Probability Abstracts 10

Click here to see the list of all abstract titles.

173. A DISCRETE FRACTAL IN Z^1

Davar Khoshnevisan

60J, 28A

In this paper, we show that the level sets of mean zero finite
variance random walks in R^1 form a discrete fractal in the sense
of Barlow and Taylor. Analogously to the Brownian motion results, the
Hausdorff dimension of the level sets in almost surely equal to 1/2.

davar@math.washington.edu (Plain TeX file available.)
		

174. SEMILINEAR STOCHASTIC EVOLUTION EQUATIONS WITH MONOTONE NONLINEARITIES

B. Z. Zangeneh

Semigroup approach is used to prove existence, uniqueness and  
boundedness of the solution of semilinear stochastic evolution  
equations with monotone nonlinearities. The existence and uniqueness  
theorem is based on Picard iteration together with results from the
the theory of deterministic semilinear evolution equations. The usual  
Granwall inequality arguments are carried out with the aid of a  
Burkholder type inequality and an Ito type "energy" inequality.
In addition diverse examples which have arisen in applications are  
shown to satisfy the hypotheses of the theorem and consequently the  
results can be applied to these examples.

zang@math.ubc.ca (Latex file available)
		

175. AN ITO--TYPE INEQUALITY AND A BURKHOLDER--TYPE INEQUALITY FOR STOCHASTIC CONVOLUTION INTEGRALS

Bijan Z. Zangeneh

 In this paper we will  prove An Ito-type"energy" inequality and a  
Burkholder-type inequality for stochastic convolution integral  
$\int_0^tU(t,s)dZ_s$ , where $U(t,s)$ is the evolution operator  
generated by $A(t)$ and $Z_t$ is an $H$-valued semimartingale and for  
each $t \in {\bf R}, A(t)$ is a closed unbounded linear operator on a  
Hilbert space which satisfies certain conditions. 
These  two theorems are the main tools for study of semilinear  
stochastic evolution equations with monotone nonlinearities. 

zang@math.ubc.ca (Latex file available)

176. PROBABILISTIC APPROACH TO THE DIRICHLET PROBLEM OF PERTURBED STABLE PROCESSES

Renming Song

In this paper $X_t$ is a symmetric stable process of index $\alpha$, $0<\alpha
<2$, on $R^d (d\ge2)$, with the characteristic function $e^{-t|z|^{\alpha}}$.
Let $F$ be a bounded function on $R^d\times R^d$ such that
(1) $F$ vanishes on the diagonal;
(2) for any $\epsilon >0$, there exists a $\delta>0$ such that $|F(x, y)|
<\epsilon$ whenever the distance between $(x, y)$ and the diagonal of $R^d
\times R^d$ is less than $\delta$;
(3) the function $\int\frac{|F(z, y)|}{|z-y|^{d+\alpha}}dy$ is bounded on $R^d$.

Put $A_t=\sum_{0<s\le t} F(X_{s-}, X_s)$.

In this paper we first proved the following gauge theorem.

GAUGE THEOREM. If $D$ is a bounded open set in $R^d$, then the gauge function
$g(x)=E^x\{E^{A(\tau_D)}\}$ is either identically infinite or bounded on $D$.

Then we proved that if $D$ is a bounded regular open set and $g$ is finite on 
$D$. Then for any $f\in \Cal D_e$ which is bounded continuous on $D^c$,
$u(x):=E^x\left(e(\tau_D)f(X(\tau_D))\right)$ is the unique continuous
solution to the Dirichlet problem of the following pseudo differential equation 
$-(-\Delta)^{\frac{\alpha}2}u(x)+\int_{R^d}\frac{e^{F(x, y)}u(y)}{|x-y|^{d+
\alpha}}dy=0$ in $D$ with the boundary function $f$.
 
song@math.ufl.edu

177. THE GAUGE THEOREM FOR A CLASS OF ADDITIVE FUNCTIONALS OF ZERO ENERGY

J. Glover, M. Rao and R. Song

Let $X_t$ be the standard Brownian motion in $R^d$ and let $u\in H^1(R^d)$ be 
a fixed bounded continuous function, then the additive functional $A_t$ defined
by $A_t=u(X_t)-u(X_0)-\int^t_0\nabla u(X_s)\cdot dX_s$ is a continuous additive
functional of zero energy. In general, $A_t$ is not of bounded variation. In 
this paper we proved the following gauge theorem for the additive functionals 
of the above type:

GAUGE THEOREM. Let $u$ be such that $|\nabla u|^2$ belongs to the Kato class. 
If $D$ is a bounded, connected open subset of $R^d$, then the function
$g(x)=E^x\{\exp(A(\tau_D))\}$ is either bounded or identically infinite
on $D$, where $\tau_D$ is the first exit time of $D$.

In this paper we also showed that the above result includes the "classical" 
gauge theorem as a special case.

song@math.ufl.edu

178. THE GENERALIZED SCHR\"ODINGER SEMIGROUPS

J. Glover, M. Rao and R. Song

This is a continuation of the paper above (PAS 177).

Suppose that $X_t$ the standard Brownian motion in $R^d$, that $u\in H^1(R^d)$
be a bounded continuous function such that $|\nabla u|^2$ belongs to the Kato 
class and $\mu$ is measure belonging to the Kato calss. Let $A^{[u]}_t$ be 
defined as $A^{[u]}_t=u(X_t)-u(X_0)-\int^t_0\nabla u(X_s)\cdot dX_s,$ and let 
$A^{\mu}_t$ be the continuous additive functional having $\mu$ as  its Revuz 
measure. Define $A_t=A^{[u]}_t+A^{\mu}_t$. The purpose of this paper is to 
study the following generalized Schr\"odinger semigroup $T_tf(x)=E^x\left\{
e^{A_t}f(X_t)\right\}$.

In this paper we proved that the semigroup $(T_t)$ has a  continuous symmetric
integral kernel $q(t, x, y)$ and we also proved that this kernel is bounded
on both sides by heat kernels. From this a lot important properties of the 
semigroup $T_t$ will follow.

song@math.ufl.edu

179. QUADRATIC FORMS CORRESPONDING TO THE GENERALIZED SCHR\"ODINGER SEMIGROUPS.

J. Glover, M. Rao, H. Sikic and R. Song

This is a continuation of the two papers above (PAS 177 and 178).

Suppose that $X_t$ the standard Brownian motion in $R^d$, that $\rho\in 
H^1(R^d)$ be a bounded continuous function such that $|\nabla\rho|^2$ belongs 
to the Kato class and $\mu$ is measure belonging to the Kato calss. Let
$A^{[\rho]}_t$ be defined as $A^{[\rho]}_t=\rho(X_t)-\rho(X_0)-\int^t_0
\nabla \rho(X_s)\cdot dX_s$, and let $A^{\mu}_t$ be the continuous additive 
functional having $\mu$ as its Revuz measure. Define $A_t$ as the sum of the two
additive functionals above, and $T_tf(x)=E^x\left\{e^{A_t}f(X_t)\right\}$.

In this paper we first proved that the bilinear function $E$ given by
$E(u, v)=\frac12\int \nabla u(x)\cdot\nabla v(x)dx+\int_{R^d}\nabla(uv)(x)\cdot
\nabla\rho(x)dx+\int_{R^d}u(x)v(x)\mu(dx)$ is finite for all $u$, $v\in 
H^1(R^d)$ and that $(E, H^1(R^d))$ is a lower semibounded, closed quadratic 
form. Then we proved that $(\tilde \Cal E, H^1(R^d))$ is the quadratic form 
corresponding to the generalized Schr\"odinger semigroup $T_t$.

song@math.ufl.edu

180. STOCHASTIC FLOWS ON THE BOUNDARIES OF SL(n,R)

Ming Liao

We study the asymptotic stability of the stochastic flows on a class 
of compact spaces induced by a diffusion process in the special linear
group SL(n,R) or the general linear group GL(n,R). These compact spaces 
are called the boundaries of SL(n,R), which include the orthogonal 
group O(n), the flag manifold, the sphere $S^{n-1}$ and the Grassmannians.
The one point motion of the flow is a Brownian motion.
For almost every $\omega$, we determine the set $\Lambda$ of stable 
points, which is an open set whose complement has zero lebesgue 
measure. The distance between any two points in the same component of
$\Lambda$ tends to zero exponentially fast under the flow. The Lyapunov
exponents at stable points are computed explicitly. The results are
applied to the stochastic flow on the sphere $S^{n-1}$ generated by 
a stochastic differential equation which exhibits some nice symmetry. 

mliao@auducvax.bitnet  

181. RATES OF CONVERGENCE OF MEANS FOR DISTANCE-MINIMIZING SUBADDITIVE EUCLIDEAN FUNCTIONALS

Kenneth S. Alexander

  Functionals L on finite subsets A of R^d are considered for which the value
is the minimum total edge length among a class of graphs with vertex set equal
to, or in some cases containing, A.  Examples include minimal spanning trees,
the traveling salesman problem, minimal matching, and Steiner trees. 
Beardwood, Halton, and Hammersley (1959) and Steele (1981) have shown
essentially that for {X_1,..,X_n} a uniform sample form [0,1]^d,
EL{X_1,..,X_n}) grows like b * n^(d-1)/d for some constant b.  It is shown
that the difference is O(n^(d-2)/d), as conjectured by Beardwood, Halton and
Hammersley.

alexandr@mtha.usc.edu

182. SOME PATH PROPERTIES OF ITERATED BROWNIAN MOTION

Krzysztof Burdzy

Let $X^1, X^2$ and $Y$ be independent standard Brownian 
motions starting from 0. Let 
$X(t) = X^1(t)$ if $t \geq 0$,
$X(t) = X^2(t)$ if $t < 0$,
$Z(t) = X(Y(t))$ for $t \geq 0$.
(A) With probability 1, the path $\{Z(t), t \geq 0\}$
corresponds to exactly 2 pairs of paths $(X,Y)$ and
$(\hat X, \hat Y)$ which are related by the following
conditions. $Y(t) = - \hat Y(t)$ for $t \geq 0$.
$X(t) = \hat X(-t)$ for $-\infty < t < \infty\}$.
(B) (LIL) With probability 1,
$$\limsup_{t\to 0} Z(t) [t^{1/4} (\log \log (1/t))^{3/4}]^{-1} 
= 2^{5/4}3^{-3/4}.$$

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

183. THE RADIAL PART OF BROWNIAN MOTION II: ITS LIFE AND TIMES ON THE CUT LOCUS.

Michael Cranston, Wilfrid S Kendall, and Peter March

Abstract:  This paper is a sequel to Kendall (1987), which explained
how the It\^o formula for the radial part of Brownian motion $X$ on a
Riemannian manifold can be extended to hold for all time including
those times at which $X$ visits the cut locus. This extension consists
of the  subtraction of a correction term, a continuous predictable
non-decreasing process $L$ which changes only when $X$ visits the cut
locus.  In this paper we derive a representation of $L$ in terms of
measures of local time of $X$ on the cut locus.  In analytic terms we
compute an expression for the singular part of the Laplacian of the
Riemannian distance function. The work uses a relationship of the
Riemannian distance function to convexity, first described by Wu (1979)
and applied to radial parts of $\Gamma$-martingales in Kendall (1992).
 
w.s.kendall@warwick.ac.uk

184. MARKOV FUNCTIONS OF A TIME-CHANGED RECURRENT DIFFUSION

Zoran Vondracek

Let $(X_t, P_x)$ be a recurrent diffusion on the state space $E$.
A necessary and sufficient condition on the continuous function
$u:E \rightarrow R$ is given so that $u$ is a Markov function for
a time-changed diffusion $X_{{\tau}_t}$. It is shown that no
non-constant continuous real-valued function is Markov for a Brownian
motion on the Sierpinski gasket.

zoran.vondracek@olimp.irb.ac.mail.yu

185. ARC-SINE LAWS FOR LEVY PROCESSES

R. K. Getoor and M. J. Sharpe

We prove the following arc-sine law for a
L\'evy process $X$ on the real line.
Let $F_c$ denote the generalized arc-sine law on $[0,1]$ with
parameter $c$. Then $\int_0^t P^0(X_s>0)ds/t\to c$ as
$t\to\infty$ is a necessary and sufficient condition for
$\int_0^t 1_{\{X_s>0\}}ds/t$ to converge in $P^0$ law to $F_c$.
Moreover, $P^0(X_s>0)=c$ for all $t>0$ is a
necessary and sufficient condition for $\int_0^t
1_{\{X_s>0\}}ds/t$ under $P^0$ to have  law $F_c$
for all $t>0$. The proof makes no use of Spitzer's
corresponding theorem for random walks. In fact, we show how
to derive Spitzer's theorem in a very simple way from the
L\'evy process version. We also use the theorem to derive
distributional limit theorems for functionals of the form
$\int_0^t f(X_s)ds/t$. 

rgetoor@ucsd.edu or mjsharpe@ucsd.edu  (AMS TeX file available)

186. A DISCRETE ANALOGUE OF A THEOREM OF MAKAROV

Gregory F. Lawler

A theorem of Makarov states that the harmonic measure (from infinity) of a
compact, connected subset of $R^2$ is supported on a set of Hausdorff dimension
one. This paper gives an analogue of this theorem for discrete harmonic
measure, i.e., the hitting measure of simple random walk.  It is proved that
for any $1/2 < \alpha < 1$, $0 < \beta < \alpha - 1/2$, there is a constant
$k = k(\alpha,beta)$ such that for any connected subset $A$ of $Z^2$ of
radius $n$,
$$ H_A \{x : n^{-1} e^{-(\ln n)^\alpha} \leq H_A(x) 
   \leq n^{-1} e^{(\ln n)^\alpha} \} \geq 1 - k(\ln n)^{-\beta}, $$
where $H_A$ denotes harmonic measure.

jose@math.duke.edu