Probability Abstracts 49
This document contains abstracts 1194-1228.
They have been mailed on February 27, 1999.
1194. BROADCASTING ON TREES AND THE ISING MODEL
William Evans, Claire Kenyon, Yuval Peres and Leonard J. Schulman
Consider a process in which information is transmitted from
a given root node on a noisy tree network $T$. We start
with an unbiased random bit $R$ at the root of the tree,
and send it down the edges of $T$. On every edge the bit
can be reversed with probability $\epsilon$, and these
errors occur independently. The goal is to reconstruct $R$
from the values which arrive at level $n$ of the tree.
This model has been studied in information theory, genetics
and statistical mechanics. We bound the reconstruction
probability from above using the maximum flow on $T$ viewed
as a capacitated network, and from below using the
electrical conductance of $T$. For general infinite trees,
we establish a sharp threshold: The probability of correct
reconstruction tends to $1/2$ as $n \to \infty$ if
$(1-2 \epsilon)^2 < p_c(T)$, but the reconstruction
probability stays bounded away from $1/2$ if the opposite
inequality holds. Here $p_c(T)$ is the critical probability
for percolation on $T$; in particular $p_c(T)=1/b$ for the
$b+1$ regular tree. The asymptotic reconstruction problem is
equivalent to purity of the ``free boundary'' Gibbs state
for the Ising model on a tree. Thus our results extend the
work of Bleher, Ruiz and Zagrebnov (1995), who considered
the case of regular trees.
will@cs.arizona.edu Claire.Kenyon@lri.fr
peres@math.huji.ac.il schulman@cc.gatech.edu
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1195. THE GRIFFITHS SINGULARITY RANDOM FIELD
Aernout van Enter, Christian Maes, Roberto H. Schonmann and Senya Shlosman
We consider a spin system on sites of a $d$-dimensional cubic
lattice ($d\ge 2$), with the values $0,1$ or $-1$. It is built over the
Bernoulli site percolation model, with spins taking the value $0$
on empty sites, and taking values $\pm 1$ on occupied sites
according to the ferromagnetic Ising model distribution on the
occupied clusters. The Hamiltonian corresponds to the nearest
neighbor interaction under external field $h$, at inverse
temperature $\beta $, and the boundary conditions for clusters are
free. When the probability $p$ for a site to be occupied is small
enough, so that a.s. all the clusters of non-$0$ spins are finite, this
description gives rise to a unique random field. We show that when
$p$ is small, $\beta $ is large and $h=0,$ this random field is non-
Gibbsian, but is almost Gibbsian. This provides another example
of a non-Gibbsian, but almost Gibbsian, random field which
emerges naturally in a Gibbsian context. The random field that we
consider is directly related to, and motivated by, the model studied
by Griffiths in connection to what became known as the
phenomenon of Griffiths' singularities.
aenter@phys.rug.nl Christian.Maes@fys.kuleuven.ac.be
rhs@math.ucla.edu shlosman@cpt.univ-mrs.fr
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1196. THE PERCOLATION PROCESS ON A TREE WHERE INFINITE CLUSTERS ARE FROZEN
David J. Aldous
Modify the usual percolation process on the infinite binary tree
by forbidding infinite clusters to grow further. The ultimate
configuration will consist of both infinite and finite clusters.
We give a rigorous construction of a version of this process and
show that one can do explicit calculations of various quantities,
for instance the law of the time (if any) that the cluster
containing a fixed edge becomes infinite. Surprisingly, the
distribution of the shape of a cluster which becomes infinite attime
$t>1/2$ does not depend on $t$; it is always distributed as the
incipient infinite percolation cluster on the tree. Similarly, a
typical finite cluster at each time $t>1/2$ has the distribution of
A critical percolation cluster. This elaborates an observation of
Stockmayer (1942).
aldous@stat.berkeley.edu
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1197. EXTENDING THE MARTINGALE MEASURE STOCHASTIC INTEGRAL
WITH APPLICATIONS TO SPATIALLY HOMOGENEOUS S.P.D.E.'S
Robert C. Dalang
We extend the definition of Walsh's martingale measure stochastic
integral so as to be able to solve stochastic partial differential
equations whose Green's function is not a function but a Schwartz
distribution. This is the case for the wave equation in dimensions
greater than two. Even when the integrand is a distribution, the value
of our stochastic integral process is a real-valued martingale. We use
this extended integral to recover necessary and sufficient conditions
under which the linear wave equation driven by spatially homogeneous
Gaussian noise has a process solution, and this in any spatial
dimension. Under this condition, the non-linear three dimensional wave
equation has a global solution. The same methods apply to the damped
wave equation, to the heat equation and to various parabolic equations.
Robert.Dalang@epfl.ch
1198. THE GENERALIZED HYPERBOLIC MODEL:
FINANCIAL DERIVATIVES AND RISK MEASURES
Ernst Eberlein and Karsten Prause
Statistical analysis of data from the financial markets shows that
generalized hyperbolic (GH) distributions allow a more realistic
description of asset returns than the classical normal distribution.
GH distributions contain as subclasses hyperbolic as well as normal
inverse Gaussian (NIG) distributions which have recently been
proposed as basic ingredients to model price processes. We derive
an option pricing formula for GH based models using the Esscher
transform as one possibility to determine prices in an incomplete
market. The GH option pricing model is a generalization of the
hyperbolic model developed by Eberlein and Keller (1995),
Keller (1997). We compare this model with the classical
Black-Scholes model. We also propose a general recipe for
comparison of models to price derivatives with market reality. The
objectives of this research are to examine the consistency of our
model assumptions with the empirically observed price processes and
the performance of option pricing models from a practitioner's point
of view. Due to the difficulty if not impossibility to construct a
test in a strict statistical sense for pricing models, we compare
the models focussing on practically relevant criteria. Finally, we
present a simplified approach to the estimation of high dimensional
generalized hyperbolic distributions and their application to
measure risk in financial markets.
prause@stochastik.uni-freiburg.de
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1199. LE THEOREME CENTRAL LIMITE PRESQUE SUR
Mohamed Atlagh and Michel Weber
The Almost Sure Central Limit Theorem (A.S.C.L.T.) is a recent domain of
research, since first results are to be found in the seminal works of
Fisher [1987], Brosamler [1988] and Schatte [1988]. Since, the subject has
been considerably developped, and numerous extensions as well as
strenghtenings of initial results were obtained. That thema of research is
nowadays still motivating probabilists, and this is the reason of the
present survey. In its contain, we present essentiel results of the theory
as well as methods involved in their proofs.
weber@math.u-strasbg.fr
1200. SIXTY YEARS OF BERNOULLI CONVOLUTIONS
Yuval Peres, Wilhelm Schlag and Boris Solomyak
Let ${r_n}$ be i.i.d. random signs. The distribution
$\nu_s$ of the random series $\sum_n r_n s^n$ is the
infinite convolution product of the 2-point measures
$(\delta_{-s^n} + \delta_{s^n})/2$. These convolutions
have been studied since the 1930's, revealing
connections with harmonic analysis, the theory of
algebraic numbers, dynamical systems, and Hausdorff
dimension estimation. In this survey we describe some
of these connections, and the progress that has been
made so far on the fundamental open problem: For which
$s$ in $(1/2,1)$ is $\nu_s$ absolutely continuous?
Our main goal is to present an exposition of results
obtained by Erdos, Kahane and the authors on this
problem. Several related unsolved problems are
collected at the end of the paper.
peres@math.huji.ac.il schlag@math.princeton.edu solomyak@math.washington.edu
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1201. THE EQUIVALENCE OF THE LOG-SOBOLEV INEQUALITY AND
A MIXING CONDITION FOR UNBOUNDED SPIN SYSTEMS ON
THE LATTICE
Nobuo Yoshida
We consider a ferromagnetic spin system with unbounded
interactions on the $d$-dimensional integer lattice
($d \geq 1$). We prove that the following conditions
for the finite volume Gibbs states are equivalent (each
being understood to be uniform in the volume and the
boundary conditions); (1) the uniform log-Sobolev
inequality holds, (2) the spectral gap is uniformly
positive, (3) The spin-spin correlation decays exponentially.
This equivalence can be seen as an extension of a result
by D. W. Stroock and B. Zegarlinski to a class of unbounded
spin sytems.
nobuo@kusm.kyoto-u.ac.jp
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1202. PROBABILITY ON TREES: AN INTRODUCTORY CLIMB.
NOTES FROM THE 1997 SAINT FLOUR SUMMER SCHOOL
Yuval Peres
These notes were transcribed from the lectures by
Dimitris Gatzouras, and then edited by him and
by David Levin. The first 10 chapters are devoted
to basic facts about percolation on trees, branching
processes and electrical networks, with an emphasis
on several key techniques: moment estimates, use of
percolation to determine dimension, and the ``method
of random paths'' to construct flows of finite energy.
In Chapter 11, the Grimmett-Kesten-Zhang theorem on
transience of simple random walk in percolation
clusters in lattices is established. Chapters 12-13
present a regularity property of subperiodic trees,
and its application to random walks on groups. In
Chapter 14 we discuss capacity estimates for hitting
probabilities; these are used in Chapter 15 to derive
intersection-equivalence of fractal percolation with
Brownian paths. In Chapters 16-17, we study the Ising
model on a tree; it has arisen independently in
genetics, as a mutation model, and in communication
theory as a broadcasting model. The last four chapters
discuss tree-indexed processes and 'dynamical percolation'.
peres@math.huji.ac.il
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1203. PATH PROPERTIES OF CAUCHY'S PRINCIPAL VALUES RELATED TO LOCAL TIME
Endre Csaki, Miklos Csorgo, Antonia Foldes and Zhan Shi
Sample path properties of the Cauchy principal values of
Brownian and random walk local times are studied. We
establish LIL type results (without exact constants). Large
and small increments are discussed. A strong approximation
result between the above two processes is also proved.
csaki@math-inst.hu mcsorgo@math.carleton.ca
afoldes@email.gc.cuny.edu shi@ccr.jussieu.fr
1204. FIBER BROWNIAN MOTION AND THE ``HOT SPOTS'' PROBLEM
Richard F. Bass and Krzysztof Burdzy
We show that in some planar domains both extrema of the second
Neumann eigenfunction lie strictly inside the domain.
The main technical innovation is the use
of ``fiber Brownian motion,'' a process which
switches between two-dimensional and one-dimensional
evolution.
bass@math.uconn.edu burdzy@math.washington.edu
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1205. FLUCTUATIONS IN THE HOPFIELD MODEL AT THE CRITICAL TEMPERATURE
Barbara Gentz, Matthias Loewe
We investigate the fluctuations of the order parameter in the
Hopfield model of spin glasses and neural networks at the critical
temperature $1/\beta_c=1$. The number of patterns $M(N)$ is
allowed to grow with the number $N$ of spins but the growth rate is
subject to the constraint $M(N)^{15}/N\to 0$. As the system size $N$
increases, on a set of large probability the distribution of the
appropriately scaled order parameter under the Gibbs measure comes
arbitrarily close (in a metric which generates the weak topology) to a
non-Gaussian measure which depends on the realization of the random patterns.
This random measure is given explicitly by its (random) density.
gentz@wias-berlin.de lowe@eurandom.tue.nl
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1206. LARGE DEVIATIONS FOR BROWNIAN MOTION
ON THE SIERPINSKI GASKET
Gerard Ben Arous and Takashi Kumagai
We study large deviations for Brownian motion on the Sierpinski
gasket in the short time limit. Because of the subtle oscillation
of hitting times of the process, no large deviation principle
can hold. In fact our result shows that there is an infinity of
different large deviation principles for different subsequences,
with different (good) rate functions. Thus, instead of taking the
time scaling $\epsilon\to 0$, we prove that the large deviations
hold for $\epsilon^z_n\equiv (2/5)^nz$ as $n\to\infty$ using one
parameter family of rate functions $I^z~(z\in [2/5,1))$. As a
corollary, we obtain Strassen-type laws of the iterated logarithm.
kumagai@math.ubc.ca
1207. ON CERTAIN PROBABILITIES EQUIVALENT TO WIENER MEASURE,
D\'APR\`ES DUBLINS,FELDMAN, SMORODINSKY AND TSIRELSON
Walter Schachermayer
L. Dubins, J. Feldman, M. Smorodinsky and B. Tsirelson gave an
example of an equivalent measure $Q$ on standard Wiener space such
that each adapted $Q$-Brownian motion generates a strictly smaller
filtration then the original one. The construction of this important
example is complicated and technical.
We give a variant of their construction which differs in some of the
technicalities but essentially follows their ideas, hoping that some
readers may find our presentation easier to digest than the original
papers.
walter.schachermayer@fam.tuwien.ac.at
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1208. A COMPACTNESS PRINCIPLE FOR BOUNDED SEQUENCES
OF MARTINGALES WITH APPLICATIONS
Freddy Delbaen, Walter Schachermayer
For $\cal{H}^1$ bounded sequences, we introduce a technique, related to
the Kadec-Pelzynski-decomposition for $L^1$ sequences, that allows us
to prove compactness theorems.
Roughly speaking, a bounded sequence in $\cal{H}^1$ can be split into
two sequences, one of which is weakly compact, the other forms the
singular part.
If the martingales are continuous then the singular part tends to zero
in the semi-martingale topology.
In the general case the singular parts give rise to a process of bounded
variation. The technique allows to give a new proof of the optional
decomposition theorem in Mathematical Finance.
walter.schachermayer@fam.tuwien.ac.at
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1209. IS THERE A PREDICTABLE CRITERION FOR MUTUAL SINGULARITY
OF TWO PROBABILITY MEASURES ON A FILTERED SPACE?
Walter Schachermayer and Werner Schachinger
The theme of providing predictable criteria for absolute continuity and
for mutual singularity of two density processes on a filtered
probability space is extensively studied, e.g., in the monograph by
J. Jacod and A. N. Shiryaev [JS]. While the issue of absolute
continuity is settled there in full generality, for the issue of mutual
singularity one technical difficulty remained open ([JS], p210):
``We do not know whether it is possible to derive a {\sl predictable}
criterion (necessary and sufficient condition) for $P_T' \perp
P_T$,\dots''.
It turns out that to this question raised in [JS] which we also
chose as the title of this note, there are two answers: on the negative
side we give an easy example, showing that in general the answer is no,
even when we use a rather wide interpretation of the concept of
``predictable criterion''.
The difficulty comes from the fact that the density process
of a probability measure $P$ with respect to another measure $P'$
may suddenly jump to zero.
On the positive side we can characterize the set, where $P'$ becomes
singular with respect to $P$ --- provided this does not happen in a
sudden but rather in a continuous way --- as the set where the
Hellinger process diverges, which certainly is a ``predictable
criterion''. This theorem extends results in the book of
J. Jacod and A. N. Shiryaev [JS].
[JS] --- J. Jacod, A. N. Shiryaev: Limit Theorems for Stochastic
Processes. Berlin: Springer 1987.
walter.schachermayer@fam.tuwien.ac.at
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1210. THE ASYMPTOTIC ELASTICITY OF UTILITY FUNCTIONS
AND OPTIMAL INVESTMENT IN INCOMPLETE MARKETS
Dimitrii Kramkov and Walter Schachermayer
The paper studies the problem of maximizing the expected utility
of terminal wealth in the framework of a general incomplete
semimartingale model of a financial market. We show that the
{\it necessary and sufficient} condition on a utility function for
the validity of several key assertions of the theory to hold true
is the requirement that the asymptotic elasticity of the utility
function is strictly less then one.
walter.schachermayer@fam.tuwien.ac.at
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1211. A BIPOLAR THEOREM FOR SUBSETS OF $L^0_+(\Omega, \cal{F}, \bf{P})$
Werner Brannath and Walter Schachermayer
A consequence of the Hahn-Banach theorem is the classical
bipolar theorem which states that the bipolar of a subset
of a locally convex vector space equals its closed convex
hull. The space $L^0(\Omega, \cal{F}, \bf{P})$ of
real-valued random variables on a probability space
$(\Omega, \cal{F}, \bf{P})$ equipped with the topology
of convergence in measure fails to be locally convex so
that --- a priori --- the classical bipolar theorem does
not apply. In this note we show an analogue of the bipolar
theorem for subsets of the positive orthant $L^0_+(\Omega,
\cal{F}, \bf{P})$, if we place $L^0_+(\Omega, \cal{F},
\bf{P})$ in duality with itself, the scalar product now
taking values in $[0, \infty]$. In this setting the order
structure of $L^0(\Omega, \cal{F}, \bf{P})$ plays an
important role and we obtain that the bipolar of a subset
of $L^0_+(\Omega, \cal{F}, \bf{P})$ equals its closed,
convex and solid hull. In the course of the proof we show
a decomposition lemma for convex subsets of $L^0_+(\Omega,
\cal{F}, \bf{P})$ into a ``bounded'' and a ``hereditarily
unbounded'' part, which seems interesting in its own right.
walter.schachermayer@fam.tuwien.ac.at
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1212. WHEN DOES CONVERGENCE OF ASSET PRICE PROCESSES
IMPLY CONVERGENCE OF OPTION PRICES?
Friedrich Hubalek and Walter Schachermayer
We consider weak convergence of a sequence of asset price
models $(S^n)$ to a limiting asset price model $S$. A typical case
for this situation is the convergence of a sequence of binomial
models to the Black-Scholes model, as studied by Cox, Ross, and
Rubinstein.
We put emphasis on two different aspects of this convergence:
firstly we consider convergence with respect to the given
``physical'' probability measures $(P^n)$ and
secondly with respect to the ``risk-neutral'' measures
$(Q^n)$ for the asset price processes $(S^n)$. (In the case of
non-uniqueness of the risk-neutral measures also the question of
the ``good choice'' of $(Q^n)$ arises.) In particular we
investigate under which conditions the weak convergence of
$(P^n)$ to $P$ implies the weak convergence of $(Q^n)$
to $Q$ and thus the convergence of prices of derivative securities.
The main theorem of the present paper exhibits an intimate
relation of this question with contiguity properties of the
sequences of measures $(P^n)$ with respect to
$(Q^n)$ which in turn is closely connected to asymptotic
arbitrage properties of the sequence $(S^n)$ of
security price processes.
We illustrate these results with general homogeneous binomial
and some special trinomial models.
friedrich.hubalek@fam.tuwien.ac.at
walter.schachermayer@fam.tuwien.ac.at
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1213. PATHWISE UNIQUENESS FOR SPDES AND SDES VIA
CHANGE OF MEASURE
Hassan Allouba
We consider pairs of SPDEs and SDEs differing by a drift
function that may be discontinuous. We prove that it
is possible to use change of measure to transfer
pathwise uniqueness (equivalently $L^p$ stability for $p>0$)
among these pairs. We do even more by showing that such a
transfer may be carried out under an almost sure $L^2$
condition on the drift/diffusion ratio, a much less
stringent condition than Novikov's condition. This is done
for parabolic and hyperbolic SPDEs by analyzing more deeply
Girsanov's theorem for continuous orthogonal martingale
measures and some of its implications, which were proved
earlier in Allouba (1996,1998). As an application we prove
uniqueness and $L^p$ stability results for the Allen-Cahn
SPDE driven by space-time white noise. The same transfer
is shown to be valid also for ordinary SDEs.
allouba@math.umass.edu
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1214. VERTEX ORDERING AND PARTITIONING PROBLEMS FOR
RANDOM SPATIAL GRAPHS
Mathew D. Penrose
Given an ordering of the vertices of a graph, let the
induced weight for an edge be the separation of its
end-points in the ordering.Layout problems involve
choosing the ordering to minimize a cost functional
such as the sum or maximum of the edge-weights. We give
gworth rates for the costs of some of these problems on
supercritical percolation processes and supercritical
random geometric graphs, obtained by placing vertices
randomly in the unit cube and joining them whenever
at most some threshold distance apart.
mathew.penrose@durham.ac.uk
1215. ESTIMATING VACCINE COVERAGE BY USING COMPUTER ALGEBRA
D. Altmann, K. Altmann
The approach of N. Gay for estimating the coverage of a multivalent vaccine
from antibody prevalence data in certain age cohorts is improved by using
computer aided elimination theory of variables. Hereby, Gay's usage of
numerical approximation can be replaced by exact formulas which are
surprisingly nice, too.
altmann@mathematik.hu-berlin.de
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1216. A WAY OUT OF THE QUANTUM TRAP
Ph. Blanchard and A. Jadczyk
We review Event Enhanced Quantum Theory (EEQT). In Section 1 we address the
question "Is Quantum Theory the Last Word". In particular we respond to some of
recent challenging staments of H.P. Stapp. We also discuss a possible future of
the quantum paradigm - see also Section 5. In Section 2 we give a short sketch
of EEQT. Examples are given in Section 3. Section 3.3 discusses a completely
new phenomenon - chaos and fractal-like phenomena caused by a simultaneous
"measurement" of several non-commuting observables (we include picture of
Barnsley's IFS on unit sphere of a Hilbert space). In Section 4 we answer
"Frequently Asked Questions" concerning EEQT.
ajad@ift.uni.wroc.pl
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1217. PATH CROSSING EXPONENTS AND THE EXTERNAL PERIMETER IN 2D PERCOLATION
Michael Aizenman, Bertrand Duplantier and Amnon Aharony
Percolation path crossing exponents describe probabilities for $\ell$
non-overlapping traversing paths, each of either occupied sites or vacancies.
We show, for collections including at least one of each, that in 2D the
exponents are those of an $O(N=1)$ loop model. This extends the earlier
identification by Saleur and Duplantier of $k$ spanning cluster exponents, for
which $\ell=2k$. The results yield $D_{EP}=4/3$ for the fractal dimension of
the accessible external cluster perimeter, and explain the absence of narrow
gate fjords, in agreement with the original findings of Grossman and Aharony.
aizenman@princeton.edu
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1218. ELEMENTARY DERIVATION OF SPITZER'S ASYMPTOTIC LAW FOR BROWNIAN WINDINGS
AND SOME OF ITS PHYSICAL APPLICATIONS
Arkady L. Kholodenko
A simple derivation of Spitzer'z asymptotic law for Brownian windings
[Trans.Am.Math.Soc.87,187 (1958)]is presented along with its generalizations
>.These include the cases of planar Brownian walks interacting with a single
puncture and Brownian walks on a single truncated cone with variable conical
angle interacting with the truncated conical tip.Such situations are typical in
the theories of quantum Hall effect and 2+1 quantum gravity, respectively .They
also have some applications in polymer physics
string@clemson.edu
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1219. NECESSARY AND SUFFICIENT CONDITIONS FOR THE STRONG LAW OF LARGE NUMBERS
FOR U-STATISTICS
Rafa{\l} Lata{\l}a and Joel Zinn
Under some mild regularity on the normalizing sequence, we obtain necessary
and sufficient conditions for the Strong Law of Large Numbers for (symmetrized)
U-statistics. We also obtain nasc's for the a.s. convergence of series of an
analogous form.
rlatala@mimuw.edu.pl
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1220. ESCAPE PROBABILITY, MEAN RESIDENCE TIME AND GEOPHYSICAL FLUID PARTICLE
DYNAMICS
Jinqiao Duan, James R. Brannan, and Vincent J. Ervin
Stochastic dynamical systems arise as models for fluid particle motion in
geophysical flows with random velocity fields. Escape probability (from a fluid
domain) and mean residence time (in a fluid domain) quantify fluid transport
between flow regimes of different characteristic motion.
We consider a quasigeostrophic meandering jet model with random
perturbations. This jet is parameterized by the parameter $\beta = (2\Omega)/r
\cos (\theta)$, where $\Omega$ is the rotation rate of the earth, $r$ the
earth's radius and $\theta$ the latitude. Note that $\Omega$ and $r$ are fixed,
so $\beta$ is a monotonic decreasing function of the latitude. The unperturbed
jet (for $0 < \beta < 2/3$) consists of a basic flow with attached eddies. With
random perturbations, there is fluid exchange between regimes of different
characteristic motion. We quantify the exchange by escape probability and mean
residence time.
duan@math.clemson.edu
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1221. NONLOCAL DYNAMICS OF PASSIVE TRACER DISPERSION WITH RANDOM STOPPING
Jinqiao Duan, James R. Brannan, and H.Gao
We investigate the nonlocal behavior of passive tracer dispersion with random
stopping at various sites in fluids. This kind of dispersion processes is
modeled by an integral partial differential equation, i.e., an
advection-diffusion equation with a memory term. We have shown the exponential
decay of the passive tracer concentration, under suitable conditions for the
velocity field and the probability distribution of random stopping time.
duan@math.clemson.edu
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1222. LINEAR BOLTZMANN EQUATION AS THE WEAK COUPLING LIMIT OF A RANDOM
SCHRODINGER EQUATION
L. Erdos, H.-T. Yau
We study the time evolution of a quantum particle in a Gaussian random
environment. We show that in the weak coupling limit the Wigner distribution of
the wave function converges to a solution of a linear Boltzmann equation
globally in time. The Boltzmann collision kernel is given by the Born
approximation of the quantum differential scattering cross section.
lerdos@math.gatech.edu
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1223. LONGEST INCREASING SUBSEQUENCES OF RANDOM COLORED PERMUTATIONS
Alexei Borodin
We compute the limit distribution for (centered and scaled) length of the
longest increasing subsequence of random colored permutations. The limit
distribution function is a power of that for usual random permutations computed
recently by Baik, Deift, and Johansson (math.CO/9810105). In two--colored case
our method provides a different proof of a similar result by Tracy and Widom
about longest increasing subsequences of signed permutations (math.CO/9811154).
Our main idea is to reduce the `colored' problem to the case of usual random
permutations using certain combinatorial results and elementary probabilistic
arguments.
borodine@math.upenn.edu
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1224. MINIMUM DESCRIPTION LENGTH INDUCTION, BAYESIANISM, AND KOLMOGOROV
COMPLEXITY
Paul Vitanyi, Ming Li
The relationship between the Bayesian approach and the minimum description
length approach is established. We sharpen and clarify the general modeling
principles MDL and MML, abstracted as the ideal MDL principle and defined from
Bayes's rule by means of Kolmogorov complexity. The basic condition under which
the ideal principle should be applied is encapsulated as the Fundamental
Inequality, which in broad terms states that the principle is valid when the
data are random, relative to every contemplated hypothesis and also these
hypotheses are random relative to the (universal) prior. Basically, the ideal
principle states that the prior probability associated with the hypothesis
should be given by the algorithmic universal probability, and the sum of the
log universal probability of the model plus the log of the probability of the
data given the model should be minimized. If we restrict the model class to the
finite sets then application of the ideal principle turns into Kolmogorov's
minimal sufficient statistic. In general we show that data compression is
almost always the best strategy, both in hypothesis identification and
prediction.
Paul.Vitanyi@cwi.nl
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information provided by the author
click here.
1225. THE EXPECTED SIZE OF HEILBRONN'S TRIANGLES
Tao Jiang, Ming Li, Paul Vitanyi
Heilbronn's triangle problem asks for the least $\Delta$ such that $n$ points
lying in the unit disc necessarily contain a triangle of area at most $\Delta$.
Heilbronn initially conjectured $\Delta = O (1/n^2)$. As a result of concerted
mathematical effort it is currently known that there are positive constants $c$
and $C$ such that $c \log n/n^2 \leq \Delta \leq C /n^{8/7 -\epsilon}$ for
every constant $\epsilon > 0$. We resolve Heilbronn's problem in the expected
case: If we uniformly at random put $n$ points in the unit disc then (i) the
area of the smallest triangle has expectation $\Theta (1/n^3)$; and (ii) the
smallest triangle has area $\Theta (1/n^3)$ with probability almost one. Our
proof uses the incompressibility method based on Kolmogorov complexity.
paul.vitanyi@cwi.nl
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information provided by the author
click here.
1226. MODELING INTEREST RATE DYNAMICS: AN INFINITE-DIMENSIONAL APPROACH
Rama Cont
We present a family of models for the term structure of interest rates which
describe the interest rate curve as a stochastic process in a Hilbert space. We
start by decomposing the deformations of the term structure into the variations
of the short rate, the long rate and the fluctuations of the curve around its
average shape. This fluctuation is then described as a solution of a stochastic
evolution equation in an infinite dimensional space. In the case where
deformations are local in maturity, this equation reduces to a stochastic PDE,
of which we give the simplest example. We discuss the properties of the
solutions and show that they capture in a parsimonious manner the essential
features of yield curve dynamics: imperfect correlation between maturities,
mean reversion of interest rates and the structure of principal components of
term structure deformations. Finally, we discuss calibration issues and show
that the model parameters have a natural interpretation in terms of empirically
observed quantities.
cont@clipper.ens.fr
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information provided by the author
click here.
1227. TREES, NOT CUBES: HYPERCONTRACTIVITY, COSINESS, AND NOISE STABILITY
Oded Schramm and Boris Tsirelson
Noise sensitivity of functions on the leaves of a binary tree is studied, and
a hypercontractive inequality is obtained. We deduce that the spider walk is
not noise stable.
tsirel@math.tau.ac.il
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information provided by the author
click here.
1228. THE CRITICAL PARAMETER FOR THE HEAT EQUATION WITH A NOISE TERM TO BLOW
UP IN FINITE TIME
Carl Mueller
Consider the stochastic partial differential equation
u_t=u_{xx}+u^gamma dot{W},
where x in [0,J], dot{W}=dot{W}(t,x) is 2-parameter white noise, and we
assume that the initial function u(0,x) is nonnegative and not identically 0.
We impose Dirichlet boundary conditions on u. We say that u blows up in finite
time, with positive probability, if there is a finite random time T such that
P(\lim_{t->T}sup_x u(t,x)=infty)>0.
It was known that if gamma<3/2, then with probability 1, u does not blow up
in finite time. It was also known that there is a positive probability of
finite time blow-up for gamma sufficiently large.
In this paper, we show that if gamma>3/2, then there is a positive
probability that u blows up in finite time.
cmlr@troi.cc.rochester.edu
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information provided by the author
click here.
