIATIONS FOR THE BINARY SEARCH TREES

Jean Jabbour-Hattab

We establish almost sure and in mean large
deviations theorems for the structure of the external 
nodes of binary search trees (BST).
To achieve this, a parametric family of martingales is 
introduced.This family also allows us to get asymptotic 
results on the number of final external nodes of a BST.

jabbour@math.uvsq.fr

1453. GENEALOGIES AND INCREASING PROPAGATIONS OF CHAOS FOR FEYNMAN-KAC AND GENETICS MODELS

Pierre Del Moral and Laurent Miclo

A path-valued interacting particle systems model 
for the genealogical structure of genetic algorithms 
is presented. We connect the historical process and 
the distribution of the whole ancestral tree with a 
class of Feynman-Kac formulae on path space. 
We also prove increasing and uniform versions of 
propagation of chaos for appropriate particle 
block size and time horizon yielding what seems 
to be the first result of this type for this class 
of particle systems.

delmoral@cict.fr miclo@cict.fr

• To see a preprint or other information provided by the author click here.

1454. LARGE AND MODERATE DEVIATIONS FOR HOTELLING'S T^2-STATISTIC

Amir Dembo and Qi-Man Shao

Let $X, X_1, X_2, ...$ be i.i.d. $R^d$-valued random 
variables. We prove large and moderate deviations for
Hotelling's $T^2$-statistic when $X$ is in the generalized 
domain of attraction of the normal law.

amir@math.stanford.edu   shao@math.uoregon.edu

1455. HOW FAST ARE THE PARTICLES OF SUPER-BROWNIAN MOTION?

Peter Moerters

In this paper we investigate fast paths in the range and
support of super-Brownian motion in the historical
setting. In this setting each particle of super-Brownian 
motion alive at time $t$ is represented by a path 
$w:[0,t]\to\mr^d$ and the state of historical 
super-Brownian motion is a measure on the 
set of paths. Typical paths are Brownian motion paths, 
however in the uncountable collection of paths in the 
range of a super-Brownian motion there are some which 
move faster than Brownian motion. We determine the maximal
speed of all paths during a given time period $E$, 
which turns out to be a function of the packing dimension 
of $E$. A path $w$ in the support of historical 
super-Brownian motion at time $t$ is called $a$-fast if 
$\limsup_{h\downarrow 0} |w(t)-w(t-h)|/\sqrt{h \log(1/h)}
\geq a$. We calculate the Hausdorff dimension of the set of
$a$-fast paths in the support and the range of historical 
super-Brownian motion. A valuable tool in the proofs is a 
uniform dimension formula for the Brownian snake, which 
reduces dimension problems in the space of
stopped paths to dimension problems on the line.

peter@mathematik.uni-kl.de

1456. MULTIFRACTALS FORMALISMS FOR THE LOCAL SPECTRAL AND WALK DIMENSIONS

Ben M. Hambly, J. Kigami and T. Kumagai

We introduce the notions of local spectral and walk dimension for fractals. 
For a class of finitely ramified fractals we show that, if the underlying 
measure for the dynamic processes on the fractal is multifractal, then the 
local spectral and walk dimensions will also have a non-trivial multifractal 
spectrum. The multifractal spectra for both dimensions can be calculated and 
are shown to be�@transformations of the original underlying multifractal 
spectrum for the measure, but with respect to the effective resistance metric.

kumagai@i.kyoto-u.ac.jp

• To see a preprint or other information provided by the author click here.

1457. ULTRACONTRACTIVITY AND SUPERCONTRACTIVITY OF MARKOV SEMIGROUPS

Michael Röckner and Feng-Yu Wang

By using perturbation arguments, a sufficient condition is presented
for the ultracontractivity of symmetric diffusion semigroups. As a
consequence, a result suggested by D. Strook is proved: let $P_t$ be
generated by $L=\Delta+\nabla V$ with $V=-\alpha\nabla\rho^\delta$,
($\alpha>0$, $\delta>2$) on a complete connected Riemannian manifold
$M$, where $\rho$ is the Riemannian distance function from a fixed
point, then $P_t$ is ultracontractive provided the Ricci curvature is
bounded below. Furthermore, $\|P_t\|_{1\to\infty}\le\exp[\lambda_1+
\lambda_2t^{-\delta/(\delta-2)}]$ for some $\lambda_,\lambda_2>0$.
Next, it is shown that, for a diffusion semigroup (not necessarily
symmetric) with an invariant probability measure, if the curvature of
its generator is bounded from below, then the ultracontractivity is
equivalent to $\|P_t\exp[\lambda\rho^2]\|_\infty<\infty$ for any
$\lambda,t>0$. Especially, the above estimate of $\|P_t\|_{1\to\infty}$
holds if $L\rho^2\le c_1-c_2\rho^\delta$ for some $c_1,c_2>0$ and
$\delta>2$. This estimate is sharp as shown by examples given at the
end of the paper. Corresponding results are proven for
supercontractive semigroups, from which one obtains explicit examples
to show that untracontractivity is strictly stronger than
supercontractivity. Ultracontractivity and supercontractivity are also
studied by using Sobolev type inequalities and isoperimetric
inequalities. Finally, some concentration results are obtained for
reversible measures of supercontractive semigroups.

roeckner@mathematik.uni-bielefeld.de fwang@mathematik.uni-bielefeld.de

1458. ELLIPTIC EQUATIONS FOR INVARIANT MEASURES ON RIEMANNIAN MANIFOLDS: EXISTENCE AND REGULARITY OF SOLUTIONS

Vladimir Bogachev, Michael Röckner and Feng-Yu Wang

We obtain sufficient conditions in terms of Lyapunov functions for the
existence of invariant measures for diffusions on finite dimensional
manifolds and prove some global regularity results for such
measures. These results are extended to countable products of finite
dimensional manifolds. A new concept of a weak elliptic equation for
measures on infinite dimensional manifolds is introduced. As an
application, we obtain some a priori estimates for Gibbs measures on
countable products of manifolds and prove a new existence result for
such measures.

vbogach@mech.math.msu.su roeckner@mathematik.uni-bielefeld.de 
fwang@mathematik.uni-bielefeld.de

1459. ELLIPTIC EQUATIONS FOR INFINTE DIMENSIONAL PROBABILITY DISTRIBUTIONS AND LYAPUNOV FUNCTIONS

Vladimir Bogachev and Michael Röckner

A new concept of a weak elliptic equation for probability
distributions on infinite dimensional spaces is introduced.
A suitable notion of a Lyapunov function is defined for
weak elliptic equations. Sufficient conditions in terms of
Lyapunov functions are given for the existence of solutions
for weak elliptic equations. Applications to Gibbs measures
are discussed.

vbogach@mech.math.msu.su roeckner@mathematik.uni-bielefeld.de 

1460. L^p-ANALYSIS OF FINITE AND INFINITE DIMENSIONAL DIFFUSION OPERATORS

Michael Röckner

This paper consists of lectures given at the C.I.M.E.
summer school on Kolmogorov equations held at Cetraro in
1998. The purpose of these lectures was to present an
approach to Kolmogorov equations in infinite dimensions
which is based on an $L^p(\mu)$-analysis of the
corresponding diffusion operator w.r.t. suitably chosen
measures. The main ideas and aims are explained, and an as
complete as possible presentation is given of what has been
achieved in this respect over the last few years.

roeckner@mathematik.uni-bielefeld.de 

1461. EMPIRICAL TESTING OF THE INFINITE SOURCE POISSON DATA TRAFFIC MODEL

C.A. Guerin,  H. Nyberg,  O. Perrin,  S. Resnick,  
H. Rootz\'en  and  C. St\u{a}ric\u{a}

The infinite source Poisson model is a fluid queue
approximation of network data transmission that assumes
that sources begin constant rate transmissions of data at
Poisson time points for random lengths of time. This model
has been a popular one as analysts attempt to provide 
explanations for observed features in 
telecommunications data such as self-similarity, 
long range dependence and
heavy tails. We survey some features 
of this model in cases where transmission length  
distributions have (a) tails so 
heavy that means are infinite, (b) heavy tails with 
finite  mean and infinite variance and  
(c) finite variance. We 
survey the self-similarity properties of various descriptor 
processes in this model and then present analyses of four 
data sets which show that certain features of the model
 are consistent with the data while others are contradicted. 
The data sets are 
1) the Boston University 1995 study of web sessions, 
2) the UC Berkeley home IP HTTP data collected 
in November 1996, 
3) traces collected in end of 1997 at a Customer 
Service Switch in Munich, and 
4) detailed data from a corporate Ericsson 
WWW server from October 1998.

caguerin@loe.u-3mrs.fr  Henrik.Nyberg@era-t.ericsson.se  
perrin@cict.fr   sid@orie.cornell.edu  
rootzen@math.chalmers.se  starica@math.chalmers.se 

• To see a preprint or other information provided by the author click here.

1462. OPTION VALUATION UNDER STOCHASTIC VOLATILITY

Alan L. Lewis

This book provides an advanced treatment of option
pricing. Providing largely original research not available
elsewhere, it covers the new generation of option models
where both the stock price and its volatility follow
diffusion processes. 
   The mathematical level is very informal and presumes
only a familiarity with stochastic calculus at the level
it is usually applied in finance.  
   One important property of this class of models is that 
option prices, relative to a numeraire, are often not 
martingales, but only strictly local martingales.
   These new models help explain important features of 
real-world option pricing such as the 'term structure of 
implied volatility'. The asymptotic term structure theory
is shown to be equivalent to an eigenvalue problem.   

alanlewis@home.com

• To see a preprint or other information provided by the author click here.

1463. SOLUTIONS TO A CLASS OF MULTIDIMENSIONAL SPDE'S

 A.L. Piatnitski, H.Z. Zhao and W.A. Zheng

Our main result is
 
THEOREM Given a positive (or negative), bounded and Lipschitz
continuous function $G$ and a function $\phi \in
C_0((0,T]\times R^d)$ for some $T>0$. Let $u_0$ be a
non-negative and Lipschitz continuous $C_0(R^d)$  function.
Then there is a measure-valued random variable $V(dx, ds)$
such that for any $q\in C_0^2(R^d)$  there is a
semimartingale $Q^{(q)}_t$ satisfying for all
$t\leq T$ the relations
\begin{eqnarray}
\int_0^t\int _{R^d}q(x)V(dx,ds)=\int_0^tQ^{(q)}_sds,
\end{eqnarray}
and 
\begin{eqnarray}
Q^{(q)}_t-\int_{R^d}q(x)u_0(x)dx={1\over 2}\int_0^t\int_{R^d}
\Delta q(x) V(dx,ds)+M^{(q)}(t),
\end{eqnarray}
where $M^{(q)}(t)$ is a martingale with respect to ${\cal
F}_t$ with its bracket given by
\begin{eqnarray}
<M^{(q)},M^{(q)}>_t=\int _0^t\int _{R^d}
q^2(x)G^2(\int_0^s\int_{R^d}\phi (s-r,x-z)V(dz,dr))V(dx,ds).
\end{eqnarray}

Our proof consists of several steps. We first discretize and 
localize the space-time white noise and consider the equation
corresponding to the discretized noise (or so-called colored
noise). For the stochastic reaction diffusion equations
with colored noise, we prove there exists a mild solution
which is continuous in space and time for almost all
$\omega$, moreover the solution can be
represented by the Feynman-Kac formula.
While we denote by $\{ u_{n,k}(t,x)\}$ the discretized
solutions, we show that the laws of $\{
u_{n,k}(t,x)dxdt\}_{n,k}$ form a tight sequence and any of
their limits is the law of some measure-valued random 
variable. Then we identify the limit as the weak solution of
the desired nonlinear SPSE given by the integral equations
(\ref{spde0})--(\ref{spde1}) with space-time white noise. 

H.Zhao@lboro.ac.uk   wzheng@uci.edu

1464. FLUCTUATIONS AND THEIR GLAUBER DYNAMICS IN LATTICE SYSTEMS

Sergio Albeverio, Alexei Daletskii, Yuri G. Kondratiev and Michael Röckner

We consider a class of unbounded spin systems (containing,
in particular, anharmonic classical crystals) and construct
the stochastic dynamics in the state of macroscopic
fluctuations starting from a given microscopic stochastic
time evolution.

albeverio@uni-bonn.de kondratiev@uni-bonn.de roeckner@mathematik.uni-bielefeld.de

1465. STOCHASTIC DIFFERENTIAL EQUATIONS FOR DIRICHLET PROCESSES

Richard F. Bass and Zhenqing Chen

We consider the stochastic differential equation
   $dX_t=a(X_t)dW_t+ b(X_t) dt$,
where $W$ is a one-dimensional Brownian motion. We 
formulate the notion of  solution and prove strong 
existence and pathwise uniqueness results when $a$ 
is in $C^{1/2}$ and $b$ is only a generalized 
function, for example, the distributional derivative 
of a Holder function or of a function of bounded 
variation. When $b=aa'$, that is, when the generator 
of the SDE is the divergence form operator 
   ${1\over 2}{d\over dx}(a^2 {d\over dx})$,
a result on  non-existence of a strong solution
and  non-pathwise uniqueness is given as well as
a result which characterizes when a solution is
a semimartingale or not. We also consider extensions 
of the notion of Stratonovich integral.

bass@math.uconn.edu   zchen@math.washington.edu

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1466. LOGARITHMIC SOBOLEV INEQUALITIES FOR UNBOUNDED SPIN SYSTEMS REVISITED

Michel Ledoux

We analyze recent proofs of decay of correlations
and logarithmic Sobolev inequalities for unbounded spin
systems in the perturbative regime developed
by B. Zegarlinski, B. Helffer, Th. Bodineau, N. Yoshida.
We investigate to this task a simple analytic model
together with a new $L^1$ bound on the correlations.
Proofs are short and self-contained.

ledoux@cict.f

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1467. LEVY PROCESSES IN STOCHASTIC DIFFERENTIAL GEOMETRY

David Applebaum

We survey constructions of Levy processes in Lie groups, 
symmetric spaces and Riemannian manifolds. In general, 
Fourier analytic techniques are not available in these 
structures and so we study processes through the generator 
of the associated semigroup or by using stochastic 
differential equations.

dba@maths.ntu.ac.uk

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1468. ISOTROPIC LEVY PROCESSES ON PRINCIPAL FIBRE BUNDLES

David Applebaum

If P is a principal fibre bundle with structure group G
and base M, we show that isotropic Levy processes in P
can be described entirely in terms of such processes
in M and G. The factorisation which expresses the above
precisely is in terms of solutions of SDEs in the 
orthonormal frame bundle O(P) and requires that the 
driving Levy process in Euclidean space splits into two 
independent components.

dba@maths.ntu.ac.uk

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1469. ON A CLASS OF NON-TRANSLATION INVARIANT FELLER SEMIGROUPS IN LIE GROUPS

David Applebaum

We consider a class of Feller semigroups on Lie groups
which fail to commute with left translation due to the
existence of a cocycle h which is identically one for 
Levy processes. Under certain conditions, we are able 
to show that the infinitesimal generator of such 
a semigroup has the Levy-Khintchine-Hunt form but 
with variable characteristics, thus we obtain an 
extension of classical work in Euclidean space 
by Courrege.

dba@maths.ntu.ac.uk

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1470. THE INTERLACING CONSTRUCTION FOR STOCHASTIC FLOWS OF DIFFEOMORPHISMS ON EUCLIDEAN SPACES

David Applebaum and Fuchang Tang

We investigate stochastic differential equations 
driven by infinite dimensional semimartingales 
with jumps and show that both the solution flow and
the derivative flow can be represented as almost sure
limits of sequences that consist of random motion
with continuous sample paths which are interlaced 
with random jumps. We are thus able to obtain 
new transparent proofs of the diffeomorphism property
of such flows.

dba@maths.ntu.ac.uk  Fuchang.Tang@ntu.ac.uk

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1471. GREEDY LATTICE ANIMALS: NEGATIVE VALUES AND UNCONSTRAINED MAXIMA

Amir Dembo,  Alberto Gandolfi and Harry Kesten

Let $\{ X_v, v \in Z^d \}$ be i.i.d. random variables, and
$S(\xi) = \sum_{v \in \xi} X_v$ be the weight of a lattice
animal $\xi$. Let $N_n = \max \{ S(\xi): | \xi | = n$ and $\xi$
contains the origin$\}$ and
$G_n = \max \{ S(\xi): \xi \subseteq [-n, n]^d \}$.

We show that, regardless of the negative tail of the
distribution of $X_v$, if $\E(X_v^+)^d(\log^+(X_v^+))^{d + a}$
is finite for some $a > 0$, then first,
$\lim_n \; n^{-1} N_n = N$ exists, is
finite and constant a.e.; and, second, there is a transition in
the asymptotic behavior of $G_n$ depending on the sign of $N$:
if $N>0$ then $G_n \approx n^d$, and if $N<0$ then
$G_n \leq c n$, for some $c>0$.
The exact behavior of $G_n$ in this last case depends on the
 positive tail of the distribution of $X_v$; we show that if it
is nontrivial and has exponential  moments, then $G_n \approx \log n$,
with a transition from $G_n \approx n^d$ occurring in general
not as predicted by  large deviations estimates.
Finally, if $x^d(1- F(x) ) \to \infty$
as $x \to \infty$, then no transition takes place.

amir@math.stanford.edu, gandolfi@axp.mat.uniroma2.it, kesten@math.cornell.edu

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1472. UNIQUENESS FOR GIBBS STATES OF QUANTUM LATTICES IN SMALL MASS REGIME

Sergio Albeverio, Yuri G. Kondratiev, Yuri Kozitsky and Michael Röckner

The model of interacting quantum particles performing anharmonic
one-dimensional oscillations around their unstable equilibrium
position, which form the $d$-dimensional simple cubic lattice $\mathbb
Z^d$, is considered. For this model, it is proved that for every fixed
value of the temperature $\beta^{-1}$, there exists a positive
$m_\ast(\beta)$ such that for the values of the physcal mass of the
particle $m\in(0,m_\ast(\beta))$, the set of tempered Gibbs states
consits of exactly one element.

albeverio@uni-bonn.de kondratiev@uni-bonn.de
jkozi@golem.umcs.lublin.pl roeckner@mathematik.uni-bielefeld.de

1473. EXPONENTIAL MIXING FOR CLASSICAL CONTINUOUS SYSTEMS

Yu. G. Kondratiev, R. A. Minlos, M. Röckner and G. V. Shchepan'uk

Using the method of cluster expansions, we prove exponential
$L_{2+\epsilon}$--mixing ($\epsilon$ arbitrarily small and positive)
for Gibbs measures corresponding to dilute continuous systems of
particles interacting via a stable, exponentially decreasing pair
potential.

kondratiev@uni-bonn.de minl@ippi.ras.ru
roeckner@mathematik.uni-bielefeld.de shchepan@physik.uni-bielefeld.de

1474. RAY HÖLDER-CONTINUITY FOR FRACTIONAL SOBOLEV SPACES IN INFINITE DIMENSIONS AND APPLICATIONS

Jiagana Ren and Michael Röckner

We prove Hölder-continuity on rays in the directions of vectors in the
(generalized) Cameron-Martin space for functions in Sobolev spaces in
$L^p$ of fractional order $\alpha\in(1/p,1)$ over infinite dimensional
linear spaces. The underlying measures are required to satisfy some
easy standard structural assumptions only. Apart from Wiener measure
they include Gibbs measures on a lattice and Euclidean interacting
quantum fields in infinite volume. A number of applications, e.g. the
two-dimensional polymer measure, are presented. In particular,
irreducibility of the Dirichlet form associated with the latter
measure is proved without restrictions on the coupling constant.

jgren@public.wuhan.cngb.com roeckner@mathematik.uni-bielefeld.de

1475. $L^1$--UNIQUENESS OF MEASURABLE STATE SPACES: A CLASS OF EXAMPLES

Michael Röckner

We give an analytic proof for $L^1$--uniqeness of a class of diffusion
operators on arbitrary measurable spaces. In particular, this
generalizes a recent result, proved by probabilistic means by L. Wu,
to the non-symmetric case. The method we use is based on the classical
DuHamel formula which is applicable here due to results on
(non-symmetric) Dirichlet forms on merely measurable state spaces, and
a recent result by W. Stannat.

roeckner@mathematik.uni-bielefeld.de

1476. UNIQUENESS OF INVARIANT MEASURES AND ESSENTIAL $m$-DISSIPATIVITY OF DIFFUSION OPERATORS ON $L^p$

Vladimir I. Bogachev, Michael Röckner and Wilhelm Stannat

It is proved that there exists at most one probability measure $\mu$
on $\mathbb R^d$, so that $L^\ast\mu=0$, where
$L=a^{ij}\partial_i\partial_j+b^i\partial_i$, provided
$(L,C^\infty_0(\mathbb R^d))$ is essentially $m$--dissipative on
$L^1(\mathbb R^d,\nu)$ for at least one $\nu$, so that
$L^\ast\nu=0$. Here it is assumed that $(a^{ij})$ is non-degenerate,
$a^{ij}\in H^{p,1}_{loc}$, and $b^i\in L^p_{loc}$. We also present a
whole class of examples (even for $a^{ij}=\delta^{ij}$), where
$L^\ast\mu=0$ has more than one solution. Furthermore, recent related
results are reviewed.

vbogach@mech.math.msu.su roeckner@mathematik.uni-bielefeld.de
stannat@mathematik.uni-bielefeld.de

1477. PROBABILISTIC REPRESENTATIONS AND HYPERBOUND ESTIMATES FOR SEMIGROUPS

Michael Röckner and Tu-Sheng Zhang

In this paper, we study lower order perturbations of a symmetric
second order differential operator generating a hypercontractive
semigroup. We give a probabilistic representation of the in general
not sub-Markovian semigroup associated with the perturbed operator and
prove that the perturbed semigroup is also hypercontractive under some
exponential integrability conditions on the coefficients.

roeckner@mathematik.uni-bielefeld.de tz@tommy.hsh.no

1478. APPROXIMATION OF HUNT PROCESSES BY MULTIVARIATE POISSON PROCESSES

Zhi-Ming Ma, Michael Röckner and Wei Sun

We prove that arbitrary Hunt processes on a general state space con be
approximated by multivariate Poisson processes starting from each
point of the state space. The key point is that no additional
regularity assumption on the state space and on the underlying
transition semigroup is used.

mazm@sun.ihep.ac.cn roeckner@mathematik.uni-bielefeld.de
sunw@math.sinica.edu.tw