Probability Abstracts 20

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318. RATE OF EXPLOSION OF THE AMPEREAN AREA OF THE PLANAR BROWNIAN LOOP

Wendelin Werner

We study the asymptotic behaviour of approximations of the Amperean area
(i.e. the integral on the plane of the squared index function of the loop)
which is almost surely infinite. This is related to some statistical
physics problems in which this quantity appears.

W.J.J.Werner@statslab.cam.ac.uk (Tex-file available)

319. ON LEVY'S UNIFORM LAW FOR BRIDGES

Frank B. Knight


Let $X_t$ be a real L\'evy process, $X_0=0$, and let either 
(a) $Y_t=X_t - tX_1$, $0\leq t\leq 1$, or (b) $Y_t = (X_t\mid X_1=0)$
(under appropriate extra assumptions). If $X_t$ is a Brownian motion,
L\'evy showed that both $S(0)(:= \int^1_0 I_{(-\infty,0)} (Y_s)ds)$
and AM(Y) ($:=$ unique argument of maximum) are uniformly distributed
on $(0,1)$. We give simple necessary and sufficient conditons for the
same two conclusions under (a), and also under (b) if we assume
Condition (C) of $0$. Kallenberg (that the characteristic function is
integrable for $t>0$). Generally speaking the answer is:  ``always,
except in exceptional cases.''

britt@symcom.math.uiuc.edu

320. ON A DECOMPOSITION OF ADDITIVE FUNCTIONALS IN THE STRICT SENSE FOR A SYMMETRIC MARKOV PROCESS

M. Fukushima

For a symmetric Markov process, the decomposition theorem of the 
associated additive functional into its martingale part and energy zero
part has been formulated admitting an exceptional set of starting points
of zero capacity due to its reliance on the potential theory of the
Dirichlet form.  Under the assumption of the absolute continuity of the 
transition probability, we present a necessary and sufficient condition
on the energy measure for the decomposition to be refined to a strict
one holding for every starting point.

fuku@sigmath.osaka-u.ac.jp

321. LOAD BALANCING IN INFINITE NETWORKS

Bruce Hajek

A set of locations and a set of consumers are given, and to each consumer there
corresponds a subset of the locations.  Each consumer has a demand, which
is a load to be distributed among the locations corresponding to the consumer.
The load at a location is the sum of the loads placed on the location by all
consumers.  Consumers assign load in such a way that no single consumer can
reassign its load to make the total load more balanced.
The model is motivated by the performance study of dynamic
allocation algorithms in large systems.

The set of possible balanced load vectors is examined for infinite
networks with deterministic or random demands.  The balanced load
vector is shown to be unique for rectangular lattice networks, and
a method for computing the load distribution is explored for tree
networks.  An FKG type inequality is proved.  The concept of load
percolation is introduced and is shown to be associated with infinite
sets of locations with identical load.

b-hajek@uiuc.edu

322. MARTIN CAPACITY FOR MARKOV CHAINS AND RANDOM WALKS IN VARYING DIMENSION

Itai Benjamini, Robin Pemantle, and Yuval Peres

The probability that a transient Markov chain, or a Brownian path, 
will ever visit a given set $\Lambda$, is classically estimated using
the capacity of $\Lambda$ with respect to the Green kernel $G(x,y)$.
We show that  replacing the Green kernel by the Martin kernel 
 $G(x,y)/G(0,y)$ yields
improved estimates, which are exact up to a factor of 2. These estimates
are applied to random walks on lattices, and also to explain a connection
found by R. Lyons between capacity and percolation on trees. Similar
methods are used to establish  recurrence criteria for 
sums of independent random variables which take values in Euclidean lattices
of varying dimension.

peres@stat.berkeley.edu  (LaTeX file available)

323. LABYRINTH DIMENSION OF BROWNIAN TRACE

• Click here to see the paper.

Krzysztof Burdzy

The labyrinth dimension of a set K is defined
as the supremum of Hausdorff dimensions of self-avoiding
continuous paths contained in K. The labyrinth dimension
of the trace X[0,1] of 2-dimensional Brownian motion X
is equal to 2.

burdzy@math.washington.edu (AmSTeX file available)

324. THE FAVORITE POINT OF A POISSON PROCESS

Davar Khoshnevisan  and Thomas M. Lewis

We consider the most visited site, X(t), of 
a Poisson process up to time t. Our point of 
departure from the literature on maximal spacings 
is our asymptotic analysis of where the maximal 
spacings occur (i.e., the size of X(t)) and not 
the size of the spacings. In particular, we provide
a complete desrcription of the growth of the favorite 
points process, X(t).

davar@math.utah.edu

325. SUR LA LOI DU LOGARITHME ITERE DE CHUNG POUR LE MOUVEMENT BROWNIEN ITERE (Chung's law of the iterated logarithm for iterated Brownian motion)

Davar Khoshnevisan and Thomas M. Lewis

Let X and Y be independent Brownian motions. For 
any positive t define Z(t)=X(|Y(t)|). In this 
paper, we prove the analogue of Chung's law of 
the iterated logarithm for the process, Z.

tlewis@frmnvax1.bitnet

326. PERCOLATION OF LEVEL SETS FOR TWO-DIMENSIONAL RANDOM FIELDS WITH LATTICE SYMMETRY

Kenneth S. Alexander and Stanislav A. Molchanov

 Percolation methods are used to analyze the sizes of the connected
components of level sets {x: Q(x) = h} and sets {x: Q(x) <= h} for several
classes of random fields Q on R^2 with lattice symmetry.  In typical cases
there is a sharp transition at a critical value of h from exponential
boundedness for such components to the existence of an unbounded component. 
In some examples, however, there is a nondegenerate interval of values of h
where components are bounded but not exponentially so, and in other cases
each level set may be single infinite line which visits every region of the
lattice.

alexandr@mtha.usc.edu

327. APPROXIMATION OF SUBADDITIVE FUNCTIONS AND CONVERGENCE RATES IN LIMITING-SHAPE RESULTS

Kenneth S. Alexander

   For a nonnegative subadditive function h on Z^d, with limiting
approximation g(x) = lim h(nx)/n, it is of interest to obtain bounds on the
discrepancy between g(x) and h(x), typically of order |x|^a for some a < 1. 
We give a general method for obtaining such bounds when h(x) is an expected
value or probability associated with an optimal random path from 0 to x. 
First, a more natural and seemingly weaker result is established:  every x is
in a bounded multiple of the convex hull of the set of sites satisfying a
similar bound.  Then we show that this convex-hull property implies the
desired |x|^a bound for all x.  Applications include rates of convergence in
limiting-shape results for first-passage percolation (standard and oriented)
and longest common subsequences, and bounds on the error in the
exponential-decay approximation to the off-axis connectivity function for
subcritical Bernoulli bond percolation on the integer lattice.

alexandr@mtha.usc.edu

328. WIENER'S TEST WITH APPLICATION TO `THORNS' FOR THE DIFFUSION ON $H^{2d+1}$ GOVERNED BY KOHN'S LAPLACIAN

K. Bruce Erickson

In the Heisenberg group $H^{2d+1}$ we can define a degenerate 
diffusion process $W$ with It\^o differential $dW=(dZ,\,dT)$, 
where Z is a d-dimensional complex Brownian motion and $dT = 
2\Im \{Z\cdot d{\bar Z}\}$.  This diffusion process, governed 
by the subelliptic operator known as Kohn's Laplacian, has left 
invariant transition probabilities.   We derive the analogue of 
Wiener's test for the regularity of the origin for a given set 
in $H^{2d+1}$ and apply it to find an itegral test for regularity 
of the origin for a thorn symmetric about the $t$-axis.

erickson@math.washington.edu

329. THE DISCONNECTION EXPONENT FOR SIMPLE RANDOM WALK

Gregory F. Lawler and Emily E. Puckette

Let $S(t)$ denote a simple random walk in $Z^2$ with integer time
$t$.  The disconnection exponent $\overline{\gamma}$ is defined by
saying the probability that the path of $S$ starting at $0$
stopped at the circle of radius $n$ disconnects $0$ from
infinity decays like $n^{-\overline{\gamma}}$.  We prove
that the disconnection exponent is well-defined and equals the
disconnection exponent for Brownian motion which is known to
exist.

jose@math.duke.edu    (LaTex file available)

330. CONCEPTUAL PROOFS OF L LOG L CRITERIA FOR MEAN BEHAVIOUR OF BRANCHING PROCESSES

Russell Lyons, Robin Pemantle, and Yuval Peres.


The Kesten-Stigum Theorem is a fundamental criterion for the rate
of growth of a supercritical branching process, showing that an $L \log L$ 
condition is decisive. In critical and subcritical cases, 
results of Kolmogorov and later authors
give the rate of decay of the probability that the
process survives at least $n$ generations.  
We  give conceptual proofs of these theorems based on comparisons
of Galton-Watson measure to another measure on the space of trees.
This approach also explains Yaglom's
exponential limit law for conditioned critical branching processes via
a simple characterization of the exponential distribution.

rdlyons@silver.ucs.indiana.edu    (plain-Tex file or hardcopy available)