Probability Abstracts 75
This document contains abstracts 2238-2306.
They have been mailed on June 21, 2003.
2238. RAPID MIXING IN MARKOV CHAINS
Ravi Kannan
A wide class of ``counting'' problems have been studied in Computer Science.
Three typical examples are the estimation of - (i) the permanent of an $n\times
n$ 0-1 matrix, (ii) the partition function of certain $n-$ particle Statistical
Mechanics systems and (iii) the volume of an $n-$ dimensional convex set. These
problems can be reduced to sampling from the steady state distribution of
implicitly defined Markov Chains with exponential (in $n$) number of states.
The focus of this talk is the proof that such Markov Chains converge to the
steady state fast (in time polynomial in $n$). A combinatorial quantity called
conductance is used for this purpose. There are other techniques as well which
we briefly outline. We then illustrate on the three examples and briefly
mention other examples.
Kannan@cs.yale.edu
2239. DERIVATION OF THE LEROUX SYSTEM AS THE HYDRODYNAMIC LIMIT OF A
TWO-COMPONENT LATTICE GAS
Jozsef Fritz, Balint Toth
The long time behavior of a couple of interacting asymmetric exclusion
processes of opposite velocities is investigated in one space dimension. We do
not allow two particles at the same site, and a collision effect (exchange)
takes place when particles of opposite velocities meet at neighboring sites.
There are two conserved quantities, and the model admits hyperbolic (Euler)
scaling; the hydrodynamic limit results in the classical Leroux system of
conservation laws, \emph{even beyond the appearence of shocks}. Actually, we
prove convergence to the set of entropy solutions, the question of uniqueness
is left open. To control rapid oscillations of Lax entropies via logarithmic
Sobolev inequality estimates, the symmetric part of the process is speeded up
in a suitable way, thus a slowly vanishing viscosity is obtained at the
macroscopic level. Following earlier work of the first author the stochastic
version of Tartar--Murat theory of compensated compactness is extended to
two-component stochastic models.
balint@math.bme.hu
2240. PHASE TRANSITIONS IN PHYLOGENY
Elchanan Mossel
We apply the theory of markov random fields on trees to derive a phase
transition in the number of samples needed in order to reconstruct phylogenies.
We consider the Cavender-Farris-Neyman model of evolution on trees, where all
the inner nodes have degree at least 3, and the net transition on each edge is
bounded by e. Motivated by a conjecture by M. Steel, we show that if 2 (1 - 2
e) (1 - 2e) > 1, then for balanced trees, the topology of the underlying tree,
having n leaves, can be reconstructed from O(log n) samples (characters) at the
leaves. On the other hand, we show that if 2 (1 - 2 e) (1 - 2 e) < 1, then
there exist topologies which require at least poly(n) samples for
reconstruction.
Our results are the first rigorous results to establish the role of phase
transitions for markov random fields on trees as studied in probability,
statistical physics and information theory to the study of phylogenies in
mathematical biology.
mossel@stat.berkeley.edu
2241. ANOTHER LOOK AT RANDOM INFINITE DIVISIBILITY
S. Satheesh
The drawbacks in the formulations of random infinite divisibility in Sandhya
(1991, 1996), Gnedenko and Korolev (1996), Klebanov and Rachev (1996), Bunge
(1996) and Kozubowski and Panorska (1996) are pointed out. For any given
Laplace transform, we conceive random infinite divisibility w.r.t to a class of
probability generating functions derived from the Laplace transform itself.
This formulation overcomes the said drawbacks, and the class of probability
generating functions is useful in transfer theorems for sums and maximums in
general. Generalizing the concepts of attraction (and partial attraction) in
the classical and the geometric summation setup to our formulation we show that
the domains of attraction (and partial attraction)in all these setups are same.
ssatheesh@sancharnet.in
2242. CONTINUITY OF THE ITO-MAP FOR HOELDER ROUGH PATHS WITH APPLICATIONS TO
THE SUPPORT THEOREM IN HOELDER NORM
Peter K. Friz
Lyons' Rough Path theory is currently formulated in p-variation topology. We
extend his main-result, the Universal Limit Theorem, to a stronger Hoelder
topology. Several approximations to Brownian Rough Paths are studied. As
application of our approach, we obtain the celebrated support theorem (in
Hoelder-topology) as immediate (!) corollary.
frizpete@cims.nyu.edu
2243. MEASURING AND HEDGING FINANCIAL RISKS IN DYNAMICAL WORLD
Nicole El Karoui
Financial markets have developed a lot of strategies to control risks induced
by market fluctuations. Mathematics has emerged as the leading discipline to
address fundamental questions in finance as asset pricing model and hedging
strategies. History began with the paradigm of zero-risk introduced by Black &
Scholes stating that any random amount to be paid in the future may be
replicated by a dynamical portfolio. In practice, the lack of information leads
to ill-posed problems when model calibrating. The real world is more complex
and new pricing and hedging methodologies have been necessary. This challenging
question has generated a deep and intensive academic research in the 20 last
years, based on super-replication (perfect or with respect to confidence level)
and optimization. In the interplay between theory and practice, Monte Carlo
methods have been revisited, new risk measures have been back-tested. These
typical examples give some insights on how may be used mathematics in financial
risk management.
elkaroui@cmapx.polytechnique.fr
2244. A NUMERAIRE-FREE AND ORIGINAL PROBABILITY BASED FRAMEWORK FOR FINANCIAL
MARKETS
Jia-An Yan
In this paper, we introduce a numeraire-free and original probability based
framework for financial markets. We reformulate or characterize fair markets,
the optional decomposition theorem, superhedging, attainable claims and
complete markets in terms of martingale deflators, present a recent result of
Kramkov and Schachermayer (1999, 2001) on portfolio optimization and give a
review of utility-based approach to contingent claim pricing in incomplete
markets.
jayan@mail.amt.ac.cn
2245. CHAINS WITH COMPLETE CONNECTIONS AND ONE-DIMENSIONAL GIBBS MEASURES
Roberto Fernandez, Gregory Maillard
We discuss the relationship between discrete-time processes (chains) and
one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin)
systems, possibly with a grammar (exclusion rule). We establish conditions for
a stochastic process to define a Gibbs measure and vice versa. Our conditions
generalize well known equivalence results between ergodic Markov chains and
fields, as well as the known Gibbsian character of processes with exponential
continuity rate. Our arguments are purely probabilistic; they are based on the
study of regular systems of conditional probabilities (specifications).
Furthermore, we discuss the equivalence of uniqueness criteria for chains and
fields and we establish bounds for the continuity rates of the respective
systems of finite-volume conditional probabilities. As an auxiliary result we
prove a (re)construction theorem for specifications starting from single-site
conditioning, which applies in a more general setting (general spin space,
specifications not necessarily Gibbsian).
roberto.fernandez@univ-rouen.fr
2246. CHAINS WITH COMPLETE CONNECTIONS: GENERAL THEORY, UNIQUENESS, LOSS OF
MEMORY AND MIXING PROPERTIES
Roberto Fernandez, Gregory Maillard
We introduce an statistical mechanical formalism for the study of
discrete-time stochastic processes with which we prove: (i) General properties
of extremal chains, including triviality on the tail $\sigma$-algebra,
short-range correlations, realization via infinite-volume limits and
ergodicity. (ii) Two new sufficient conditions for the uniqueness of the
consistent chain. The first one is a transcription of a criterion due to
Georgii for one-dimensional Gibbs measures, and the second one corresponds to
Dobrushin criterion in statistical mechanics. (iii) Results on loss of memory
and mixing properties for chains in the Dobrushin regime. These results are
complementary of those existing in the literature, and generalize the Markovian
results based on the Dobrushin ergodic coefficient.
roberto.fernandez@univ-rouen.fr
2247. ASPECTS OF RANDOMIZATION IN INFINITELY DIVISIBLE AND MAX-INFINITELY
DIVISIBLE LAWS
S. Satheesh
Continuing the study reported in Satheesh (2001),(arXiv:math.PR/0304499 dated
01May2003) here we study certain aspects of randomization in infinitely
divisible (ID) and max-infinitely divisible (MID) laws. They generalize ID and
MID laws. In particular we study mixtures of ID & MID laws, its relation to
random sums & random maximums, corresponding stationary processes & extremal
processes and some of their properties. It is shown that mixtures of ID laws
and mixtures of MID laws appear as limits of random sums and random maximums
respectively. We identify a class of probability generating functions for N,
the random sample size. A method to construct class-L laws is given.
ssatheesh@sancharnet.in
2248. RANDOM COMBINATORIAL STRUCTURES: THE CONVERGENT CASE
A.D.Barbour, B.Granovsky
This paper studies the distribution of the component spectrum of
combinatorial structures such as uniform random forests, in which the classical
generating function for the numbers of (irreducible) elements of the different
sizes converges at the radius of convergence; here, this property is expressed
in terms of the expectations of {\it independent} random variables $Z_j$,
$j\ge1$, whose joint distribution, conditional on the event that $\sum_{j=1}^n
jZ_j = n$, gives the distribution of the component spectrum for a random
structure of size $n$. For a large class of such structures, we show that the
component spectrum is asymptotically composed of $Z_j$ components of size $j$,
$j\ge1$, with the remaining part, of size $n-\sum_{j\ge1} Z_j$, being made up
of a single, giant component.
mar18aa@techunix.technion.ac.il
2249. MARKOV CHAIN SAMPLING FOR NON-LINEAR STATE SPACE MODELS USING EMBEDDED
HIDDEN MARKOV MODELS
Radford M. Neal
I describe a new Markov chain method for sampling from the distribution of
the state sequences in a non-linear state space model, given the observation
sequence. This method updates all states in the sequence simultaneously using
an embedded Hidden Markov model (HMM). An update begins with the creation of a
``pool'' of K states at each time, by applying some Markov chain update to the
current state. These pools define an embedded HMM whose states are indexes
within this pool. Using the forward-backward dynamic programming algorithm, we
can then efficiently choose a state sequence at random with the appropriate
probabilities from the exponentially large number of state sequences that pass
through states in these pools. I show empirically that when states at nearby
times are strongly dependent, embedded HMM sampling can perform better than
Metropolis methods that update one state at a time.
radford@utstat.toronto.edu
2250. INFINITE DIVISIBILITY AND MAX-INFINITE DIVISIBILITY WITH RANDOM SAMPLE
SIZE
S. Satheesh and E. Sandhya (CUSAT, India,
Continuing the study reported in Satheesh (2001),(arXiv:math.PR/0304499 dated
01 May 2003) and Satheesh (2002)(arXiv:math.PR/0305030 dated 02May 2003), here
we study generalizations of infinitely divisible (ID) and max-infinitely
divisible (MID) laws. We show that these generalizations appear as limits of
random sums and random maximums respectively. For the random sample size N, we
identify a class of probability generating functions. Necessary and sufficient
conditions that implies the convergence to an ID (MID) law by the convergence
to these generalizations and vise versa are given. The results generalize those
on ID and random ID laws studied previously in Satheesh (2001) and those on
geometric MID laws studies in Rachev and Resnick (1991). We discuss attraction
and partial attraction in this generalization of ID and MID laws.
ssatheesh@sancharnet.in
2251. POISSON CALCULUS FOR SPATIAL NEUTRAL TO THE RIGHT PROCESSES
Lancelot F. James
In this paper we consider classes of nonparametric priors on spaces of
distribution functions and cumulative hazards that are based on extensions of
the neutral to the right concept. In particular we extend the definition of NTR
processes from the real line to classes of distributions on general spaces.
Representations of the posterior distributions are give using a different type
of calculus than traditionally used in the Bayesian literature. The techniques
are applied to progeressively more complex models. Refinements are then given
which describes the underlying properties of spatial NTR models analagous to
those developed for the Dirihclet process. The analysis yields accessible
moment formulae and characterizations of the the posterior distribution and
relavant marginal distributions. In the homogeneous case this work turns out to
be connected to and overlap with recent work on regenerative compositions
defined by a suitable discretisation of subordinators. The results also have
connections to other related work on exponential functionals of subordinators.
In addition we develop results for spatial NTR mixture models and identify a
class of species sampling models derived from spatial NTR processes.
lancelot@ust.hk
2252. GLAUBER DYNAMICS ON THE CYCLE IS MONOTONE
Serban Nacu
We study heat-bath Glauber dynamics for the ferromagnetic Ising model on a
finite cycle (a graph where every vertex has degree two). We prove that the
relaxation time $\tau_2$ is an increasing function of any of the couplings
$J_{xy}$. We also prove some further inequalities, and obtain exact asymptotics
for $\tau_2$ at low temperatures.
serban@stat.berkeley.edu
2253. A NOTE ON MAXIMUM AND MINIMUM STABILITY OF CERTAIN DISTRIBUTIONS
S. Satheesh and N. U. Nair
In the context of stability of the extremes of a random variable X with
respect to a positive integer valued random variable N we discuss the cases (i)
X is exponential (ii) non-geometric laws for N (iii) identifying N for the
stability of a given X and (iv) extending the notion to a discrete random
variable X.
ssatheesh@sancharnet.in
2254. ON THE STABILITY OF GEOMETRIC EXTREMES
S. Satheesh and N. U. Nair
Possible reasons for the uniqueness of the positive geometric law in the
context of stability of random extremes are explored here culminating in a
conjecture characterizing the geometric law. Our reasoning comes closer in
justifying the geometric law in similar contexts discussed in Arnold et al.
(1986) and Marshall & Olkin (1997) and supplement their arguments.
ssatheesh@sancharnet.in
2255. SMALL DEVIATIONS FOR FRACTIONAL STABLE PROCESSES
Mikhail. A. Lifshits and Thomas Simon
Let R be a symmetric a-stable Riemann-Liouville process with Hurst parameter
H > 0. Consider ||.|| a translation invariant, b-self-similar, and
p-pseudo-additive functional semi-norm. We show that if H > (b + 1/p) and c =
(H - b - 1/p), then x log P [ log ||R|| < c log x ] -> - k < 0, when x -> 0,
with k finite in the Gaussian case a = 2. If a < 2, we prove that k is finite
when R is continuous and H > (b + 1/p + 1/a). We also show that under the above
assumptions, x log P [ log ||X|| < c log x ] -> - k < 0 when x -> 0, where k is
finite and X is the linear a-stable fractional motion with Hurst parameter 0 <
H < 1 (if a = 2, then X is the classical fractional Brownian motion). These
general results recover many cases previously studied in the literature, and
also prove the existence of new small deviation constants, both in Gaussian and
Non-Gaussian frameworks.
simon@maths.univ-evry.fr
2256. TOWARDS DEAD TIME INCLUSION IN NEURONAL MODELING
A. Buonocore, G. Esposito, V. Giorno, C. Valerio
A mathematical description of the refractoriness period in neuronal diffusion
modeling is given and its moments are explicitly obtained in a form that is
suitable for quantitative evaluations. Then, for the Wiener, Ornstein-Uhlenbeck
and Feller neuronal models, an analysis of the features exhibited by the mean
and variance of the first passage time and of refractoriness period is
performed.
luigi.ricciardi@unina.it
2257. ROBUST ESTIMATORS UNDER THE IMPRECISE DIRICHLET MODEL
Marcus Hutter
Walley's Imprecise Dirichlet Model (IDM) for categorical data overcomes
several fundamental problems which other approaches to uncertainty suffer from.
Yet, to be useful in practice, one needs efficient ways for computing the
imprecise=robust sets or intervals. The main objective of this work is to
derive exact, conservative, and approximate, robust and credible interval
estimates under the IDM for a large class of statistical estimators, including
the entropy and mutual information.
marcus@idsia.ch
2258. A SUPPLEMENT TO THE BOSE-DASGUPTA-RUBIN (2002) REVIEW OF INFINITELY
DIVISIBLE LAWS AND PROCESSES
S Satheesh
This paper proves that if a discrete distribution is infinitely divisible
(ID) then it has a mass at the origin, which also implies why certain ID
discrete laws do not have gaps in its support. We argue that discrete laws also
can be stable and such laws do have domain of attraction. Then we give certain
recent developments and references not reported in the Bose Dasgupta Rubin
(2002) review in Sankhya, and some examples in the topics; infinitely
divisibility and stability of discrete laws, random infinite divisibility,
operator stable laws, class-L laws, Goldie-Steutel result, max-infinite
divisibility and stability, simulation, alternate stable laws, applications and
free probability theory.
ssatheesh@sancharnet.in
2259. ON THE MAXIMUM SATISFIABILITY OF RANDOM FORMULAS
Dimitris Achlioptas, Assaf Naor and Yuval Peres
Maximum satisfiability is a canonical NP-hard optimization problem that
appears empirically hard for random instances. Let us say that a Conjunctive
normal form (CNF) formula consisting of $k$-clauses is $p$-satisfiable if there
exists a truth assignment satisfying $1-2^{-k}+p 2^{-k}$ of all clauses
(observe that every $k$-CNF is 0-satisfiable). Also, let $F_k(n,m)$ denote a
random $k$-CNF on $n$ variables formed by selecting uniformly and independently
$m$ out of all possible $k$-clauses. It is easy to prove that for every $k>1$
and every $p$ in $(0,1]$, there is $R_k(p)$ such that if $r >R_k(p)$, then the
probability that $F_k(n,rn)$ is $p$-satisfiable tends to 0 as $n$ tends to
infinity. We prove that there exists a sequence $\delta_k \to 0$ such that if
$r <(1-\delta_k) R_k(p)$ then the probability that $F_k(n,rn)$is
$p$-satisfiable tends to 1 as $n$ tends to infinity. The sequence $\delta_k$
tends to 0 exponentially fast in $k$.
peres@stat.berkeley.edu
2260. OPTIMAL NONLINEAR PREDICTION OF RANDOM FIELDS ON NETWORKS
Cosma Rohilla Shalizi
It is increasingly common to encounter time-varying random fields on networks
(metabolic networks, sensor arrays, distributed computing, etc.). This paper
considers the problem of optimal, nonlinear prediction of these fields, showing
from an information-theoretic perspective that it is formally identical to the
problem of finding minimal local sufficient statistics. I derive general
properties of these statistics, show that they can be composed into global
predictors, and explore their recursive estimation properties. For the special
case of discrete-valued fields, I describe a convergent algorithm to identify
the local predictors from empirical data, with minimal prior information about
the field, and no distributional assumptions.
cshalizi@umich.edu
2261. BROWNIAN BEADS
Balint Virag
We show that the past and future of half-plane Brownian motion at certain
cutpoints are independent of each other after a conformal transformation. Like
in Ito's excursion theory, the pieces between cutpoints form a Poisson process
with respect to a local time. The size of the path as a function of this local
time is a stable subordinator whose index is given by the exponent of the
probability that a stretch of the path has no cutpoint. The index is computed
and equals 1/2.
balint@math.mit.edu
2262. LAW OF LARGE NUMBERS FOR THE ASYMMETRIC SIMPLE EXCLUSION PROCESS
E. Andjel, P. A. Ferrari, A. Siqueira
We consider simple exclusion processes on Z for which the underlying random
walk has a finite first moment and a non-zero mean and whose initial
distributions are product measures with different densities to the left and to
the right of the origin. We prove a strong law of large numbers for the number
of particles present at time t in an interval growing linearly with t.
pablo@ime.usp.br
2263. ON THE UNIQUENESS OF THE BRANCHING PARAMETER FOR A RANDOM CASCADE
MEASURE
G. Molchan
An independent random cascade measure is specified by a random generator, a
vector of dimension c with non-negative components. The dimension c is called
the branching cascade parameter. It is shown under certain restrictions that,
if this measure has two generators with a.s. positive components, and the ratio
ln c_1/ln c_2 for their branching parameters is an irrational number, then this
measure is a Lebesgue measure. In other words, when c is a power of an integer
number p and the p is minimal for c, then a cascade measure that has the
property of intermittency specifies p uniquely.
ansobol@obs-nice.fr
2264. ON THE ASYMPTOTIC BEHAVIOR OF FIRST PASSAGE TIME DENSITIES FOR
STATIONARY GAUSSIAN PROCESSES
E. Di Nardo, A.G. Nobile, E. Pirozzi, L.M. Ricciardi
Making use of a Rice-like series expansion, for a class of stationary
Gaussian processes the asymptotic behavior of the first passage time
probability density function through certain time-varying boundaries, including
periodic boundaries, is determined. Sufficient conditions are then given such
that the density asymptotically exhibits an exponential behavior when the
boundary is either asymptotically constant or asymptotically periodic.
luigi.ricciardi@unina.it
2265. SPHERES AND MINIMA
Igor Rivin
We write down a one-dimensional integral formula and compute large-n
asymptotics for the expectation of the absolute value of the smallest component
of a unit vector in n-dimensional Euclidean space.
The method is general, and allows to write the mean over the sphere of an
homogeneous function in terms of an expectation of a function of independent,
identically distributed Gaussians. We also write down an asymptotic formula for
the minimum of a large number of identical independent positive random
variables.
irivin@math.princeton.edu
2266. STATE TAMENESS: A NEW APPROACH FOR CREDIT CONSTRAINS
Jaime A. Londo\~no
We propose a new definition for tameness within the model of security prices
as It\^o processes that is risk-aware. We give a new definition for arbitrage
and characterize it. We then prove a theorem that can be seen as an extension
of the second fundamental theorem of asset pricing, and a theorem for valuation
of contingent claims of the American type. The valuation of European contingent
claims and American contingent claims that we obtain does not require the full
range of the volatility matrix. The formulas obtain to price American
contingent claims are closer in spirit to a computational approach.
jalondon@sigma.eafit.edu.co
2267. THE POISSON-DIRICHLET LAW IS THE UNIQUE INVARIANT DISTRIBUTION FOR
UNIFORM SPLIT-MERGE TRANSFORMATIONS
Persi Diaconis, Eddy Mayer-Wolf, Ofer Zeitouni, Martin Zerner
We consider a Markov chain on the space of (countable) partitions of the
interval [0,1], obtained first by size biased sampling twice (allowing
repetitions) and then merging the parts (if the sampled parts are distinct) or
splitting the part uniformly (if the same part was sampled twice). We prove a
conjecture of Vershik stating that the Poisson-Dirichlet law with parameter
theta=1 is the unique invariant distribution for this Markov chain.
Our proof uses a combination of probabilistic, combinatoric, and
representation-theoretic arguments.
zeitouni@ee.technion.ac.il
2268. EVOLVING SETS, MIXING AND HEAT KERNEL BOUNDS
Ben Morris and Yuval Peres
We show that a new probabilistic technique, recently introduced by the first
author, yields the sharpest bounds obtained to date on mixing times of Markov
chains in terms of isoperimetric properties of the state space (also known as
conductance bounds or Cheeger inequalities). We prove that the bounds for
mixing time in total variation obtained by Lovasz and Kannan, can be refined to
apply to the maximum relative deviation $|p^n(x,y)/\pi(y)-1|$ of the
distribution at time $n$ from the stationary distribution $\pi$. We then extend
our results to Markov chains on infinite state spaces and to continuous-time
chains. Our approach yields a direct link between isoperimetric inequalities
and heat kernel bounds; previously, this link rested on analytic estimates
known as Nash inequalities.
peres@stat.berkeley.edu
2269. HYDRODYNAMIC LIMIT OF ASYMMETRIC EXCLUSION PROCESSES UNDER DIFFUSIVE
SCALING IN $D\GE 3$
C. Landim, R. M. Sued and G. Valle
We consider the asymmetric exclusion process. We start from a profile which
is constant along the drift direction and prove that the density profile, under
a diffusive rescaling of time, converges to the solution of a parabolic
equation.
landim@impa.br
2270. RECONSTRUCTION THRESHOLDS ON REGULAR TREES
James B. Martin
We consider a branching random walk with binary state space and index set
$T^k$, the infinite rooted tree in which each node has k children (also known
as the model of "broadcasting on a tree"). The root of the tree takes a random
value 0 or 1, and then each node passes a value independently to each of its
children according to a 2x2 transition matrix P. We say that "reconstruction is
possible" if the values at the d'th level of the tree contain non-vanishing
information about the value at the root as $d\to\infty$. Adapting a method of
Brightwell and Winkler, we obtain new conditions under which reconstruction is
impossible, both in the general case and in the special case $p_{11}=0$. The
latter case is closely related to the "hard-core model" from statistical
physics; a corollary of our results is that, for the hard-core model on the
(k+1)-regular tree with activity $\lambda=1$, the unique simple invariant Gibbs
measure is extremal in the set of Gibbs measures, for any k.
martin@liafa.jussieu.fr
2271. TOWARDS THE MODELING OF NEURONAL FIRING BY GAUSSIAN PROCESSES
E. Di Nardo, A.G. Nobile, E. Pirozzi and L.M. Ricciardi
This paper focuses on the outline of some computational methods for the
approximate solution of the integral equations for the neuronal firing
probability density and an algorithm for the generation of sample-paths in
order to construct histograms estimating the firing densities. Our results
originate from the study of non-Markov stationary Gaussian neuronal models with
the aim to determine the neuron's firing probability density function. A
parallel algorithm has been implemented in order to simulate large numbers of
sample paths of Gaussian processes characterized by damped oscillatory
covariances in the presence of time dependent boundaries. The analysis based on
the simulation procedure provides an alternative research tool when closed-form
results or analytic evaluation of the neuronal firing densities are not
available.
luigi.ricciardi@unina.it
2272. ON THE SPEED OF A PLANAR RANDOM WALK AVOIDING ITS PAST CONVEX HULL
Martin P. W. Zerner
We consider a random walk in the plane which takes steps uniformly
distributed on the unit circle centered around the walker's current position
but avoids the convex hull of its past positions. This model has been
introduced by Angel, Benjamini and Virag. We show a large deviation estimate
for the distance of the walker from the origin, which implies that the walker
has positive lim inf speed.
zerner@math.stanford.edu
2273. SMALL DEVIATIONS IN P-VARIATION NORM FOR MULTIDIMENSIONAL LEVY PROCESSES
T. Simon
Let Z be an Rd-valued Levy process with strong finite p-variation for some
p<2. We prove that the ''decompensated'' process Y obtained from Z by
annihilating its generalized drift has a small deviations property in
p-variation. This property means that the null function belongs to the support
of the law of Y with respect to the p-variation distance. Thanks to the
continuity results of T. J. Lyons/D. R. E. Williams, this allows us to prove a
support theorem with respect to the p-Skorohod distance for canonical SDE
driven by Z without any assumption on Z, improving the results of H. Kunita. We
also give a criterion ensuring the small deviation property for Z itself,
noticing that the characterization under the uniform distance, which we had
obtained in a previous paper, no more holds under the p-variation distance.
simon@maths.univ-evry.fr
2274. SMALL DEVIATIONS IN P-VARIATION FOR STABLE PROCESSES
T. Simon
Let $\{Z_t, t\geq 0\}$ be a strictly stable process on $\R$ with index
$\alpha\in (0,2]$. We prove that for every $p > \alpha$, there exists $\gamma =
\gamma (\alpha, p)$ and $\k = \k (\alpha, p)\in (0, +\infty)$ such that
$$\lim_{\ee\downarrow 0}\ee^{\gamma}\log\pb\lcr ||Z||_{p}\leq \ee \rcr = -
\k,$$ where $||Z||_{p}$ stands for the strong $p$-variation of $Z$ on $[0,1]$.
The critical exponent $\gamma (\alpha, p)$ takes a different shape according as
$|Z|$ is a subordinator and $p >1$, or not. The small ball constant $\k
(\alpha, p)$ is explicitly computed when $p \leq 1$, and a lower bound on $\k
(\alpha, p)$ is easily obtained in the general case. In the symmetric case and
when $p > 2$, we can also give an upper bound on $\k (\alpha, p)$ in terms of
the Brownian small ball constant under the $(1/p)$-H\"older semi-norm. Along
the way, we remark that the positive random variable $||Z||^p_{p}$ is not
necessarily stable when $p > 1$, which gives a negative answer to an old
question of P.~E.~Greenwood.
simon@maths.univ-evry.fr
2275. QUENCHED LARGE DEVIATIONS FOR ONE DIMENSIONAL NONLINEAR FILTERING
E. Pardoux, O. Zeitouni
Consider the standard, one dimensional, nonlinear filtering problem for a
diffusion processe $\Xi_t$ observed in small additive white noise. Denote by
$q^\eps_1(\cdot)$ the density of the law of $\Xi_1$ conditioned on
$\sigma(Y_t^\eps: 0\leq t\leq 1)$. We provide "quenched" large deviation
estimates for the random family of measures $q^\eps_1(x)dx$: there exists a
continuous, explicit mapping $\bar \JJ : \reels^2\to\reels$ such that for
almost all $B_\cdot,V_\cdot$, $\bar \JJ(\cdot,X_1)$ is a good rate function and
for any measurable $G\subset \reels$, $$-\inf_{x\in G^o} \bar \JJ(x,X_1) \leq
\liminfe \eps \log \int_G q_1^\eps(x) dx \leq \limsupe \eps \log \int_G
q_1^\eps(x) dx \leq -\inf_{x\in \bar G} \bar \JJ(x,X_1) .$$
zeitouni@ee.technion.ac.il
2276. A LARGE-DEVIATION THEOREM FOR TREE-INDEXED MARKOV CHAINS
Amir Dembo, Peter Morters, Scott Sheffield
Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical
subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed
by taking all descendants of a single vertex. We prove a large deviation
principle in $n$, with explicit rate function, for the empirical subtree
measures of multitype Galton-Watson trees conditioned to have exactly $n$
vertices. In the process, we extend the notions of shift-invariance and
specific relative entropy--as typically understood for Markov fields on
deterministic graphs such as $\mathbb Z^d$--to Markov fields on random trees.
We also develop single-generation empirical measure large deviation principles
for a more general class of random trees including trees sampled uniformly from
the set of all trees with $n$ vertices.
scott@math.stanford.edu
2277. TRANSFER THEOREMS AND ASYMPTOTIC DISTRIBUTIONAL RESULTS FOR M-ARY SEARCH
TREES
James Allen Fill and Nevin Kapur
We derive asymptotics of moments and identify limiting distributions, under
the random permutation model on m-ary search trees, for functionals that
satisfy recurrence relations of a simple additive form. Many important
functionals including the space requirement, internal path length, and the
so-called shape functional fall under this framework. The approach is based on
establishing transfer theorems that link the order of growth of the input into
a particular (deterministic) recurrence to the order of growth of the output.
The transfer theorems are used in conjunction with the method of moments to
establish limit laws. It is shown that (i) for small toll sequences $(t_n)$
[roughly, $t_n =O(n^{1 / 2})$] we have asymptotic normality if $m \leq 26$ and
typically periodic behavior if $m \geq 27$; (ii) for moderate toll sequences
[roughly, $t_n = \omega(n^{1 / 2})$ but $t_n = o(n)$] we have convergence to
non-normal distributions if $m \leq m_0$ (where $m_0 \geq 26$) and typically
periodic behavior if $m \geq m_0 + 1$; and (iii) for large toll sequences
[roughly, $t_n = \omega(n)$] we have convergence to non-normal distributions
for all values of m.
nevin@jhu.edu
2278. CRITICAL PERCOLATION IN ANNULI AND $SLE_6$
Julien Dubedat
Building on the identification of the scaling limit of the critical
percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE
characterization for the crossing probability of an annulus.
dubedat@clipper.ens.fr
2279. A PROBABILISTIC REPRESENTATION FOR THE VORTICITY OF A 3D VISCOUS FLUID
AND FOR GENERAL SYSTEMS OF PARABOLIC EQUATIONS
B. Busnello, F. Flandoli, M. Romito
A probabilistic representation formula for general systems of linear
parabolic equations, coupled only through the zero-order term, is given. On
this basis, an implicit probabilistic representation for the vorticity in a 3D
viscous fluid (described by the Navier-Stokes equations) is carefully analysed,
and a theorem of local existence and uniqueness is proved.
romito@math.unifi.it
2280. VITESSE DANS LE THEOREME LIMITE CENTRAL POUR CERTAINS PROCESSUS
STATIONNAIRES FORTEMENT DECORRELES
Stephane Le Borgne and Francoise Pene
We prove a central limit theorem with speed $n^{-1/2}$ for stationary
processes satisfying a strong decorrelation hypothesis. The proof is a
modification of the proof of a theorem of Rio. It is elementary but quite long
and technical.
stephane.leborgne@univ-rennes1.fr
2281. ECOLOGICAL SUCCESSION MODEL
Nicolas Lanchier
We introduce a new interacting particle system intended to model an example
of ecological succession involving two species: the bracken and the european
beech. The objective is to exhibit phase transitions by proving that there
exist three possible evolutions of the system to distinct ecological balances.
More precisely, the whole population may die out, the european beech may
conquer the bracken, and coexistence of both species may occur.
Nicolas.Lanchier@univ-rouen.fr
2282. CHORDAL LOEWNER FAMILIES AND UNIVALENT CAUCHY TRANSFORMS
Robert O. Bauer
We study chordal Loewner families in the upper half-plane and show that they
have a parametric representation. We show one, that to every chordal Loewner
family there corresponds a unique measurable family of probability measures on
the real line, and two, that to every measurable family of probability measures
on the real line there corresponds a unique chordal Loewner family. In both
cases the correspondence is being given by solving the chordal Loewner
equation. We use this to show that any probability measure on the real line
with finite variance and mean zero has univalent Cauchy transform if and only
if it belongs to some chordal Loewner family. If the probability measure has
compact support we give two further necessary and sufficient conditions for the
univalence of the Cauchy transform, the first in terms of the transfinite
diameter of the complement of the image domain of the reciprocal Cauchy
transform, and the second in terms of moment inequalities corresponding to the
Grunsky inequalities.
rbauer@math.uiuc.edu
2283. THE ARCTIC CIRCLE BOUNDARY AND THE AIRY PROCESS
Kurt Johansson
We prove that the, appropriately rescaled, boundary of the north polar region
in the Aztec diamond converges to the Airy process. The proof uses certain
determinantal point processes given by the extended Krawtchouk kernel. We also
prove a version of Propp's conjecture concerning the structure of the tiling at
the center of the Aztec diamond.
kurtj@math.kth.se
2284. LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES
James Allen Fill and Nevin Kapur
Additive tree functionals represent the cost of many divide-and-conquer
algorithms. We derive the limiting distribution of the additive functionals
induced by toll functions of the form (a) n^\alpha when \alpha > 0 and (b) log
n (the so-called shape functional) on uniformly distributed binary search
trees, sometimes called Catalan trees. The Gaussian law obtained in the latter
case complements the central limit theorem for the shape functional under the
random permutation model. Our results give rise to an apparently new family of
distributions containing the Airy distribution (\alpha = 1) and the normal
distribution [case (b), and case (a) as $\alpha \downarrow 0$]. The main
theoretical tools employed are recent results relating asymptotics of the
generating functions of sequences to those of their Hadamard product, and the
method of moments.
nevin@jhu.edu
2285. RATES OF CONVERGENCE FOR CONSTRAINED DECONVOLUTION PROBLEM
Denis Belomestny
Let $X$ and $Y$ be two independent identically distributed random variables
with density $p(x)$ and $Z=\alpha X+\beta Y$ for some constants $\alpha>0$ and
$\beta>0$. We consider the problem of estimating $p(x)$ by means of the samples
from the distribution of $Z$. Non-parametric estimator based on the sync kernel
is constructed and asymptotic behaviour of the corresponding mean integrated
square error is investigated.
db@izks.uni-bonn.de
2286. PERTURBED MARKOV CHAINS
Eilon Solan, Nicolas Vieille
We study irreducible time-homogenous Markov chains with finite state space in
discrete time. We obtain results on the sensitivity of the stationary
distribution and other statistical quantities with respect to perturbations of
the transition matrix. We define a new closeness relation between transition
matrices, and use graph-theoretic techniques, in contrast with the matrix
analysis techniques previously used.
eilons@post.tau.ac.il
2287. GLAUBER DYNAMICS OF CONTINUOUS PARTICLE SYSTEMS
Yu. Kondratiev, E. Lytvynov
This paper is devoted to the construction and study of an equilibrium
Glauber-type dynamics of infinite continuous particle systems. This dynamics is
a special case of a spatial birth and death process. On the space $\Gamma$ of
all locally finite subsets (configurations) in ${\Bbb R}^d$, we fix a Gibbs
measure $\mu$ corresponding to a general pair potential $\phi$ and activity
$z>0$. We consider a Dirichlet form $ \cal E$ on $L^2(\Gamma,\mu)$ which
corresponds to the generator $H$ of the Glauber dynamics. We prove the
existence of a Markov process $\bf M$ on $\Gamma$ that is properly associated
with $\cal E$. In the case of a positive potential $\phi$ which satisfies
$\delta{:=}\int_{{\Bbb R}^d}(1-e^{-\phi(x)}) z dx<1$, we also prove that the
generator $H$ has a spectral gap $\ge1-\delta$. Furthermore, for any pure Gibbs
state $\mu$, we derive a Poincar\'e inequality. The results about the spectral
gap and the Poincar\'e inequality are a generalization and a refinement of a
recent result by L. Bertini, N. Cancrini, and F. Cesi.
lytvynov@wiener.iam.uni-bonn.de
2288. A STOCHASTIC HEISENBERG INEQUALITY
C. Mueller, A. Stan
An analogue of the Fourier transform will be introduced for all square
integrable continuous martingale processes whose quadratic variation is
deterministic. Using this transform we will formulate and prove a stochastic
Heisenberg inequality.
cmlr@math.rochester.edu
2289. PROBABILISTIC EVENTS AND PHYSICAL REALITY: A COMPLETE ALGEBRA OF PROBABILITY
Paolo Rocchi, Leonida Gianfagna
This contribution derives from a rather extensive
study on the foundations of probability.
We start by discussing critically the two main models of the random event in Probability Theory
and cast light over a number of incongruities.
We conclude that the argument of probability is the critical knot of the probability foundations
and put forward the structure of levels for the partially determinate event.
The structural model enables us to define the probability and to attune its subjective
and objective interpretations.
PAOLOROCCHI@it.ibm.com Leonida.Gianfagna@it.ibm.com
- To see a preprint or other
information provided by the author
click here.
2290. SERIES REPRESENTATION AND SIMULATION OF MULTIFRACTIONAL
LEVY MOTIONS
C\'eline Lacaux
This paper introduces a method of generating Real Harmonizable
Multifractional L\'evy Motions, in short RHMLMs. The simulation
of these fields is closely related to that of infinitely indivisible laws
or L\'evy processes. In the case where the control measure of the
RHMLM is finite, one uses generalized shot noises series. An estimation
of the error is also given. Otherwise the RHMLM $X_h$ is split into
two independent RHMLMs $X_{\veps,1}$ and $X_{\veps,2}$. More
precisely, $X_{\veps,2}$ is a RHMLM whose control measure is finite.
Then it can be rewritten as a generalized shot noise series. The asymptotic
behaviour of $X_{\veps,1}$ as $ \veps \to 0_+$ is further elaborated.
Sufficient conditions to approximate $X_{\veps,1}$ by a Multifractional
Brownian Motion are given. The error rate in term of Berry-Esseen bounds
is then discussed. At last examples of simulation are given.
Celine.Lacaux@math.ups-tlse.fr
- To see a preprint or other
information provided by the author
click here.
2291. HARNACK INEQUALITIES FOR NON-LOCAL OPERATORS OF VARIABLE ORDER
Richard F. Bass and M. Kassmann
We consider harmonic functions with respect to the operator
L u(x)=\int [u(x+h)-u(x)-1_{(|h|\leq 1)} h\cdot \nabla u(x)] n(x,h) dh.
Under suitable conditions on n(x,h) we establish a Harnack
inequality for functions that are nonnegative and harmonic in a domain.
The operator L is allowed to be anisotropic and of variable order.
bass@math.uconn.edu kassmann@math.uconn.edu
- To see a preprint or other
information provided by the author
click here.
2292. DERRIDA'S GENERALIZED RANDOM ENERGY MODELS.
1. POISSON CASCADES AND EXTREMAL PROCESSES
Anton Bovier and Irina Kurkova
This is the first of a series of four
papers in which we present a full rigorous analysis of a class of
spin glass models introduces by Derrida under the name of
Generalized Random Energy Models (GREM). They are based on
Gaussian random processes on the hypercube $\{-1,1\}^N$ with a
hierarchical correlation structure. In this first paper we prove
some result concerning the convergence of the extremal process to
Poisson cascade processes which will form the basis of the
analysis of the Gibbs measures to be presented in the sequel.
bovier@wias-berlin.de, kourkova@ccr.jussieu.fr
- To see a preprint or other
information provided by the author
click here.
2293. DERRIDA'S GENERALIZED RANDOM ENERGY MODELS.
2. GIBBS MEASURES AND PROBABILITY CASCADES.
Anton Bovier and Irina Kurkova
This is the second in a series of four
papers in which we present a full rigorous analysis of Derrida's
Generalized Random Energy Models (GREM). In this paper we still
consider only models with finitely many levels. In this context we
present two ways to prove the convergence of the Gibbs measures
(in a suitable representation). The first is direct and shows
convergence to the probability cascades introduced by Ruelle. The
second approach is indirect and use the so-called Ghirlanda-Guerra
identities, that allow to control the distribution of the Gibbs
measures via recursive identities for their moments.
bovier@wias-berlin.de, kourkova@ccr.jussieu.fr
- To see a preprint or other
information provided by the author
click here.
2294. DERRIDA'S GNERALIZED RANDOM ENERGY MODELS.
3. MODELS WITH CONTINUOUS HIERARCHIES.
Anton Bovier and Irina Kurkova
This is the third of a series of four
papers in which we present a rigorous analysis of Derrida's
Generalized
Random Energy Models (GREM). Here we study the general case
of models with a ``continuum of hierarchies''. We prove the
convergence of the free energy and give
explicite
formulas for the free energy and the two-replica distribution function.
Then we introduce the empirical distance distribution to
describe effectively the Gibbs measures.
We shhow that its limit is uniquely determined via the
Ghirlanda-Guerra identities up to the mean of the replica distribution
function.
Finally, we show that suitable
discretizations of the limiting random measure can be described by
the same objects in suitably constructed GREM's.
bovier@wias-berlin.de, kourkova@ccr.jussieu.fr
- To see a preprint or other
information provided by the author
click here.
2295. DERRIDA'S GENERALIZED RANDOM ENERGY MODELS.
4. CONTINUOUS STATE BRANCHING AND COALESCENTS.
Anton Bovier and Irina Kurkova
In this paper we conclude our analysis of
Derrida's Generalized
Random Energy Models (GREM) by identifying the thermodynamic limit
with a one-parameter family of probability measures related
to a continuous state branching process introduced by Neveu.
Using a construction introduced by Bertoin and Le Gall in terms
of a coherent family of subordinators related to Neveu's branching process,
we show how the Gibbs geometry of
the limiting Gibbs measure is given in terms
of the genealogy of this process
via a deterministic time-change. This
construction is fully universal in that all different models (characterized
by the covariance of the underlying Gaussian process) differ only through that
time change, which in turn is expressed in terms of Parisi's overlap
distribution. The proof uses strongly the Ghirlanda-Guerra identities that
impose the structure of Neveu's process as the only possible asymptotic
random mechanism.
bovier@wias-berlin.de, kourkova@ccr.jussieu.fr
- To see a preprint or other
information provided by the author
click here.
2296. SYMMETRIC STABLE PROCESSES IN CONES
Rodrigo Ba\~nuelos and Krzysztof Bogdan
We study exponents of integrability of the first exit
time from generalized cones for conditioned
rotation invariant stable L\'evy processes.
The key step in our proof is to identify, for generalized cones,
the Martin kernel with pole at infinity as a
homogeneous function of a certain degree which depends
on the geometry of the cone and on the paramenter of
the stable process. From this we estimate the distribution of the
exit time and even the transition probabilities of the process
in the cone. Along the way, we introduce a 'spherical fractional
Laplacian' and derive some of its spectral properties.
banuelos@math.purdue.edu bogdan@im.pwr.wroc.pl
- To see a preprint or other
information provided by the author
click here.
2297. THE CAUCHY PROCESS AND THE STEKLOV PROBLEM
Rodrigo Ba\~nuelos and Tadeusz Kulczycki
Let X(t) be a Cauchy process in $R^d$, $d\geq 1$.
We investigate some of the fine spectral theoretic
properties of the semigroup of this process killed upon
leaving a domain D. We establish a connection between
the semigroup of this process and a mixed boundary value
problem for the Laplacian in one dimension higher, known
as the 'Mixed Steklov Problem.' Using this and a combination
of analytic and probabilistic techniques we derive
a variational characterization for the eigenvalues of the
Cauchy process in D. This characterization leads to many
detailed properties of the eigenvalues and eigenfunctions
for the Cauchy process inspired by those for Brownian
motion. These results are new even in the simplest
geometric setting of the interval (-1, 1) where we
obtain more precise information on the size of
the second and third eigenvalues and on the
geometry of their corresponding eigenfunctions.
Such results, although trivial for the Brownian motion,
take considerable work to prove for the Cauchy process
and remain open for general symmetric stable processes.
The paper also contains many other general
properties of the eigenfunctions, such as real
analyticity, which even though well known in the case
of the Laplacian, are not rarely available for more
general symmetric stable processes.
banuelos@math.purdue.edu tkulczyc@kac.im.pwr.wroc.pl
- To see a preprint or other
information provided by the author
click here.
2298. AN INEQUALITY FOR POTENTIALS AND THE `HOT-SPOTS' PROBLEM
Rodrigo Ba\~nuelos and Michael Pang
Consider a nonnegative continuous potential $V$ on the
half disk $D^+=\{z=x+iy: y>0, |z|<1\}$
for which $r^2V(re^{i\theta})$ is nondecreasing as
a function of $r$ for every fixed $0<\theta<\pi$.
We prove a 'hot-spots' inequality for the distribution of
$\int_0^{\tau_{D^+}} V(B_s) ds$ where $B_s$ is
the Brownian motion reflected on the top
portion of the boundary and killed on the lower portion
and $\tau_{D^+}$ is its lifetime. This inequality, by the
conformal invariance of Brownian motion and asymptotics
of heat kernels, implies the 'hot-spots' conjecture of Jeff
Rauch for certain symmetric convex domains in the plane.
Related questions concerning 'hot-spots' for mixed boundary
value problems are raised in this paper.
banuelos@math.purdue.edu pangm@math.missouri.edu
- To see a preprint or other
information provided by the author
click here.
2299. CONVEX REARRANGEMENTS, GENERALIZED LORENZ CURVES, AND
CORRELATED GAUSSIAN DATA
Youri Davydov, Davar Khoshnevisan, Zhan Shi, and Ricardas Zitikis
We propose a statistical index for measuring the
fluctuations of a stochastic process $\xi$. This index
is based on the generalized Lorenz curves and (modified)
Gini indices of econometric theory.
When $\xi$ is a fractional Brownian motion with
Hurst index
$\alpha\in(0,1)$, we develop a complete picture of the
asymptotic theory of our index. In particular, we
show that the asymptotic behavior
of our proposed index depends critically on whether
$\alpha\in(0,\frac34)$, $\alpha=\frac34$, or
$\alpha\in(\frac34,1)$. Furthermore, in the
first two cases, there is a Gaussian limit law,
while the third case has an explicit limit law that
is in the second Wiener chaos.
Youri.Davydov@univ-lille1.fr davar@math.utah.edu zhan@proba.jussieu.fr zitikis@stats.uwo.ca
- To see a preprint or other
information provided by the author
click here.
- Or
here.
2300. BROWNIAN MOTION WITH KILLING AND REFLECTION AND THE 'HOT SPOTS' PROBLEM
Rodrigo Ba\~nuelos, Michael Pang and Mihai Pascu
We investigate the 'hot-spots' property for the survival time
probability of Brownian motion with killing and reflection in
planar convex domains whose boundary consists of two curves,
one of which is an arc of a circle, intersecting at acute angles.
This leads to the 'hot-spots' property for the mixed
Dirichlet-Neumann eigenvalue problem in the domain
with Neumann conditions on one of the curves and
Dirichlet conditions on the other
banuelos@math.purdue.edu pangm@math.missouri.edu pascu@math.purdue.edu
- To see a preprint or other
information provided by the author
click here.
2301. ASYMPTOTICS FOR HITTING TIMES
Michal Kupsa, Yves Lacroix
We give a complete description of asymptotic (pseudo) distribution
functions for hitting
times in ergodic aperiodic
dynamical systems. Denote such a system by $(X,T,\mu )$.
For $U\subset X$ measurable, let
$\tau_U(x)=\inf\{k\ge 1:T^kx\in U\}$
denote the hitting time of $x$ to $U$.
The random variable $\tau_U$ is $\mu$-a.s.
well defined by Poincar\'e's recurrence theorem,
as soon as $\mu (U)>0$.
Kac's return time theorem asserts that
the expectation of the normalized random
variable $\mu (U)\tau_U$ equals one on
the induced probability space on $U$.
We investigate weak limits of the
distribution functions
of the variables $\mu (U_n)\tau_{U_n}$
when $\mu (U_n)\to 0$.
Let $\cal F$ denote the set of functions $F:R\to [0,1]$, that are
increasing, null on $]-\infty ,0]$, continuous and concave. We prove that
in any dynamical system $(X,T,\mu )$ as above, for any $F\in\cal F$,
there exists a sequence $(U_n)_{n\ge 1}$ of measurable subsets in $X$
with $\mu (U_n)\to 0$, and if $F_{U_n}$ denotes the distribution function
of the variable $\mu (U_n)\tau_{U_n}$ on the probability space $(X,\mu
)$, then $F_{U_n}$ converges weakly to $F$. It is also proved that any
such weak limit (an asymptotic for hitting times) must belong to the
functional class
$\cal F$.
yves.lacroix@u-picardie.fr kupsa@cpt.univ-mrs.fr
2302. RANDOM PARTITIONS APPROXIMATING THE COALESCENCE
OF LINEAGES DURING A SELECTIVE SWEEP
Jason Schweinsberg and Rick Durrett
When a beneficial mutation occurs in a population, the new, favored allele
may spread to the entire population. This process is known as a selective
sweep. Suppose we sample $n$ individuals at the end of a selective sweep.
If we focus on a site on the chromosome that is close to the location of
the beneficial mutation, then many of the lineages will likely be
descended from the individual that had the beneficial mutation, while others
will be descended from a different individual because of recombination
between the two sites. We introduce two approximations for the effect of
a selective sweep. The first one is simple but not very accurate:
flip $n$ independent coins with probability $p$ of heads and say that the
lineages whose coins come up heads are those that are descended from the
individual with the beneficial mutation. A second approximation, which is
related to Kingman's paintbox construction, replaces the coin flips by
integer-valued random variables and leads to very accurate results.
jasonsch@math.cornell.edu rtd1@cornell.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
2303. TRAPS FOR REFLECTED BROWNIAN MOTION
Krzysztof Burdzy, Zhen-Qing Chen and Donald Marshall
Consider an open set $D\in\R^d$, $d\geq 2$, and a closed ball
$B\subset D$. Let $\E^xT_B$ denote the expectation of the hitting
time of $B$ for reflected Brownian motion in $D$ starting from
$x\in D$. We say that $D$ is a trap domain if $\sup_x \E^x T_B =
\infty$. We fully characterize simply connected planar trap
domains using a geometric condition. We give a number of (less
complete) results for multidimensional domains. We discuss the
relationship between trap domains and some other potential
theoretic properties of $D$ such as compactness of the 1-resolvent
of the Neumann Laplacian. In addition, we give an answer to an
open problem raised by Davies and Simon in 1984 about the possible
relationship between intrinsic ultracontractivity for the
Dirichlet Laplacian in a domain $D$ and compactness of the
1-resolvent of the Neumann Laplacian in $D$.
burdzy@math.washington.edu zchen@math.washington.edu marshall@math.washington.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
2304. LEVY PROCESSES: CAPACITY AND HAUSDORFF DIMENSION
Davar Khoshnevisan and Yimin Xiao
We use the recently-developed multiparameter theory of additive
Lévy processes to establish novel connections between an arbitrary
Lévy process X in $R^d$, and a new class of energy forms and their
corresponding capacities. We then apply these connections to solve
two long-standing problems in the folklore of the theory of Lévy processes.
First, we compute the Hausdorff dimension of the image X(G) of a nonrandom
linear Borel set G in $R_+$, where X is an arbitrary Lévy process in $R^d$.
Our work completes the various earlier efforts of Taylor (1953), McKean (1955),
Blumenthal and Getoor (1960; 1961), Millar (1971), Pruitt (1969),
Pruitt and Taylor (1969), Hawkes (1971; 1978; 1998), Hendricks (1972; 1973),
Kahane (1983; 1985b), Becker-Kern, Meerschaert, and Scheffler (2003),
and Khoshnevisan, Xiao, and Zhong (2003a), where dimX(G) is computed under
various conditions on G, X, or both.
We next solve the following problem (Kahane, 1983): When X is an isotropic
stable process, what is an analytic necessary and sufficient condition
on any two disjoint Borel sets F,G in $R^d$ such that with positive probability
X(F) intersects X(G)? Prior to this article, this was understood only in the case
that X is Brownian motion (Khoshnevisan, 1999). Here, we present a solution
to Kahane's problem for an arbitrary L\evy process X provided the distribution
of X(t) is mutually absolutely continuous with respect to the Lebesgue measure
on $R^d$ for all t>0.
As a third application of these methods, we compute the Hausdorff dimension
and capacity of the preimage $X^{-1}(F)$ of a nonrandom Borel set F in $R^d$
under very mild conditions on the process X. This completes the work of
Hawkes (1998) that covers the special case where X is a subordinator.
davar@math.utah.edu xiao@stt.msu.edu
- To see a preprint or other
information provided by the author
click here.
2305. EXACT $L_2$-SMALL BALL BEHAVIOR OF INTEGRATED
GAUSSIAN PROCESSES AND SPECTRAL ASYMPTOTICS
OF BOUNDARY VALUE PROBLEMS
A.I. Nazarov and Ya.Yu. Nikitin
We find the exact small deviation asymptotics for the
$L_2$-norm of certain $m$-times integrated Gaussian processes
closely connected with the Wiener process and the Ornstein --
Uhlenbeck process. Using a general approach from the spectral
theory of linear differential operators we are able to obtain
rather precise asymptotics of eigenvalues in corresponding
boundary value problems. We improve, in particular, recent results
of Gao et al. (2003) on the small ball asymptotics of $m$-times
integrated Wiener process and prove their conjecture. Moreover,
exact small ball asymptotics for the $m$-times integrated
Brownian bridge, the $m$-times integrated Ornstein -- Uhlenbeck
process and related processes appear for the first time.
an@AN4751.spb.edu Yakov.Nikitin@pobox.spbu.ru
- To see a preprint or other
information provided by the author
click here.
2306. LOGARITHMIC $L_2$-SMALL BALL ASYMPTOTICS
FOR SOME FRACTIONAL GAUSSIAN PROCESSES
A.I. Nazarov and Ya.Yu. Nikitin
We find the logarithmic $L_2$-small ball asymptotics of
some Gaussian processes related to the fractional Brownian motion,
fractional Ornstein -- Uhlenbeck process (fOU) and their
integrated analogues. To that end we use general theorems on
spectral asymptotics of integral operators obtained by Birman and
Solomyak (1970) combining them with the classical theorem of Weyl.
In the simplest case of fractional Brownian motion we generalize
the result of Bronski (2003). We consider also the fractional
L\'evy's Brownian motion as well as the multiparameter fOU process
on the bounded domain.
an@AN4751.spb.edu Yakov.Nikitin@pobox.spbu.ru
- To see a preprint or other
information provided by the author
click here.
