Probability Abstracts 54

1403. VALUES OF BROWNIAN INTERSECTION EXPONENTS I: HALF-PLANE EXPONENTS

Gregory F. Lawler, Oded Schramm, Wendelin Werner

This paper proves conjectures originating in the physics literature regarding
the intersection exponents of Brownian motion in a half-plane. For instance,
suppose that B and B' are two independent planar Brownian motions started from
distinct points in a half-plane H. Then, as $t \to \infty$, $$ P [ B[0,t] \cap
B'[0,t] = \emptyset, B[0,t] \cup B'[0,t] \subset H ] = t^{-5/3 + o(1)}.$$ The
proofs use ideas and tools developped by the authors in previous papers. We
prove that one of the stochastic Loewner evolution processes (with parameter 6,
that we will call SLE_6 and which has been conjectured to correspond to the
scaling limit of critical percolation cluster boundaries) satisfies the
``conformal restriction property''. We establish a generalization of Cardy's
formula (for crossings of a rectangle by a percolation cluster) for SLE_6, from
which the exact values of intersection exponents for SLE_6 followd. Since this
process satisfies the conformal restriction property, the Brownian intersection
exponents can be determined from the SLE_6 intersection exponents.
  Results about intersection exponents in the whole plane will appear in a
subsequent paper.

schramm@wisdom.weizmann.ac.il

• This article is available from the xxx mathematics archive as math.PR/9911084.

1404. RIGOROUS PROBABILISTIC ANALYSIS OF EQUILIBRIUM CRYSTAL SHAPES

T.Bodineau, D.Ioffe and Y.Velenik

The rigorous microscopic theory of equilibrium crystal shapes has made
enormous progress during the last decade. We review here the main results which
have been obtained, both in two and higher dimensions. In particular, we
describe how the phenomenological Wulff and Winterbottom constructions can be
derived from the microscopic description provided by the equilibrium
statistical mechanics of lattice gases. We focus on the main conceptual issues
and describe the central ideas of the existing approaches.

velenik@math.tu-berlin.de

• This article is available from the xxx mathematics archive as math.PR/9911106.

1405. SPLITTING: TANAKA'S SDE REVISITED

Jon Warren

The weak solution of Tanaka's SDE is not a function of the driving Brownian
motion, and therefore it has no Wiener chaos expansion. However in some sense
explained here it has a generalised chaos expansion involving infinite products
of stochastic differentials accumulating at the minimum of the Brownian path.
This is related to the existence of a non-classical noise richer than the usual
white noise.

warren@stats.warwick.ac.uk

• This article is available from the xxx mathematics archive as math.PR/9911115.

1406. COALESCENCE OF SKEW BROWNIAN MOTIONS

Martin Barlow, Krzysztof Burdzy, Haya Kaspi and Avi Mandelbaum

We prove that two skew Brownian motions with the same skewness parameter
(different from 0) and driven by the same Brownian motion coalesce a.s.

burdzy@math.washington.edu

• This article is available from the xxx mathematics archive as math.PR/9911136.

1407. PERFECT SIMULATION FOR INTERACTING POINT PROCESSES, LOSS NETWORKS AND ISING MODELS

Roberto Fernandez, Pablo A. Ferrari, Nancy Garcia

We present a perfect simulation algorithm for measures that are absolutely
continuous with respect to some Poisson process and can be obtained as
invariant measures of birth-and-death processes. Examples include area- and
perimeter-interacting point processes (with stochastic grains), invariant
measures of loss networks, and the Ising contour and random cluster models. The
algorithm does not involve any coupling - hence it is not tied up to
monotonicity requirements - and it directly provides samples of the
\emph{infinite-volume} measure. The algorithm is based on a two-step procedure:
(i) a perfect-simulation scheme for (space-time) marked Poisson processes (free
birth-and-death process, free loss networks), and (ii) a ``cleaning'' algorithm
that trims out this process according to the interaction rules of the target
process. The first step involves the perfect generation of ``ancestors'' of a
given object, that is of predecessors that may have an influence on the
birth-rate under the target process. The second step, and hence the whole
procedure, is feasible if these ``ancestors'' form a finite set with
probability one. We present a sufficiency criteria for this condition, based on
the absence of infinite clusters for an associated (backwards) oriented
percolation model. The criteria is expressed in terms of the subcriticality of
a majorizing multi-type branching process, whose corresponding parameter yields
bounds for errors due to ``user-impatience bias''. The approach has previously
been used, as an alternative to cluster expansion techniques, to extract
properties of the invariant measures involved.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/9911162.

1408. CONVERGENCE TO THE MAXIMAL INVARIANT MEASURE FOR A ZERO-RANGE PROCESS WITH RANDOM RATES

Enrique D. Andjel, Pablo A. Ferrari, Herve Guiol, Claudio Landim

We consider a one-dimensional totally asymmetric nearest-neighbor zero-range
process with site-dependent jump-rates - an environment. For each environment p
we prove that the set of all invariant measures is the convex hull of a set of
product measures with geometric marginals. As a consequence we show that for
environments p satisfying certain asymptotic property, there are no invariant
measures concentrating on configurations with critical density bigger than
$\rho^*(p)$, a critical value. If $\rho^*(p)$ is finite we say that there is
phase-transition on the density. In this case we prove that if the initial
configuration has asymptotic density strictly above $\rho^*(p)$, then the
process converges to the maximal invariant measure.

herve@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/9911205.

1409. HYDRODYNAMICS FOR TOTALLY ASYMMETRIC $K$-STEP EXCLUSION PROCESSES

Herve Guiol, Krishnamurthi Ravishankar, Ellen Saada

We describe the hydrodynamic behavior of the $k$-step exclusion process.
Since the flux appearing in the hydrodynamic equation for this particle system
is neither convex nor concave, the set of possible solutions include in
addition to entropic shocks and continuous solutions those with contact
discontinuities. We finish with a limit theorem for the tagged particle.

herve@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/9911213.

1410. THE ASYMMETRIC SIMPLE EXCLUSION PROCESS WITH MULTIPLE SHOCKS

Pablo A. Ferrari, L. Renato G. Fontes, M. Eulalia Vares

We consider the one dimensional totally asymmetric simple exclusion process
with initial product distribution with densities $0 \leq \rho_0 < \rho_1 <...<
\rho_n \leq 1$ in $(-\infty,c_1\ve^{-1})$,
$[c_1\ve^{-1},c_2\epsilon^{-1}),...,[c_n \ve^{-1}, + \infty)$, respectively.
The initial distribution has shocks (discontinuities) at $\epsilon^{-1}c_k$,
k=1,...,n and we assume that in the corresponding macroscopic Burgers equation
the n shocks meet in $r^*$ at time $t^*$. The microscopic position of the
shocks is represented by second class particles whose distribution in the scale
$\epsilon^{-1/2}$ is shown to converge to a function of n independent Gaussian
random variables representing the fluctuations of these particles ``just before
the meeting''. We show that the density field at time $\ve^{-1}t^*$, in the
scale $\ve^{-1/2}$ and as seen from $\ve^{-1}r^*$ converges weakly to a random
measure with piecewise constant density as $\ve \to 0$; the points of
discontinuity depend on these limiting Gaussian variables. As a corollary we
show that, as $\epsilon\to 0$, the distribution of the process at site
$\epsilon^{-1}r^*+\ve^{-1/2}a$ at time $\epsilon^{-1}t^*$ tends to a non
trivial convex combination of the product measures with densities $\rho_k$, the
weights of the combination being explicitly computable.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/9911237.

1411. RATE OF CONVERGENCE TO EQUILIBRIUM OF SYMMETRIC SIMPLE EXCLUSION PROCESSES

P. A. Ferrari, A. Galves, C. Landim

We give bounds on the rate of convergence to equilibrium of the symmetric
simple exclusion process in $\Z^d$. Our results include the existent results in
the literature. We get better bounds and larger class of initial states via a
unified approach. The method includes a comparison of the evolution of n
interacting particles with n independent ones along the whole time trajectory.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/9912008.

1412. CESARO MEAN DISTRIBUTION OF GROUP AUTOMATA STARTING FROM MEASURES WITH SUMMABLE DECAY

Pablo A. Ferrari, Alejandro Maass, Servet Martinez, Peter Ney

Consider a finite Abelian group (G,+), with |G|=p^r, p a prime number, and F:
G^N -> G^N the cellular automaton given by {F(x)}_n= A x_n + B x_{n+1} for any
n in N, where A and B are integers relatively primes to p. We prove that if P
is a translation invariant probability measure on G^Z determining a chain with
complete connections and summable decay of correlations, then for any w=
(w_i:i<0) the Cesaro mean distribution of the time iterates of the automaton
with initial distribution P_w --the law P conditioned to w on the left of the
origin-- converges to the uniform product measure on G^N. The proof uses a
regeneration representation of P.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/9912135.

1413. POISSON APPROXIMATION FOR LARGE-CONTOURS IN LOW-TEMPERATURE ISING MODELS

Pablo A. Ferrari, Pierre Picco

We consider the contour representation of the infinite volume Ising model at
low temperature. Fix a subset V of Z^d, and a (large) N such that calling
G_{N,V} the set of contours of length at least N intersecting V, there are in
average one contour in G_{N,V} under the infinite volume "plus" measure. We
find bounds on the total variation distance between the law of the contours of
lenght at least N intersecting V under the "plus" measure and a Poisson
process. The proof builds on the Chen-Stein method as presented by Arratia,
Goldstein and Gordon. The control of the correlations is obtained through the
loss-network space-time representation of contours due to Fernandez, Ferrari
and Garcia.

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/9912136.

1414. PROBABILITY LAWS RELATED TO THE JACOBI THETA AND RIEMANN ZETA FUNCTION AND BROWNIAN EXCURSIONS

P. Biane, J. Pitman, M. Yor

This paper reviews known results which connect Riemann's integral
representations of his zeta function, involving Jacobi's theta function and its
derivatives, to some particular probability laws governing sums of independent
exponential variables. These laws are related to one-dimensional Brownian
motion and to higher dimensional Bessel processes. We present some
characterizations of these probability laws, and some approximations of
Riemann's zeta function which are related to these laws.

philippe.biane@ens.fr

• This article is available from the xxx mathematics archive as math.PR/9912170.

1415. FRACTIONAL BROWNIAN MOTIONS AS HIGHER-ORDER FRACTIONAL DERIVATIVES OF BROWNIAN LOCAL TIMES

Endre Csaki, Zhan Shi and Marc Yor

Fractional derivatives ${\cal D}^\gamma$ of Brownian
local times are well defined for all $\gamma<3/2$. We
show that, in the weak convergence sense, these fractional
derivatives admit themselves derivatives which feature all
fractional Brownian motions. Strong approximation results
are also developed as counterparts of limit theorems for
Brownian additive functionals which feature the fractional
derivatives of Brownian local times.

csaki@math-inst.hu  zhan@proba.jussieu.fr  secret@proba.jussieu.fr

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

1416. GENERALIZATIONS OF BOLD PLAY IN RED AND BLACK

Marcus Pendergrass and Kyle Siegrist

The strategy of bold play in the game of red and black 
leads to a number of interesting mathematical properties: 
the player's fortune follows a deterministic map, before 
the transition that ends the game; the bold strategy can
be 're-scaled' to produce new strategies with the same win 
probability; the win probability is a continuous function 
of the initial fortune, and in the fair case, equals the 
initial fortune. We consider several Markov chains in more
general settings and study the extent to which these 
properties are preserved.  In particular, we study two 
'$k$-player' models.

marcus@hiwaay.net  siegrist@math.uah.edu

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

1417. A PARTIAL INTRODUCTION TO FINANCE

Philip Protter

We present an introduction to mathematical finance theory 
for mathematicians.  We present the basics of European call 
and put options, versions of the two fundamental theorems, 
invariance of the numeraire, and we show the connection
between American put options and backwards stochastic 
differential equations.  The approach is to start with an 
abstract setting and then introduce hypotheses as needed 
to develop the theory.

protter@math.purdue.edu

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

1418. A FLEMING-VIOT PARTICLE REPRESENTATION OF DIRICHLET LAPLACIAN

Krzysztof Burdzy, Robert Holyst and Peter March

We consider a model with a large number $N$ of particles which move 
according to independent Brownian motions. A particle which leaves 
a domain $D$ is killed; at the same time, a different particle splits
into two particles. For large $N$, the particle distribution
density converges to the normalized heat equation solution in $D$
with Dirichlet boundary conditions.
The stationary distributions converge as $N\to \infty$
to the first eigenfunction of the Laplacian in $D$ with
the same boundary conditions.

burdzy@math.washington.edu

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

1419. NARROW-BAND ANALYSIS OF NONSTATIONARY PROCESSES

Peter M. Robinson and D.Marinucci

The behaviour of the averaged periodograms and 
cross-periodograms of a broad class of nonstationary
processes is studied. The processes include nonstationary
ones that are fractional of any order, as well as
asymptotically stationary fractional ones, and the 
cross-periodogram can involve two nonstationary processes
of possibly different orders, or a nonstationary and an
asymptotically stationary one. The averaging takes 
place either over the whole frequency band, or on 
one that degenerates slowly to zero frequency as sample
size increases. In some cases it is found to make 
no asymptotic difference, and in particular we indicate 
how the behaviour of the mean and variance changes 
across the two-dimensional space of integration orders. 
The results employ only local-to-zero assumptions on the
spectra of the underlying weakly stationary sequences.

marinucc@scec.eco.uniroma1.it

1420. CENTRAL LIMIT THEOREMS FOR SOME GRAPHS IN COMPUTATIONAL GEOMETRY

Mathew D. Penrose and J. E. Yukich

We develop a general methodology for establishing
central limit theorems (CLT's) for functionals of graphs
in computational geometry. We prove a general CLT for
functionals of point sets, obtained by taking a large
sample from a uniform distribution on the unit cube
in d dimensions.  Using this general result we obtain
CLT's for the total edge length, the total number
of components, and the total number of edges of graphs
such as the k nearest neighbors graph, the sphere of 
influence graph, and the Voronoi graph.

mathew.penrose@durham.ac.uk  joseph.yukich@lehigh.edu

1421. A REMARK ON THE NOTION OF ROBUST PHASE TRANSITIONS

Aernout C.D. van Enter

We point out that the high-q Potts model on a regular 
lattice at its transition temperature provides an 
example of a non-robust -- in the sense recently proposed
by Pemantle and Steif-- phase transition

aenter@phys.rug.nl

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

1422. CONVEX GEOMETRIC INCLUSION-EXCLUSION IDENTITIES AND BONFERRONI INEQUALITIES WITH APPLICATIONS TO SYSTEM RELIABILITY ANALYSIS AND RELIABILITY COVERING PROBLEMS

Klaus Dohmen

This paper establishes a connection between the theory of 
convex geometries, the principle of inclusion-exclusion 
and the theory of abstract tubes. In particular, it is 
shown that convex geometries give rise to improved 
inclusion-exclusion identities and via abstract tubes to 
improved Bonferroni inequalities. Thus, several results 
from the literature are rediscovered in a concise and 
unified way. Our general results are applied in identifying 
a new class of hypergraphs for which the reliability 
covering problem can be solved in polynomial time.

Informatik-Bericht Nr. 132, HU Berlin, 1999.

dohmen@informatik.hu-berlin.de

1423. ASYMPTOTICS FOR LINEAR RANDOM FIELDS

Domenico Marinucci and Suren Poghosyan

We prove that partial sums of linear multiparameter
stochastic processes can be represented as partial sums
of independent innovations plus components that are
uniformly of smaller order. This representation is
exploited to establish functional central limit
theorems and strong approximations for random fields.

marinucc@scec.eco.uniroma1.it

1424. PROBABILISTIC APPROACH TO THE STRONG FELLER PROPERTY

Bohdan Maslowski and Jan Seidler 

A new probabilistic method, based on the Girsanov
theorem, for establishing the strong Feller property 
of diffusion processes in both finite and infinite 
dimensional spaces is proposed. Applications to second 
order stochastic differential equations, stochastic delay 
equations and stochastic partial differential equations of 
parabolic type are discussed, with a twofold aim: both to 
extend some older results, usually by
weakening the assumptions on the drift term, and to 
obtain simpler proofs of them.

maslow@math.cas.cz  seidler@math.cas.cz

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

1425. A FUNCTIONAL CENTRAL LIMIT THEOREM FOR MARKOV ADDITIVE PROCESSES

J.L. Steichen

A functional central limit theorem for a certain class of
time-homogeneous continuous-time Markov processes (X,Y) is 
proved. The process X is a positive recurrent Markov process 
on a countable-state space and the process Y has conditionally 
independent increments given X.  The pair (X,Y) is then called 
a Markov additive process.  This paper unifies and generalizes 
several functional central limit theorems for different types 
of Markov additive processes.  The variance parameter for the 
limit process is calculated and several examples, including an 
application to queueing theory, are given.

steichen@member.ams.org

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

1426. ALMOST SURE CONVERGENCE OF WEIGHTED SERIES OF CONTRACTIONS

Fakhr eddine Boukhari and Michel Weber

That paper is devoted to the study of properties almost sure convergence of
series of contractions (of an arbitrary Hilbert space) with random weights.
It is a continuation of a previous work Peskir-Schneider-Weber in which
only convergence in operator norm was investigated. We find conditions
ensuring the existence of universal sets on which these series are
converging almost everywhere, whatsoever the contraction. It is also a
continuation of work by Schneider-Weber in which an analog problem
concerning ergodic averages was considered. Some variants of the problem
are also examined. The proofs of the results rely on uniform estimates of
random polynomials established in a recent paper by the second named author
and proved by means of metric entropy methods.

boukhari@math.u-strasbg.fr      weber@math.u-strasbg.fr

1427. ESTIMATING RANDOM POLYNOMIALS BY MEANS OF METRIC ENTROPY METHODS

Michel Weber

The purpose of this paper is to indicate how easy, classical metric
entropy methods arising from the theory
of stochastic processes, apply to get uniform estimates for random
polynomials like the well-known
Salem-Zygmund's bound. As an application, we give a criterion for uniform
convergence of some random
Fourier series.

weber@math.u-strasbg.fr

1428. OCCUPATION TIME FLUCTUATIONS IN BRANCHING SYSTEMS

D.A. Dawson  L.G. Gorostiza and A. Wakolbinger

We consider particle systems in locally compact Abelian groups with
particles moving according to a process with symmetric stationary
independent increments and undergoing one and two levels of critical
branching. We obtain long time fluctuation limits for the occupation
time process of the one--and two--level systems. 
We give complete results for the case of finite variance branching, 
where the fluctuation limits are Gaussian random fields, and partial results
 for an example of infinite variance branching, where the fluctuation
limits are stable random fields. The asymptotics of the occupation
time fluctuations are determined by the Green potential operator $G$
of the individual particle motion and its powers $G^2, G^3$, and by the growth
as $t\rightarrow\infty$ of the operator $G_t=\int^t_0T_sds$ and its
powers, where $T_t$ is the semigroup of the motion. The results are
illustrated with two examples of motions: the symmetric 
$\alpha$--stable L\'evy process in $R^d$ $(0<\alpha\leq2)$, and the so 
called $c$--hierarchical random walk in the hierarchical group of 
order $N$ ($0<c<N$).  We show that the two motions have analogous 
asymptotics of $G_t$ and its powers that depend on an order parameter 
$\gamma$ for their transience/recurrence behavior. 
This parameter is $\gamma=d/\alpha-1$ for the $\alpha$--stable motion, 
and $\gamma=\log c/\log (N/c)$ for the $c$--hierarchical random walk.  
As a consequence of these analogies, the asymptotics of the occupation time 
fluctuations of the corresponding branching particle systems are also analogous.
In the case of the $c$--hierarchical random walk, however, the growth of 
$G_t$ and its powers is modulated by oscillations on a logarithmic time scale.

ddawson@fields.utoronto.ca  gortega@servidor.unam.mx wakolbin@math.uni-frankfurt.de

1429. A PATHWISE VERSION OF SPITZER'S LAW

Peter Moerters

Spitzer's law describes the long term asymptotics behaviour of the 
distribution of the winding number of a Brownian motion in the plane.
The pathwise result of this paper shows that an analogous behaviour can
be seen at every typical Brownian path by considering the winding number
at a random time chosen according to the logarithmic laws of order three.

peter@mathematik.uni-kl.de