Probability Abstracts 4

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60. INTERNAL DIFFUSION LIMITED AGGREGATION

M. Bramson, D. Griffeath, and G. Lawler

We study the asymptotic shape of the occupied region for an interacting
lattice system proposed recently by Diaconis and Fulton.  In this model
particles are repeatedly dropped at the origin of the $d$-dimensional
integers.  Each successive particle then performs an independent simple
random walk until it ``sticks'' at the first site not previously
occupied.  Our main theorem asserts that as the cluster of stuck
particles grows, it shape approaches a Euclidean ball.  The proof of
this result involves Green's function asymptotics, duality, and large
deviation bounds.  We also quantify the time scale of the model,
establish close connections with a continuous-time variant, and pose
some challenging problems concerning more detailed aspects of the
dynamics.

jose@math.duke.edu (Greg Lawler)

61. MODELLING RANDOM FLUCTUATIONS IN A BISTABLE TELECOMMUNICATIONS NETWORK

P.K. Pollett

In this paper I shall consider a model for the simplest kind of dynamic 
routeing in a circuit-switched telecommunications network, namely Random 
Alternative Routeing: if a call cannot be carried on a first-choice route, 
then a second-choice route is chosen at random from a fixed set of alternatives.
This kind of routeing can give rise to several modes of behaviour. For example,
the simple model I shall consider can exhibit bistability; the system fluctuates
between a low-blocking state, where calls are accepted readily, and a 
high-blocking state, where the likelihood of a call being accepted can be 
quite low. I shall describe a method which allows one to study the stability
of the two states. In particular, the method allows one to estimate the time 
for which these states persist.

pkp@markov.maths.uq.oz.au  (AmSTeX (Ver 3) file available)

62. ON THE RELATIONSHIP BETWEEN $mu$-INVARIANT MEASURES AND QUASISTATIONARY DISTRIBUTIONS FOR CONTINUOUS-TIME MARKOV CHAINS

M.G. Nair and P.K. Pollett

In a recent paper, van Doorn (1991) explained how quasistationary distributions 
for an absorbing birth-death process could be determined from the transition
rates of the process, thus generalizing earlier work of Cavender (1978).
In particular, he showed that a given probability distribution is 
quasistationary if and only if it is a convergent $mu$-invariant measure for
the transition rates. In this paper we shall examine the extent to which van 
Doorn's results can be extended to an arbitrary continuous-time Markov chain 
over a countable state-space consisting of an irreducible class, $C$, and an 
absorbing state, $0$, which is accessible from $C$. Some of the results we 
shall present are extensions of theorems proved for {\it honest\/} chains in 
Pollett and Vere-Jones (1991).

In Section 3 we shall show that a probability distribution on $C$ is a quasi-
stationary distribution if and only if it is a $mu$-invariant measure for the 
transition function, $P$. Then, we shall establish a relationship between 
$mu$-invariant measures for $P$ and $mu$-invariant and $mu$-subinvariant 
measures for the transition rates, $Q$, from which we shall conclude that if 
$m$ is quasistationary for $P$, then a necessary and sufficient condition 
for $m$ to be strictly $mu$-invariant for $Q$ is that $P$ satisfies the 
Kolmogorov forward equations over $C$. When the remaining forward equations 
hold, the quasistationary distribution must satisfy a set of ``residual 
equations'' involving the transition rates into the absorbing state; it was 
one of the insights of van Doorn's paper that, for birth-death proceses, these 
equations play a crucial role. The residual equations allow us to deduce that 
the value of $\mu$ for which the quasistationary distribution is $mu$-invariant 
for $P$ is uniquely determined as the ratio of the (conditional) probability 
flux into state $0$ and the probability that the chain is eventually absorbed. 
This conclusion is a corollary of more general results which establish bounds 
on the values of $\mu$ for which a {\it convergent\/} measure can be a 
$mu$-subinvariant and then $mu$-invariant measures for $P$. The remainder of 
the paper is devoted to the question of when a convergent $mu$-subinvariant 
measure, $m$, for $Q$ is a quasistationary distribution.

Section 4 establishes that a necessary and sufficient condition for $m$ to be 
a quasistationary distribution for the minimal chain is that $m$ satisfies the 
residual equations. In Section 5 we consider ``single-exit'' chains, a class 
which includes all birth-death processes. We derive a necessary and sufficient
condition for there to exist a process for which $m$ is a quasistationary
distribution. Under this condition {\it all\/} such processes can be specified 
explicitly through their resolvents. The results proved here allow us to 
conclude that the bounds for $\mu$ obtained in Section 3 are, in fact, tight.

mgn@axiom.maths.uq.oz.au, pkp@markov.maths.uq.oz.au 
(AmSTeX (Ver 3) file available)

63. LOCAL FIELD BROWNIAN MOTION

Steven N. Evans

   A local field is any locally compact, non-discrete field other
than the field of real numbers or the field of complex numbers.
There is a natural notion of Gaussian measures on a local
field vector space.  We construct and study a specific local
field Gaussian stochastic process taking values in a finite
dimensional local field vector space and indexed by another
finite dimensional local field vector space.  This process 
has a structure that strongly reflects the algebraic and 
geometric structure of the underlying index space and,
as such, plays the same role in the local field setting that
standard Brownian motion and the related multiparameter processes
 such as L\'evy's multiparameter Brownian motion play in a Euclidean 
context.  

   We investigate the theory of `additive functionals' 
and the related potential theory for this process and show that it
strongly resembles the Euclidean prototype. As a particular 
consequence of this investigation, we find that a local time process 
exists when the process hits points.  We give two intrinsic 
constructions of the local time at a given level.  These 
constructions are analogous to the dilation construction of Kingman 
and the Hausdorff measure construction of Taylor and Wendell in the 
Euclidean case.  Finally, the local time is shown to be continuous 
as a measure valued stochastic process indexed by the level at which 
it is evaluated.

evans@stat.berkeley.edu

64. INVARIANTS OF SOME PROBABILITY MODELS USED IN PHYLOGENETIC INFERENCE

Steven N. Evans and T.P. Speed

   The so-called method of invariants is a technique in the 
field of molecular evolution for inferring phylogenetic 
relations among a number of species on the basis of 
nucleotide sequence data.  An invariant is a polynomial 
function of the probability distribution defined by a 
stochastic model for the observed nucleotide sequence.  
This function has the special property that it is 
identically zero for one possible phylogeny and typically 
non-zero for another possible phylogeny.  Thus it is 
possible to discriminate statistically between two 
competing phylogenies using an estimate of the invariant.  
The advantage of this technique is that it enables such 
inferences to be made without the need for estimating 
nuisance parameters that are related to the specific 
mechanisms by which the molecular evolution occurs.
  
   For a wide class of models found in the literature, 
we present a simple algebraic formalism for recognising 
whether or not a function is an invariant and for 
generating all possible invariants.  Our work is based 
on recognising an underlying group structure and then 
using discrete Fourier analysis and the theory of random
walks on groups. 

evans@stat.berkeley.edu

65. COLLISION LOCAL TIMES AND MEASURE-VALUED PROCESSES

Martin T. Barlow, Steven N. Evans and Edwin A. Perkins

   We consider two independent Dawson-Watanabe super-Brownian 
motions, $Y^1$ and $Y^2$. These processes are diffusions 
taking values in the space of finite measures on $R^d$.
We show that if $d \le 5$ then with positive probability 
there exist times $t$ such that the closed supports of 
$Y_t^1$ and $Y_t^2$ intersect; whereas if $d>5$ then no 
such intersections occur.  For the case $d \le 5$, we 
construct a continuous, non-decreasing measure-valued
process $L(Y^1 ,Y^2)$, the ``collision local time'', 
such that the measure defined by 
$[0,t] \times B \mapsto L_t (Y^1 ,Y^2)(B)$, $B \in B ( R^d )$,
is concentrated on the set of times and places at which 
intersections occur. We give a Tanaka-like semimartingale 
decomposition of $L(Y^1 ,Y^2)$.  We also extend these results 
to a certain class of coupled measure-valued processes.
This extension will be important in a forthcoming paper where 
we use the tools developed here to construct coupled pairs of 
measure-valued diffusions with ``point interactions''.
In the course of our proofs we obtain smoothness results for 
the random measures $Y_t^i$ that are uniform in $t$.
These theorems use a nonstandard description of $Y^i$ and are
of independent interest.

evans@stat.berkeley.edu

66. NON-POLAR POINTS FOR REFLECTED BROWNIAN MOTION

Krzysztof Burdzy and Donald Marshall

For each measurable function $\theta: R \to [-\pi/2, \pi/2]$
a reflected Brownian motion in a half-plane with the oblique
angle of reflection $\theta$ is constructed. Some results
on points which may be hit by the process with positive
probability are given. A related theorem describes a new family
of "exceptional" points on paths of standard 2-dimensional
Brownian motion.

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

67. DIRICHLET FORMS ON SEPARABLE METRIC SPACES

Ichiro Shigekawa and Setsuo Taniguchi

The general theory of Dirichlet forms on locally compact state spaces has been 
developed deeply by Fukushima, Silverstein and others.  Recently various 
investigations on Dirichlet forms on infinite dimensional (hence non-locally 
compact) topological vector spaces, and much attempts to extend the general 
theory on locally compact spaces to such spaces have been made.  Our goal is to
present a general theory of Dirichlet forms on metric spaces, following the 
work by Fukushima. The investigations are made from analytic and probabilistic 
points of view and we unify them.
 
Let $X$ be a separable Lusinian metric space and $B(X)$ be its topological 
Borel field.   We fix an everywhere dense probability measure $m$ on $(X,B(X))$
and investigate a Dirichlet form $(E,F)$ on $L^2(X;m)$ with $1\in F$. 
A key hypothesis we made is that the associated 1-capacity is tight.
With assuming a few more assumptions, we have discussed on the following topics:
 1. Measures of finite energy integral and potentials.
 2. Equilibrium potentials.
 3. Beurling-Deny formula.
 4. Associated Hunt process and probabilistic interpretation of the Beurling- 
    DEny formula.
 5. Smooth measures and associated additive functionals.
 6. Time change and closable part of pre-Dirichlet form.
 
f77273a@kyu-cc.cc.kyushu-u.ac.jp

68. ON NETWORK STRUCTURE FUNCTION COMPUTATIONS

Edward C. Waymire

Some mathematical problems involving the asymptotic analysis
of rooted random tree graphs and branching patterns for large
numbers of vertices are described.  The motivation comes from
applications to hydrology and geomorphology in which one seeks 
formulae for the average behavior of various contours (level sets)
associated with river networks and which depend on only a few large
scale parameters.  While this paper is mainly an overview, a new
formula is derived in a special case as an illustration.  Scaling
problems and connections with Aldous's new theory of "self-similar"
continuum random trees are noted.

waymire@math.orst.edu (PlainTex)

69. A LARGE DEVIATION RATE AND CENTRAL LIMIT THEOREM FOR HORTON RATIOS

Stanley Xi Wang, Edward C. Waymire

Although originating in hydrology, the classical Horton analysis
is based on a geometric progression which is widely used in the
empirical analysis of branching patterns found in biology, atmospheric
science, plant pathology, etc., and more recently in tree register
allocation in computer science.  The main results of this paper are
a large deviation rate and a central limit theorem for Horton bifurcation
ratios in a standard network model.  The methods are largely self-contained.
In particular, derivations of some previously known results of the theory  
are indicated along the way.

waymire@math.orst.edu (SIAMTex)

70. MULTIFRACTAL DIMENSIONS & SCALING EXPONENTS FOR STRONGLY BOUNDED RANDOM CASCADES

Richard Holley, Edward C. Waymire

The multifractal structure of a measure refers to some notion of
dimension of the set which supports singularities of a given order
\alpha as a function of the parameter \alpha. Measures with a non-
trivial multifractal structure are commonly referred to as multi-
fractals.  Multifractal measures are being studied both theoretically
and empirically within the statistical theory of turbulence and in the
study of strange attractors of certain dynamical systems.  Conventional
wisdom suggests that various definitions of the multifractal structure
of random cascades exist and coincide.  While this is rigorously known
to be the case for certain deterministic cascade measures, the same is
not true for random cascades.  The purpose of this paper is to pursue
this theory for a class of random cascades.  In addition to providing
a new role for the modified cumulant generating function (structure
function) studied by Mandelbrot, Kahane, and Peyriere, the results have
implications for the theoretical interpretation of empirical data on
turbulence and rainfall distributions.

waymire@math.orst.edu (PlainTex)

71. BRANCHING RANDOM WALKS ON TREES

N. Madras and R. Schinazi

Let p(x,y) be the transition probability of an isotropic random
walk on a homogeneous tree where each site has d neighbors. We define a
branching random walkby letting a particle at site x give birth to a new
particle at site y at rate lambda d p(x,y), jump to y at rate nu d p(x,y),
and die at rate delta (lambda, nu, delta are positive parameters). For
fixed nu, delta, p(x,y), let lambd_2 (respectively mu_2) be the infimum
of lambda such that the process starting with one particle has positive
probability of surviving forever (respectively of having a fixed site
occupied at arbitrarily large times). We compute lambda_2 and mu_2 as
functions of nu, delta and p(x,y) proving that the branching walk on
the tree has two phase transitions in the sense that lambda_2<mu_2 (for
d larger or equal to 3). We also prove that the second phase transition has
a discontinuity.

RBSCHINAZI@COLOSPGS.BITNET

72. RANDOM STATIONARY PROCESSES

Kenneth S. Alexander and Steve Kalikow

Given a finite alphabet, there is an inductive method for constructing a
stationary measure on doubly infinite words from this alphabet.  This
construction can be randomized; the main focus in this work is on a
particular uniform randomization which intuitively corresponds to the idea of
choosing a generic stationary process.  It is shown that with probability
one, the random stationary process has zero entropy and gives positive
probability to every periodic infinite word.  Simulations suggest that
aperiodic words may also occur with positive probability.

alexandr@mtha.usc.edu

73. FINITE CLUSTERS IN HIGH-DENSITY CONTINUOUS PERCOLATION: COMPRESSION AND SPHERICALITY

Kenneth S. Alexander

A percolation process in d dimensions is considered in which the sites are a
Poisson process with intensity s and the bond between each pair of sites is
open if and only if the sites are within a fixed distance r of each other. 
The distribution of the number of sites in the cluster C of the origin is
examined.  It is shown that when s and k are large, there is a characteristic
radius R such that conditionally on |C| = k, the convex hull of C closely
approximates a ball of radius R, with high probability.  When the normal
volume k/s that k points would occupy is small, the cluster is compressed, in
that the number of points per unit volume in this R-ball is much greater than
the ambient density s.  For larger normal volumes there is less compression.

alexandr@mtha.usc.edu

74. ASYMPTOTIC BEHAVIOR OF BROWNIAN TOURISTS

R.T. Durrett and L.C.G. Rogers 

Let B_t be a Brownian motion and f be Lipschitz continuous. We consider
processes of the form

X_t = B_t +  int_0^t ds  int_0^s du  f(X_s-X_u)

where f(x) = g(x)x/|x| and g is real valued. We can think of X_t as the
trajectory of a tourist who wants to stay away from places she has visited
before or of X_t as describing a growing polymer in which newly added
units are repelled by existing ones. With this in mind one would like to
understand the rate of growth of X_t. In the physically uninteresting case
f(x) >= 0 we are able to prove X_t is of order t for compactly supported f
but we do not know how to show that X_t/t converges. We obtain precise
asymptotics when f(x) behaves like x^{-b} with 0<b<1.

rtd@CornellA.bitnet or  lcgr@maths.qmw.ac.uk

75. THE EXPONENT FOR MEAN SQUARE DISPLACEMENT OF SELF-AVOIDING RANDOM WALK ON SIERPINSKI GASKET

T. Hattori and S. Kusuoka

    We prove that the exponent for the mean square displacement of 
self-avoiding random walk on the Sierpinski Gasket is 2p, where
2p=2log(2)/log((7-sqrt(5))/2).

    More precisely, let $P_{n}$ be the probability measure
on the self-avoiding path on the pre-Sierpinski Gasket
(the SG with unit minimum length in structure)
of length $n$ and starting from $O$, with each path in equal weight.
Then for any $s>0$,
\[ \lim_{n\rightarrow\infty}(\log n)^{-1} (\log E_{n}[|w(n)|^{s}]) = ps \].
Here $E_{n}[.]$ is the expectation value with respect to $P_{n}$,
and $|w(n)|$ is the distance in Euclidean metric between the 
initial and final points of the path $w$.

hattori%tansei.cc.u-tokyo.ac.jp

76. LARGE DEVIATIONS OF MONTGOMERY TYPE AND ITS APPLICATION TO THE THEORY OF ZETA-FUNCTIONS

T. Hattori and  K. Matsumoto

Let $X_{1},X_{2},...$ be i.i.d. random variables with uniform 
distribution on $[0,1]$.  Put $X=\sum r_{n} \cos(2\pi X_{n})$,
where $r_{1},r_{2},...$ is a sequence of non-negative real numbers
with infinitely many non-zero terms, satisfying $\sum r_{n}^{2} < \infty$.
Montgomery has shown that there exist positive constants $A,B,C$ such that
\[ Prob ( X > A \sum_{n \leq N} r_{n} )
> B \exp(-C (\sum_{n \leq N} r_{n})^{2} (\sum_{n>N} r_{n}^{2})^{-1} ),\]
for any positive integer $N$, if $(r_{n})$ is a monotonically decreasing
sequence.
    In the present paper, we give a necessary and sufficient condition 
on $({r_{n})$ for such estimates to hold.  As an application,
we prove a lower bound of the asymptotic probability measure for the 
value-distribution of zeta-functions attached to certain cusp form.

hattori@tansei.cc.u-tokyo.ac.jp

77. ASYMPTOTIC VARIANCE PARAMETERS FOR THE BOUNDARY LOCAL TIMES OF REFLECTED BROWNIAN MOTION ON A COMPACT INTERVAL

R. J. Williams

A direct derivation is  given of a formula  for the normalized asymptotic 
variance parameters of the boundary local times of reflected Brownian motion 
(with drift) on a compact interval. This  formula  was previously obtained  
by Berger and Whitt using an M/M/1/C queue approximation to the reflected 
Brownian motion.  The bivariate Laplace transform of the hitting time of a 
level and the boundary local time up to that hitting time, for a 
one-dimensional reflected Brownian motion with drift, is obtained as part 
of the derivation. 

rjwilliams@ucsd.bitnet