Probability Abstracts 34

711. RAZUMIKHIN-TYPE THEOREMS ON EXPONENTIAL STABILITY OF NEUTRAL STOCHATIC DIFFERENTIAL EQUATIONS

Xuerong Mao

In our previous paper (Systems and Control Letters 26 (1995), 245--251) 
we initiated the study of exponential stability of neutral stochastic
functional differential equations. In this paper we shall further our 
study in this direction.  We should emphasize that the main technique employed in this 
paper is the well-known Razumikhin argument and is completely different from 
those used in our previous paper (1995). The results obtained there can only 
be applied to a certain class of neutral stochastic functional differential 
equations excluding neutral stochastic differential delay equations, but the 
results obtained in this paper are more general, especially can be used to
deal with neutral stochastic differential delay equations. Moreover, in 
paper (1995) we only studied the exponential stability in mean square 
but in this paper we shall also study the almost sure exponential
stability. It should be pointed out that although the results established
in this paper are applicable to more general neutral type equations, 
for a  particular type of equations discussed in paper (1995) the results there 
are sharper.  

This paper will appear in SIAM J. Math. Anal.  
Compressed postscript file available at
ftp://ftp.stams.strath.ac.uk/incoming/raznfde.ps.Z

xuerong@stams.strath.ac.uk

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712. STOCHASTIC SELF-STABILIZATION

Xuerong Mao

Given a nonlinear It\^o equation  $dx(t) = f(x(t),t) dt + u g(x(t), t) dw(t)$, 
suppose that the equation is almost surely exponentially stable provided 
the noise intensity  $u$ is large enough.  We show that if $u$ is replaced
by  $u = \int_0^t |r(s)x(s)|^p ds$, then the equation stabilizes itself in 
the sense that $\int_0^\8 |r(t)x(t)|^p dt < \8$ a.s.  Moreover, if
 $u$ is replaced by  $u = \sup_{0\le \s \le t} |r(s)x(s)|$, then the
equation stabilizes itself in the sense that $\sup_{0\le t < \infty}
|r(t)x(t)| < \8$ a.s.  By choosing various convergence rate functions $r(t)$, 
one can stabilize the system in different ways. For example, if let 
$r(t)=e^t$, one can stabilize the system almost surely exponentially.

This paper will appear in Stochastics and Stochastics Reports.

xuerong@stams.strath.ac.uk  (compressed ps file available)

713. THE SMALLEST SETS HIT BY A BROWNIAN IMAGE

Christopher J. Bishop and Yuval Peres

We show that For any analytic set $A$ on the positive axis, the infimum of the
Hausdorff dimensions of compact  sets in $R^d$ which are hit by the Brownian 
image $B_d(A)$ with positive probability, is $d-2\dimp(A)$.  Here $B_d$ is 
Brownian motion in $R^d$ with $d \geq 2$, and  $\dimp$ is packing dimension.
We also prove an analogous variational formula concerning Brownian preimages: 
 For any analytic set $K$ on the line, the infimum of the Hausdorff dimensions 
of compact sets $A$ on the positive reals that intersect the Brownian preimage
of $K$ with positive probability is  $(1-\dimp(K))/2$. Thus packing dimension 
arises naturally even if one is initially only interested in Hausdorff 
dimension. Extensions to symmetric stable processes are obtained as well.

peres@math.huji.ac.il

714. WIENER'S TEST FOR SUPER-BROWNIAN MOTION AND THE BROWNIAN SNAKE

Jean-St\'ephane Dhersin and Jean-Fran\c{c}ois Le Gall

We prove a Wiener-type criterion for the super-Brownian motion and the 
Brownian snake. If $F$ is a Borel subset of ${\bf R}^d$ and $x \in {\bf R}^d$, 
we provide a necessary and sufficient condition for super-Brownian motion 
started at $\delta_x$ to immediately hit the set $F$. Equivalently, this 
condition is necessary and sufficient for the hitting time of $F$ by the 
Brownian snake with initial point $x$ to be 0. A key ingredient of the proof 
is an estimate showing that the hitting probability of $F$ is comparable, 
up to multiplicative constants, to the relevant capacity of $F$. This 
estimate, which is of independant interest, refines previous results due 
to Perkins and Dynkin. An important role is played by additive functionals of 
the Brownian snake, which are investigated here via the potential theory of 
symmetric Markov processes. As a direct application of our probabilistic 
results, we obtain a necessary and sufficient condition for the existence 
in a domain $D$ of a positive solution of the equation $\Delta u = u^2$ 
which explodes at a given point of $\partial D$.

dhersin@proba.jussieu.fr, gall@ccr.jussieu.fr

715. IDENTIFICATION OF FILTERED WHITE NOISES AND OF ELLIPTIC GAUSSIAN RANDOM PROCESSES

Albert Benassi, Serge Cohen, Jacques Istas, and St\'ephane Jaffard

In this paper , two classes of Gaussian Processes having locally the
same fractal properties as Fractional Brownian Motion are studied. Our aim
is to give estimators of the relevant parameters of these processes from one
sample path. In the first class (i.e. Filtered White Noises), a time
dependency of the integrand of the classical Wiener Integral associated to  the
Fractional Brownian Motion is introduced. We show how to identify the
asymptotic expansion  for high frequencies of these integrands  on one
sample path. Then  the identification of the first terms of this expansion
is used to solve some ``filtering'' problems. Furthermore rates of
convergence of the estimators are then given. In the second class (i.e.
Elliptic Gaussian Random Processes), spatial modulations in the symbol of
the pseudo-differential operator characterizing the reproducing spaces of
the processes is introduced. At last identification results for 
the symbol of the Elliptic Gaussian Random Processes are deduced from
the identification results for Filtered White Noises. 

cohen@fermat.math.uvsq.fr  

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716. SMOOTHING THE MOMENT ESTIMATOR OF THE EXTREME VALUE PARAMETER

Sidney Resnick and C\u{a}t\u{a}lin St\u{a}ric\ua

Let $\{X_{n}\}$ be a sequence of i.i.d. random variables whose common
cdf F belongs to the domain of attraction of an extreme value
distribution.  A frequently used estimator of the extreme
value parameter is the moment estimator (Dekkers, Einmahl and de Haan, 1989).
Because the moment estimator is a function of
$k=k(n)$, the number of upper order statistics used in estimation and
which is only subject to the conditions $k\to\infty, k/n\to 0$, its
use in practice is problematic since there are few reliable guidelines
about how to choose k.  An
averaging technique is proven to reduce the asymptotic variance of
this estimator. As a direct consequence of the averaging the
range in which the smoothed estimator varies as a function of $k$
decreases and the successful use of the estimator is made less
dependent on the choice of $k$.

Available as TR1158 at the url below.

sid@orie.cornell.edu

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717. ELLIPTIC REGUALRITY AND ESSENTIAL SELF-ADJOINTNESS OF DIRICHLET OPERATORS ON R^n

Vladimir I. Bogachev, Nicolai V. Krylov, Michael R\"ockner

We prove a new regularity result for operators of type $L=\Delta+B\bdot \nabla +
c$, on open sets $\Omega \subset \bbbr^n$, provided $B:\Omega\to \bbbr^n$,
$c:\Omega\to \bbbr$ satisfy mild integrability conditions. As a consequence we
prove that $L=\Delta +\nabla \log \rho\,\bdot\, \nabla $ with domain $C_0^\infty
(\bbbr^n)$ is essentially self-adjoint on $L^2(\bbbr^n,\rho dx)$, if $\rho\in
H_{loc}^{1,1}(\bbbr^n)$  and $\nabla \log \rho \in L^\gamma_{loc}(\bbbr^n,dx)$ 
for some $\gamma > n$.

(SFB-343-Bielefeld-Preprint-96-038)

roeckner@mathematik.uni-bielefeld.de

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718. APPROXIMATION OF ARBITRARY DIRICHLET PROCESSES BY MARKOV CHAINS

Zhi-Ming Ma, Michael R\"ockner, Tu--Sheng Zhang

We prove that any Hunt process on a Hausdorff topological space
associated with a Dirichlet form can be approximated by a Markov chain in a
canonical way. This also gives a new and ``more explicit'' proof for the
existence of Hunt processes associated with strictly quasi-regular Dirichlet
forms on general state spaces.  

(SFB-343-Bielefeld-Preprint-96-)

roeckner@mathematik.uni-bielefeld.de

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719. SYMMETRIC INFINITELY DIVISIBLE PROCESSES WITH SAMPLE PATHS IN ORLICZ SPACES AND ABSOLUTE CONTINUITY OF INFINITELY DIVISIBLE PROCESSES

Michael Braverman and Gennady Samorodnitsky

We give necessary and sufficient conditions under which a symmetric
measurable infinitely divisible process has sample paths in an Orlicz space 
with a function  satisfying the Delta_2 condition
and, as an application, obtain necessary and sufficient conditions for
a symmetric  infinitely divisible process to have a version with absolutely 
continuous paths. 

Available at http://www.orie.cornell.edu/~gennady as TR1162.ps.Z ,
or by ftp from ftp.orie.cornell.edu as TR1162.ps.Z .

gennady@orie.cornell.edu 

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720. A MAXIMAL INEQUALITY FOR SUMS OF EXCHANGEABLE RANDOM VARIABLES

Alexander R. Pruss

Let $X_1,X_2,...,X_{2n}$ be exchangeable Banach space valued random 
variables, i.e., random variables such that all joint distributions 
are invariant under permutations of the variables. We prove that there 
is an absolute constant $c$ such that if $S_j=\sum_{i=1}^j X_i$, then
$$
P(\sup_{1\le j\le 2n} \| S_j \| > \lambda) \le c P(\| S_n \| > \lambda/c),
$$
for all $\lambda\ge 0$. This generalizes an inequality of Lata\l{}a
and Montgomery-Smith for independent and identically distributed
random variables. As a corollary, we obtain a comparison inequality
for tail probabilities of sums of arbitrary random variables over
random subsets of the indices.

pruss@math.ubc.ca

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721. A BOUNDED N-TUPLEWISE INDEPENDENT AND IDENTICALLY DISTRIBUTED COUNTEREXAMPLE TO THE CLT

Alexander R. Pruss

A sequence of random variables $X_1,X_2,X_3,...$ is said to
be $N$-tuplewise independent if $X_{i_1},X_{i_2},...,X_{i_N}$ are
independent whenever $(i_1,i_2,...,i_N)$ is an $N$-tuple of distinct
positive integers. For any fixed $N\in Z^+$, we construct a sequence
of bounded identically distributed $N$-tuplewise independent random
variables which fail to satisfy the central limit theorem.

pruss@math.ubc.ca

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722. COMPARISONS BETWEEN TAIL PROBABILITIES OF SUMS OF INDEPENDENT SYMMETRIC RANDOM VARIABLES

Alexander R. Pruss

We show how estimates for the tail probabilities of sums of
independent identically distributed random variables can be used to
estimate the tail probabilities of sums of not necessarily
identically distributed independent symmetric random variables which
are majorized by a single distribution in the sense of Gut's (1992)
weak mean domination. As an application, we prove a weak one-sided
extension of a law of large numbers of Chen (1978) to a
non-identically distributed case and show how some of Gut's (1992)
extensions of laws of large numbers of Hsu-Robbins type from
identically distributed cases to non-identically distributed cases
automatically follow from the identically distributed cases in light
of our main inequality. We also show that results of Klesov (1993)
extend to the case of weak mean domination.

pruss@math.ubc.ca

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723. A TWO-SIDED ESTIMATE IN THE HSU-ROBBINS-ERDOS LAW OF LARGE NUMBERS

Alexander R. Pruss

Let $X_1,X_2,...$ be independent identically distributed
random variables. Then, Hsu, Robbins and Erdos (1947, 1949) have
proved that
$$
S(\epsilon)=\sum_{n=1}^\infty P(|X_1+\dots+X_n| \ge \epsilon n) <
\infty, \qquad \forall\epsilon>0,
$$
if and only if $E[X_1^2]<\infty$ and $E[X_1]=0$. We prove that there
are absolute constants $C_1,C_2 \in (0,\infty)$ such that if
$X_1,X_2,...$ are independent identically distributed mean zero
random variables, then
$$
C_1 \epsilon^{-2} E[X_1^2\cdot 1_{\{ |X_1| \ge \epsilon\}}] \le
S(\epsilon) \le C_2 \epsilon^{-2} E[X_1^2 \cdot 1_{\{|X_1|\ge
\epsilon\}}],
$$
for every $\epsilon>0$.

pruss@math.ubc.ca

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724. THE RANDOM MINIMAL SPANNING TREE IN HIGH DIMENSIONS

Mathew Penrose

For the minimal spanning tree on $n$  independent uniform points
in the $d$-dimensional unit cube, the proportionate number of points
of degree $k$ is known to converge to a limit $\alpha_{k,d}$ as
$n \to \infty$. We show that  $\alpha_{k,d}$ converges to a limit
$\alpha_k$   as $d \to \infty$, for each $k$.  The limit $\alpha_k$
arose in earlier work by Aldous, as the asymptotic proportionate number
of vertices of degree $k$ in the minimum-weight spanning tree on $k$ vertices,
when the  edge-weights are taken to be independent, indentically distributed
random  variables.  We give a graphical alternative to Aldous's
characterisation of the $\alpha_k$.

Mathew.Penrose@durham.ac.uk

725. EXTREMES FOR THE MINIMAL SPANNING TREE ON NORMALLY DISTRIBUTED POINTS

Mathew Penrose

Let $n$ points be placed indepenently in $d$-dimensional space
according to the standard $d$-dimensional normal distribution.
Let $M_n$ be the longest edge-length of the minimal spanning tree
on these points; equivalently let $M_n$ be the infimum of those $r$
such that the union of balls of radius $r/2$ centred at the points
is connected.  We show that the distribution of $(2 \log n)^{1/2} M_n - b_n$
converges weakly to the Gumbel (double exponential) distribution,
where  $b_n$ are explicit constants with $b_n \sim (d-1) \log \log n$.
We also show the same result holds if $M_n$ is the
longest edge-length for the nearest neighbour graph on the points.

Mathew.Penrose@durham.ac.uk

726. OPTIMAL HEDGING STRATEGIES FOR MISSPECIFIED ASSET MODELS

Hyungsok Ahn, Adviti Muni and Glen Swindle

In this paper we construct optimal hedging strategies when the
volatility of the asset price is misspecified. The Black-Scholes
option pricing methodology requires that the model for the price
of the underlying asset be completely specified.  Often the
underlying price is taken to be a geometric Brownian motion
with a constant, known volatility.  In practice one does not
know precise values of parameters such as the volatility, and
estimates from historical prices or implied volatilities must be
used instead.  The strategies obtained here are optimal in the
$H^\infty$-control context--- namely, maximizing the utility of
the investor in a worst-case volatility scenario.

swindle@orie.cornell.edu    (hardcopy and postscript available)

727. PROBLEMS SUGGESTED IN THE WORKSHOP ON "DISCRETE PROBABILITY AND PROBLEMS FROM PHYSICS", JERUSALEM 1996.

Compiled by Yuval Peres

This is a list of unsolved problems concerning percolation and related 
probabilistic models with origin in statistical physics or computer science.

PROBLEM POSERS:  K. Alexander, N. Alon, I. Benjamini and O. Schramm, 
C. Borgs and J. Chayes, G. Grimmett, O.  Haggstrom, C. Kenyon, R. Kenyon, 
H. Kesten, N. Linial, R. Lyons, R. Pemantle, Y. Peres, D. Randall, 
L. Schulman,  J. Steif, O. Zeitouni,  Y. Zhang and U. Zwick.

This list is available in postscript at
http://www.ma.huji.ac.il/~peres/surveys.html

peres@math.huji.ac.il

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728. SPECTRAL GAP FOR AN UNRESTRICTED KAWASAKI TYPE DYNAMICS

Gustavo Posta

We give an accurate asymptotic estimate for the gap of the generator of
a particular interacting particle system.  The model we consider  
may be informally described as follows. A certain number of charged 
particles moves on the segment $[1,L]\cap\natural$ according to 
a Markovian law.  If  $\h_k\in\integer$ is the charge at a
site $k\in [1,L]\cap\natural$ one unitary charge, positive or negative, 
jumps to a neighboring site, $k\pm1$ at a rate which
depends on the charge at site $k$ and at site $k\pm1$.
The total charge $\sum_{k=1}^L\h_k$ is preserved by the dynamics,
in this sense our dynamics is similar to the Kawasaki dynamics, 
but in our case there is no restriction on the maximum charge allowed per site.
The model is equivalent to an interface dynamics connected with the 
stochastic Ising model at very low temperature:
the ``unrestricted solid on solid model''.
Thus the results we obtain may be read as results for this model.
We give necessary and sufficient conditions to ensure that
gap shrinks as $L^{-2}$, independently of the total charge.
We follow the method outlined in some papers by Yau 
where a similar spectral gap is proved for the original Kawasaki dynamics.

postagus@mat.uniroma1.it    (TeX file available)

729. CONSTRUCTION OF MARKOVIAN COALESCENTS

Steven N. Evans and Jim Pitman 

Partition-valued and measure-valued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass $m$ into a 
finite or countably infinite number of masses with sum $m$, and whose evolution is determined by the following intuitive prescription: each pair of masses of 
magnitudes $x$ and $y$ runs the risk of a binary collision to form a single 
mass of magnitude $x+y$ at rate  $\kappa(x,y)$, for some nonnegative, symmetric collision rate kernel $\kappa(x,y)$.  Such processes with finitely many masses 
have been used to model polymerization, coagulation, condensation, and the 
evolution of galactic clusters by gravitational attraction.  With a suitable 
metric on the state space, and under appropriate restrictions on $\kappa$ and 
the initial distribution of mass, it is shown that such processes can be 
constructed as Feller or Feller-like processes. 
A number of further results are obtained for the {\em additive coalescent} 
with collision kernel $\kappa(x,y) = x + y$. This process, which arises from the
evolution of tree components in a random graph process, has
asymptotic properties related to the stable subordinator of index $1/2$.

evans@stat.Berkeley.EDU    pitman@stat.Berkeley.EDU  

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730. ON $L_p$-ALGEBRAIC CONVERGENCE AND $L_p$-EXPONENTIAL CONVERGENCE FOR MARKOV PROCESSES

Xin Sheng Zhang

Exponential convergence to equilibrium in the $L_2$ sense for Markov
processes has been studied by many authors. Recently, Liggett introduced
the algebraic convergence in the $L_2$ sense for Markov Processes.(see,
The Annals of Probbility Vol. 19, No. 3, 935-959) This work is concerned
with the relationship betwwen these two kinds of covergence in the $L_p$
sense. Under very mild conditions, we prove that a Markov process 
$X_t$ converges exponentially to its equilibrium in the $L_p$ sense if 
and only if $X_t$ converges algebraicly to its equilibrium in the $L_p$ 
sense and the order of $L_p$-algebraic convergence is great then $1$.

xszhang@mailserv.stu.edu.cn

731. ON THE CRITICAL VALUE FOR PERCOLATION OF MINIMUM-WEIGHT TREES IN THE MEAN-FIELD DISTANCE MODEL

David J Aldous

Consider the complete $n$-graph with independent exponential (mean $n$)
edge-weights.  Let $M(c,n)$ be the maximal size of subtree for which the 
average edge-weight is at most $c$.
It is shown that $M(c,n)$ transitions from $o(n)$ to $\Omega(n)$
around some critical value $c(0)$, which can be specified in terms
of a fixed point of a mapping on probability distributions.

aldous@stat.berkeley.edu

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732. ON GENERATORS OF INFINITE DIMENSIONAL DIFFUSIONS

A. Chojnowska-Michalik and B. Goldys

We give sufficient conditions for the generators of semilinear Hilbert
space valued diffusions to be continuously imbedded into some Sobolev and
Orlicz spaces built on the Gaussian invariant measure of its linearized
version. As a byproduct we derive a purely analytic
version of the Girsanov theorem for such diffusions. We give also
necessary and sufficient conditions for the generator 
of a nonsymmetric Hilbert space-valued Ornstein-Uhlenbeck process to
satisfy the Logarithmic Sobolev Inequality. 

beng@alpha.maths.unsw.edu.au

733. RANDOM MINIMAL SPANNING TREE AND PERCOLATION ON THE N-CUBE

Mathew Penrose


The $N$-cube is a graph with $2^N$ vertices and $N2^{N-1}$ edges.
Suppose independent uniform random edge-weights are assigned,
and let $T$ be the spanning tree of minimal (total) weight.
Then the weight of $T$ is asymptotic to
$N^{-1}2^N \sum_{i=1}^{\infty}i^{-3}$ as $N \to \infty$.
Asymptotics are also given for the local structure of $T$, 
and for the distribution of its $k$-th largest edge-weight,
$k$ fixed.

Mathew.Penrose@durham.ac.uk

734. ON THE QUASI-REGULARITY OF SEMI-DIRICHLET FORMS

P. J. Fitzsimmons

We prove that if a right Markov process is associated with a semi-Dirichlet
form, then the form is necessarily quasi-regular. As applications, we
develop the theory of Revuz measures in the semi-Dirichlet context and we
show that quasi-regularity is invariant with respect to time change.

pfitzsim@ucsd.edu {Hard copy, plainTex file, and postscript file available}

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735. MARKOV PROCESSES WITH EQUAL CAPACITIES

P. J. Fitzsimmons

Let X and X* be right Markov processes in weak duality with respect to
a sigma-finite measure m. Let (Y,Y*,mu) be a second ``dual pair''
with the same state space E as (X,X*,m).  Let Cap_X and Cap_Y be the
0-order capacities associated with (X,X*,m) and (Y,Y*,mu), and let V
and V* denote the (0-order) potential kernels for Y and Y*. Assuming that
singletons are polar with respect to both X and Y, and that semipolar sets
are of capacity zero (for either dual pair) we show that if
Cap_X(B)=Cap_Y(B) for every Borel subset of E then there is a strictly
increasing  continuous additive functional D(t), t\ge 0, of (X,X*,m) such
that
(1) U_D(x,dy)+U*_D(x,dy)=V(x,dy)+V*(x,dy),
with the exception of capacity-zero set of x's, where U_D (resp.
U*_D) is the potential kernel of the time-changed process X(D^{-1}(t))
(resp. \hat X*(D^{-1}(t)) ), t\ge 0.  In particular, if both X and Y are
symmetric processes, then the equality of the capacities Cap_X and Cap_Y
implies that X and Y are time changes of one another. The derivation of (1)
rests on a generalization of a formula of Choquet concerning the
``differentiation'' of capacities. In the symmetric case, our main result
extends a theorem of Glover, Hansen and Rao.

pfitzsim@ucsd.edu {Hard copy, plainTex file, and postscript file available}

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