Probability Abstracts 76

This document contains abstracts 2307-2341. They have been mailed on August 31, 2003.

2307. THE MONGE-KANTOROVITCH PROBLEM AND MONGE-AMPERE EQUATION ON WIENER SPACE 

D. Feyel and A.S. Ustunel

We give the solution of the Monge-Kantorovitch problem on the Wiener space
for the singular Wasserstein metric which is defined with respect to the
distance of the underlying Cameron-Martin space. We show, under the hypothesis
that this distance is finite, the existence and the uniquness of the solutions,
that they are supported by the graphs of the weak derivatives of H-convex
Wiener functionals. then we prove the more general situation, where the
measures are not even necessarily absolutely continuous w.r.to the Wiener
measure. We give sufficient conditions for the hypothesis about the Wassestein
distance is finite with the help of the Girsanov theorem. Finally we give the
solutions of the Monge-Ampere equation using the classical Jacobi
representation and/or the Ito parametrization of the Wiener space.

ustunel@enst.fr

2308. CHERNOFF'S BOUND FORMS 

M. Grendar, Jr. and M. Grendar

Chernoff's bound binds a tail probability (ie. $Pr(X \ge a)$, where $a \ge
EX$). Assuming that the distribution of $X$ is $Q$, the logarithm of the bound
is known to be equal to the value of relative entropy (or minus
Kullback-Leibler distance) for $I$-projection $\hat P$ of $Q$ on a set
$\mathcal{H} \triangleq \{P: E_PX = a\}$. Here, Chernoff's bound is related to
Maximum Likelihood on exponential form and consequently implications for the
notion of complementarity are discussed. Moreover, a novel form of the bound is
proposed, which expresses the value of the Chernoff's bound directly in terms
of the $I$-projection (or generalized $I$-projection).

umergren@savba.sk

2309. A PHASE TRANSITION FOR THE METRIC DISTORTION OF PERCOLATION ON THE HYPERCUBE 

Omer Angel, Itai Benjamini

Let H_n be the hypercube {0,1}^n, and let H_{n,p} denote the same graph with
Bernoulli bond percolation with parameter p=n^-\alpha. It is shown that at
\alpha=1/2 there is a phase transition for the metric distortion between H_n
and H_{n,p}. For \alpha<1/2, asymptotically there is a map from H_n to H_{n,p}
with constant distortion (depending only on \alpha). For \alpha>1/2 the
distortion tends to infinity as a power of n. We indicate the similarity to the
existence of a non-uniqueness phase in the context of infinite nonamenable
graphs.

omer@wisdom.weizmann.ac.il

2310. RANDOM SETS AND INVARIANTS FOR (TYPE II) CONTINUOUS TENSOR PRODUCT SYSTEMS OF HILBERT SPACES 

Volkmar Liebscher

In a series of papers Tsirelson constructed from measure types of random sets
and generalised random processes a new range of examples for continuous tensor
product systems of Hilbert spaces introduced by Arveson for classifying
$E_0$-semigroups. This paper establishes the converse: Each continuous tensor
product systems of Hilbert spaces comes with measure types of distributions of
random (closed) sets in [0,1] or $R_+$. These measure types are stationary and
factorise over disjoint intervals. In a special case of this construction, the
corresponding measure type is an invariant of the product system and the range
of the invariant is characterized.
  Moreover, based on a detailed study of this kind of measure types, we
construct for each stationary factorizing measure type a continuous tensor
product systems of Hilbert spaces such that this measure type arises as the
before mentioned invariant.
  The measure types of the above described kind are connected with
representations of the corresponding $L^\infty$-spaces. This leads to direct
integral representations of the elements of a given product system which
combine well under tensor products. Using this structure in a constructive way,
we can relate to any (type III) product system a product system of type $II_0$
preserving isomorphy classes. Thus, the classification of type III product
systems reduces to that of type II ones.
  Further, we show that all consistent measurable structures on an algebraic
continuous tensor product systems of Hilbert spaces yield isomorphic product
systems. Thus the measurable structure of a continuous tensor product systems
of Hilbert spaces is essentially determined by its algebraic one.

liebscher@gsf.de

2311. NONCOLLIDING BROWNIAN MOTIONS AND HARISH-CHANDRA FORMULA 

Makoto Katori and Hideki Tanemura

We consider a system of noncolliding Brownian motions introduced in our
previous paper, in which the noncolliding condition is imposed in a finite time
interval $(0,T]$. This is a temporally inhomogeneous diffusion process whose
transition probability density depends on a value of $T$, and in the limit $T
\to \infty$ it converges to a temporally homogeneous diffusion process called
Dyson's model of Brownian motions. It is known that the distribution of
particle positions in Dyson's model coincides with that of eigenvalues of a
Hermitian matrix-valued process, whose entries are independent Brownian
motions. In the present paper we construct such a Hermitian matrix-valued
process, whose entries are sums of Brownian motions and Brownian bridges given
independently of each other, that its eigenvalues are identically distributed
with the particle positions of our temporally inhomogeneous system of
noncolliding Brownian motions. As a corollary of this identification we derive
the Harish-Chandra formula for an integral over the unitary group.

katori@phys.chuo-u.ac.jp

2312. EXTRA HEADS AND INVARIANT ALLOCATIONS 

Alexander E. Holroyd and Yuval Peres

Let \Pi be an ergodic simple point process on R^d, and let \Pi^* be its Palm
version. Thorisson proved that there exists a shift coupling of \Pi and \Pi^*;
that is, one can select a (random) point Y of \Pi such that translating \Pi by
-Y yields a configuration whose law is that of \Pi^*. We construct shift
couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no
shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a
deterministic translation-invariant rule to allocate sets of equal area
(forming a plane partition) to the points of \Pi. The construction is based on
the Gale-Shapley stable marriage algorithm. Next, let \Gamma be an ergodic
random element of {0,1}^Z^d, and let \Gamma^* be \Gamma conditioned on
\Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head
scheme. We show that there exists an extra head scheme which is a function of
\Gamma if and only if the marginal E \Gamma_0 is the reciprocal of an integer.
When the law of \Gamma is product measure and d is at least 3, we prove that
there exists an extra head scheme X satisfying E \exp c|X|^d < \infty; this
answers a question of Holroyd and Liggett.

holroyd@math.ubc.ca

2313. CONTROLLING ROUGH PATHS 

Massimiliano Gubinelli

We formulate indefinite integration with respect to an irregular function as
an algebraic problem and provide a criterion for the existence and uniqueness
of a solution. This allows us to define a good notion of integral with respect
to irregular paths with Hoelder exponent greater than 1/3 (e.g. samples of
Brownian motion) and study the problem of the existence, uniqueness and
continuity of solution of differential equations driven by such paths. We
recover Young's theory of integration for Hoelder continuous functions and the
main results of Lyons' theory of rough paths.

m.gubinelli@dma.unipi.it

2314. ASYMPTOTIC ANALYSIS BY THE SADDLE POINT METHOD OF THE ANICK-MITRA-SONDHI MODEL 

Diego Dominici and Charles Knessl

We consider a fluid queue where the input process consists of N identical
sources that turn on and off at exponential waiting times. The server works at
the constant rate c and an on source generates fluid at unit rate. This model
was first formulated and analyzed by Anick, Mitra and Sondhi. We obtain an
alternate representation of the joint steady state distribution of the buffer
content and the number of on sources. This is given as a contour integral that
we then analyze for large N. We give detailed asymptotic results for the joint
distribution, as well as the associated marginal and conditional distributions.
In particular, simple conditional limits laws are obtained. These shows how the
buffer content behaves conditioned on the number of active sources and vice
versa. Numerical comparisons show that our asymptotic results are very accurate
even for N=20.

ddomin1@uic.edu

2315. ENTROPIC REPULSION OF AN INTERFACE IN AN EXTERNAL FIELD 

Yvan Velenik

We consider an interface above an attractive hard wall in the complete
wetting regime, and submitted to the action of an external increasing, convex
potential, and study its delocalization as the intensity of this potential
vanishes. Our main motivation is the analysis of critical prewetting, which
corresponds to the choice of a linear external potential.
  We also present partial results on critical prewetting in the two dimensional
Ising model, as well as a few (weak) results on pathwise estimates for the pure
wetting problem for effective interface models.

velenik@cmi.univ-mrs.fr

2316. PRODUCTS OF BETA MATRICES AND STICKY FLOWS 

Yves Le Jan, Sophie Lemaire

A discrete model of Brownian sticky flows on the unit circle is described: it
is constructed with products of Beta matrices on the discrete torus. Sticky
flows are defined by their "moments'' which are consistent systems of
transition kernels on the unit circle. Similarly, the moments of the discrete
model form a consistent system of transition matrices on the discrete torus. A
convergence of Beta matrices to sticky kernels is shown at the level of the
moments.

sophie.lemaire@math.u-psud.fr

2317. CONVERGENCE TO EQUILIBRIUM FOR FINITE MARKOV PROCESSES, WITH APPLICATION TO THE RANDOM ENERGY MODEL 

Pierre Mathieu and Pierre Picco 

We estimate the distance in total variation between the law of a finite state
Markov process at time t, starting from a given initial measure, and its unique
invariant measure. We derive upper bounds for the time to reach the
equilibrium. As an example of application we consider a special case of finite
state Markov process in random environment: the Metropolis dynamics of the
Random Energy Model. We also study the process of the environment as seen from
the process.

pierre.mathieu@cmi.univ-mrs.fr

2318. WIENER CHAOS AND THE COX-INGERSOLL-ROSS MODEL 

M. R. Grasselli and T. R. Hurd

In this we paper we recast the Cox--Ingersoll--Ross model of interest rates
into the chaotic representation recently introduced by Hughston and Rafailidis.
Beginning with the ``squared Gaussian representation'' of the CIR model, we
find a simple expression for the fundamental random variable X. By use of
techniques from the theory of infinite dimensional Gaussian integration, we
derive an explicit formula for the n-th term of the Wiener chaos expansion of
the CIR model, for n=0,1,2,.... We then derive a new expression for the price
of a zero coupon bond which reveals a connection between Gaussian measures and
Ricatti differential equations.

grasselli@icarus.math.mcmaster.ca

2319. ASYMPTOTIC BEHAVIOUR OF WATERMELONS 

Florent Gillet

A watermelon is a set of $p$ Bernoulli paths starting and ending at the same
ordinate, that do not intersect. In this paper, we show the convergence in
distribution of two sorts of watermelons (with or without wall condition) to
processes which generalize the Brownian bridge and the Brownian excursion in
$\mathbb{R}^p$. These limit processes are defined by stochastic differential
equations. The distributions involved are those of eigenvalues of some
Hermitian random matrices. We give also some properties of these limit
processes.

gillet@iecn.u-nancy.fr

2320. ON THE FIRST-VISIT-TIME PROBLEM FOR BIRTH AND DEATH PROCESSES WITH CATASTROPHES 

A. Di Crescenzo, V. Giorno, A.G. Nobile, L.M. Ricciardi

For a birth-death process subject to catastrophes, defined on the state-space
$S=\{r,r+1,r+2,...\}$, with $r$ a positive integer or zero, the first-visit
time to a state $k\in S$ is considered and the Laplace transform of its
probability density function is determined, use of which is then made to obtain
mean and variance. The Laplace transform of the probability density function of
the first effective catastrophe occurrence time and its expected value are also
obtained. Some extensions to time-non-homogeneous processes are then provided.
Finally, certain additional results concerning the determination of the
steady-state distribution and the representation of the transition
probabilities are worked out, while some applications to particular birth-death
processes are shown in the Appendix.

luigi.ricciardi@unina.it

2321. FOCK SPACE DECOMPOSITION OF LEVY PROCESSES 

R. F. Streater

We show that the general L\'{e}vy process can be embedded in a suitable Fock
space, classified by cocycles of the real line regarded as a group, ${\bf R}$.
The formula of de Finetti corresponds to coboundaries. Kolmogorov's processes
correspond to cocycles of which the derivatives are cocycles of the Lie algebra
of ${\bf R}$. L\'{e}vy's formula gives the most general cocycle possible.

rf.streater@kcl.ac.uk

2322. APPROXIMATION PROBABILITIES, THE LAW OF QUASISTABLE MARKETS, AND PHASE TRANSITIONS FROM THE "CONDENSED" STATE 

V. P. Maslov 

For common people, in contrast to brokers, bankers, and those who play on
rising and falling prices of stocks, the stock market law is based on the
simple fact that the depositors aim for financial profit at any given concrete
stage. The common depositor cannot cause any significant variations in prices.
This concept suggests an analogy with the quasistable physics, i.e.,
thermodynamics, in the situation in which the temperature varies slowly along
with the external conditions. Therefore, in the quasistable market, we can see
phase transitions similar to those in the situation of the Bose-condesate in
thermodynamics. We stress the positive role of information for common
depositors and the possibility of changing bonds of large denomination into
bonds of small denomination.

pm@miem.edu.ru

2323. ON SPECTRA OF NOISES ASSOCIATED WITH HARRIS FLOWS 

Jon Warren and Shinzo Watanabe

We study the noise, in the sense of Tsirelson, generated by Harris flows. A
criterion is given for the noise to be non-white, and in this case we study the
associated spectral sets.

warren@stats.warwick.ac.uk

2324. REGENERATIVE COMPOSITION STRUCTURES 

Alexander Gnedin and Jim Pitman

A new class of random composition structures (the ordered analog of Kingman's
partition structures) is defined by a regenerative description of component
sizes. Each regenerative composition structure is represented by a process of
random sampling of points from an exponential distribution on the positive
halfline, and separating the points into clusters by an independent
regenerative random set. Examples are composition structures derived from
residual allocation models, including one associated with the Ewens sampling
formula, and composition structures derived from the zero set of a Brownian
motion or Bessel process. We provide characterisation results and formulas
relating the distribution of the regenerative composition to the L{\'e}vy
parameters of a subordinator whose range is the corresponding regenerative set.
In particular, the only reversible regenerative composition structures are
those associated with the interval partition of $[0,1]$ generated by excursions
of a standard Bessel bridge of dimension $2 - 2 \alpha$ for some $\alpha \in
[0,1]$.

gnedin@math.uu.nl

2325. BOUNDARY TRACE OF REFLECTING BROWNIAN MOTIONS 

Itai Benjamini, Zhen-Qing Chen and Steffen Rohde

A uniform dimensional result for normally reflected Brownian motion (RBM) in
a large class of non-smooth domains is established. Exact Hausdorff dimensions
for the boundary occupation time and the boundary trace of RBM are given.
Extensions to stable-like jump processes and to symmetric reflecting diffusions
are also mentioned.

itai@wisdom.weizmann.ac.il

2326. SPECTRAL MEASURE OF LARGE RANDOM HANKEL, MARKOV AND TOEPLITZ MATRICES 

Wlodzimierz Bryc, Amir Dembo, Tiefeng Jiang

We study the limiting spectral measure of large symmetric random matrices of
linear algebraic structure.
  For Hankel, Toeplitz and Markov matrices generated by i.i.d. random variables
of zero mean and unit variance, scaling the eigenvalues by the square root of n
we prove the almost sure, weak convergence of the spectral measures to
universal, non-random, symmetric distributions of unbounded support. We show
that limiting spectral measure of Hankel matrices is not unimodal and that
limiting spectral measure of Toeplitz matrices is not normal. The moments of
these limiting spectral measures are the sums of volumes of solids related to
Eulerian numbers. The limiting spectral measure of the Markov matrices has a
bounded smooth density given by the free convolution of the semi-circle and
normal densities.
  For symmetric Markov matrices generated by i.i.d. random variables of mean m
and finite variance, scaling the eigenvalues by n we prove the almost sure,
weak convergence of the spectral measures to the atomic measure at -m. If m=0,
and the fourth moment is finite, we prove that the spectral radius scaled by
the square root of (2nlog n) converges almost surely to one.

brycw@math.uc.edu

2327. GLAUBER DYNAMICS ON TREES: BOUNDARY CONDITIONS AND MIXING TIME 

Fabio Martinelli, Alistair Sinclair and Dror Weitz

We give the first comprehensive analysis of the effect of boundary conditions
on the mixing time of the Glauber dynamics in the so-called Bethe
approximation. Specifically, we show that spectral gap and the log-Sobolev
constant of the Glauber dynamics for the Ising model on an n-vertex regular
tree with plus-boundary are bounded below by a constant independent of n at all
temperatures and all external fields. This implies that the mixing time is
O(log n) (in contrast to the free boundary case, where it is not bounded by any
fixed polynomial at low temperatures). In addition, our methods yield simpler
proofs and stronger results for the spectral gap and log-Sobolev constant in
the regime where there are multiple phases but the mixing time is insensitive
to the boundary condition. Our techniques also apply to a much wider class of
models, including those with hard-core constraints like the antiferromagnetic
Potts model at zero temperature (proper colorings) and the hard--core lattice
gas (independent sets).

martin@mat.uniroma3.it

2328. ON DYNAMICAL GAUSSIAN RANDOM WALKS 

D. Khoshnevisan, D. A. Levin and P. J. Mendez-Hernandez

Motivated by the recent work of Benjamini, Haggstrom, Peres, and Steif (2003)
on dynamical random walks, we: Prove that, after a suitable normalization, the
dynamical Gaussian walk converges weakly to the Ornstein-Uhlenbeck process in
classical Wiener space; derive sharp tail-asymptotics for the probabilities of
large deviations of the said dynamical walk; and characterize (by way of an
integral test) the minimal envelop(es) for the growth-rate of the dynamical
Gaussian walk. This development also implies the tail capacity-estimates of
Mountford (1992) for large deviations in classical Wiener space. The results of
this paper give a partial affirmative answer to the problem, raised in
Benjamini et al (2003, Question 4), of whether there are precise connections
between the OU process in classical Wiener space and dynamical random walks.

levin@math.utah.edu

2329. CONFORMAL RESTRICTION AND RELATED QUESTIONS 

Wendelin Werner

This paper is based on mini-courses given in July 2003. Its goal is to give a
self-contained sketchy and heuristic survey of the recent results concerning
conformal restriction, that were initiated in our joint work with Greg Lawler
and Oded Schramm, and further investigated in the last year in joint work with
Roland Friedrich, Greg Lawler, and by Julien Dubedat. These notes can be viewed
as complementary to my Saint-Flour notes.

wendelin.werner@math.u-psud.fr

2330. Q-REPRESENTATION OF REAL NUMBERS AND FRACTAL PROBABILITY DISTRIBUTIONS 

Sergio Albeverio, Volodymyr Koshmanenko, Mykola Pratsiovytyi
  and Grygoriy Torbin

A $\widetilde{Q}-$representation of real numbers is introduced as a
generalization of the $p-$adic and $Q-$representations. It is shown that the
$\widetilde{Q}-$representation may be used as a convenient tool for the
construction and study of fractals and sets with complicated local structure.
Distributions of random variables $\xi$ with independent
$\widetilde{Q}-$symbols are studied in details. Necessary and sufficient
conditions for the probability measures $\mu_\xi $ associated with $\xi$ to be
either absolutely continuous or singular (resp. pure continuous, or pure point)
are found in terms of the $\widetilde{Q}-$representation. In addition the
metric-topological properties for the distribution of $\xi $ are investigated.
A number of examples are presented.

torbin@imath.kiev.ua

2331. ON THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR COUNTABLE ERGODIC MARKOV CHAINS 

Stefano Isola

Using elementary methods, we prove that for a countable Markov chain $P$ of
ergodic degree $d > 0$ the rate of convergence towards the stationary
distribution is subgeometric of order $n^{-d}$, provided the initial
distribution satisfies certain conditions of asymptotic decay. An example,
modelling a renewal process and providing a markovian approximation scheme in
dynamical system theory, is worked out in detail, illustrating the
relationships between convergence behaviour, analytic properties of the
generating functions associated to transition probabilities and spectral
properties of the Markov operator $P$ on the Banach space $\ell_1$. Explicit
conditions allowing to obtain the actual asymptotics for the rate of
convergence are also discussed.

stefano.isola@unicam.it

2332. IMAGE MEASURES OF INFINITE PRODUCT MEASURES AND GENERALIZED BERNOULLI CONVOLUTIONS 

Albeverio Sergio, Torbin Grygoriy

We examine measure preserving mappings $f$ acting from a probability space
$(\Omega, F,\mu) $ into a probability space $% (\Omega ^{*},F^{*},\mu ^{*}) ,$
where $\mu ^{*}=\mu (f^{-1})$. Conditions on $f$, under which $f$ preserves the
relations ''to be singular'' and ''to be absolutely continuous'' between
measures defined on $(\Omega, F) $ and corresponding image measures, are
investigated.
  We apply the results to investigate the distribution of the random variable
$% \xi =\sum\limits^{\infty}_{k=1} \xi_k\lambda ^k,$ where $% \lambda \in
(0;1),$ and $\xi_k$ are independent not necessarily identically distributed
random variables taking the values $i$ with probabilities $% p_{ik}$ ,$i=0,1.$
  We also studied in details the metric-topological and fractal properties of
the distribution of a random variable $\psi = \sum\limits^{\infty}_{k=1} \xi
_ka_k,$ where $a_k>0$ are terms of the convergent series.

torbin@imath.kiev.ua

2333. THE MAXIMUM QUEUE LENGTH FOR HEAVY TAILED SERVICE TIMES 

M.F.M. Nuyens

In this article we study the maximum queue length $M$ in a busy cycle in the
M/G/1 queue. Assume that the service times have a logconvex density. For such
(heavy tailed) service time distributions the Foreground Background service
discipline is optimal. This discipline gives service to those customer(s) that
have received the least amount of service so far. It will be shown that under
this discipline $M$ has an exponentially decreasing tail. From the behaviour of
$M$ we obtain asymptotics for the maximum queue length $M(t)$ over the interval
$(0,t)$ for $t\to\infty$

mnuyens@science.uva.nl

2334. ON THE CHERNOFF BOUND FOR EFFICIENCY OF QUANTUM HYPOTHESIS TESTING 

Vladislav Kargin

The paper estimates the Chernoff rate for the efficiency of quantum
hypothesis testing. For both joint and separable measurements, approximate
bounds for the rate are given if both states are mixed and exact expressions
are derived if at least one of the states is pure. The efficiency of tests with
separable measurements is found to be close to the efficiency of tests with
joint measurements. The results are illustrated by a test of quantum
entanglement.

ys2040@columbia.edu

2335. DIFFUSIVE LONG-TIME BEHAVIOR OF KAWASAKI DYNAMICS 

Filippo Cesi, Nicoletta Cancrini and Cyril Roberto

If $P_t$ is the semigroup associated with the
Kawasaki dynamics on $\Z^d$ and $f$ is a  local
function on the configuration space, then the
variance with respect to the invariant measure
$\mu$ of $P_t f$ goes to zero, as t goes to infinity,
faster than $t^{-d/2+\e}$, with $\e$ arbitrarily
small. The fundamental assumption is a mixing
condition on the interaction of Dobrushin and
Schlosman type.

filippo.cesi@roma1.infn.it

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

2336. CONSTRUCTION OF DIFFUSION PROCESSES ON FRACTALS, D-SETS, AND GENERAL METRIC MEASURE SPACES 

Takashi Kumagai and Karl-Theodor Sturm

We give a sufficient condition to construct non-trivial $\mu$-symmetric diffusion 
processes on a locally compact metric measure space $(M,\rho,\mu)$. 
These processes are associated with local regular Dirichlet forms which are obtained 
as continuous parts of $\Gamma$-limits for approximating non-local Dirichlet forms. 
For various fractals, we can use existing estimates to verify our assumptions. This shows
that our general method of constructing diffusions can be applied to these fractals. 

kumagai@kurims.kyoto-u.ac.jp   sturm@uni-bonn.de

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

2337. HITTING AND RETURN TIMES 

Nicolai Haydn, Yves Lacroix, and Sandro Vaienti

Given an ergodic dynamical
system $(X,T,\mu )$, and $U\subset X$ measurable with $\mu (U)>0$, let
$\mu (U)\tau_U(x)$ denote the normalized hitting time of $x\in X$ to
$U$. We prove that given a sequence $(U_n)$ with $\mu (U_n)\to 0$, 
the distribution function of the normalized hitting
time to $U_n$ converges weakly to some pseudo-distribution $F$ if and
only if the distribution function of the normalized return time converges
weakly to some distribution function $\tilde F$, and that in the
converging case, 
$$
F(t)=\int_0^t(1-\tilde F(s))ds,\; t\ge 0.
$$
This shows, among other consequences, 
that the asymptotic for return times is exponential if and only if the
one for hitting times is too. 

lacroix@univ-tln.fr nhaydn@math.usc.edu vaienti@cpt.univ-mrs.fr

2338. RANDOM SAMPLING OF MULTIVARIATE TRIGONOMETRIC POLYNOMIALS 

Richard F. Bass and Karlheinz Gr\''ochenig

A central problem in the theory of sampling is to
reconstruct the function given the data. For example,
if one is given points $(x_i, p(x_i))$, $i=1, ..., r$,
and one is told that $p$ is a multivariate 
trigonometric polynomial with integer frequencies that is band-limited,
that is, the frequencies lie in $[-M,M]^d$, can one determine
the coefficients of $p$? This problem is well-understood in
 dimension 1, but very poorly understood in higher dimensions,
and in fact very few rigorous results are known. Nevertheless
there are algorithms (which involve solving a system of
 linear equations of the form $Ax=y$ where the 
matrix $A$ is a block Toeplitz matrix) that work very well in practice.

We consider the case where the sampling points $x_i$ are chosen
randomly according to some distribution. We show that under
very mild conditions the matrix $A$ is invertible a.s. We give
large deviations estimates for the condition number of the
matrix $A$. Finally we prove a law of the iterated logarithm
for the condition number of $A$ as $r$ gets large. The LIL
shows that the condition number is worse than the optimal
spacing of the $x_i$ only by a factor of $\log r$, which 
gives some explanation of why the algorithms that are
used work well in practice.

bass@math.uconn.edu  groch@math.uconn.edu

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

2339. BRANCHING-COALESCING PARTICLE SYSTEMS 

S.R. Athreya and Jan M. Swart

We study the ergodic behaviour of systems of particles
performing independent random walks, binary splitting,
coalescence and deaths. Such particle systems are dual
to systems of linearly interacting Wright-Fisher diffusions,
used to model a population with resampling, selection and 
mutations. We use this duality to prove that the upper
invariant measure of the particle system is the only
homogeneous nontrivial invariant law and the limit started
from any homogeneous nontrival initial law. An interesting
tool in our proofs is an ergodic theorem for countable groups
that need not be amenable.

athreya@isid.ac.in swart@mi.uni-erlangen.de

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  • Or here.

2340. AN EXAMPLE AND A CONJECTURE CONCERNING SCALING LIMITS OF SUPERDIFFUSIONS 

Janos Englander

Consider the superdiffusion corresponding to the
semilinear operator $Lu+\beta u-\alpha u^2$ on $\mathbb R^d$. In
\cite{ET}, the existence of the random measure
$$\lim_{t\uparrow\infty} e^{-\lambda_c t}
 X_t (\mathrm{d}x)$$ was shown under appropriate  spectral
theoretical assumptions. In the same paper we asked whether the
probability of having a zero limit is the same as the probability
of finite time extinction.  In this note we give an answer in the
negative by showing a counterexample.

englander@pstat.ucsb.edu

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  • Or here.

2341. LARGE DEVIATIONS FOR THE GROWTH RATE OF THE SUPPORT OF SUPERCRITICAL SUPER-BROWNIAN MOTION 

Janos Englander

We prove a large deviation result for the growth rate of the
support of the $d$-dimensional (strictly dyadic) branching
Brownian motion $Z$ and the $d$-dimensional (supercritical)
super-Brownian motion $X$. We show  that the probability that $Z$
($X$)
 remains in a smaller than typical ball up to time $t$ is
exponentially small in $t$ and we compute the cost function. The
cost function turns out to be the same for $Z$ and $X$. In the
proof we use a decomposition result due to Evans and O'Connell and
elementary probabilistic arguments. Our method also provides a
short alternative proof for the lower estimate of  the large time
growth rate of the support of $X$, first obtained by Pinsky by pde
methods.

englander@pstat.ucsb.edu

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