Probability Abstracts 6

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100. ON A SYMMETRY OF TURBULENCE

Scott Peckham and Ed Waymire

The spectrum of singularities of a measure is a dimension curve
$f(\alpha)$ which measures the size of the set of singularities
of order $\alpha.$  A new symmetry in the empirical singularity
spectrum for certain turbulence data is first identified. Then
necessary and sufficient conditions on the distribution of the
cascaded random variables in a class of random cascades are
obtained which yield a symmetric singularity spectrum.

waymire@math.orst.edu (Plain TeX file available)

101. TRIMMED SUMS OF CERTAIN I.I.D. BANACH SPACE VALUED RANDOM VECTORS

Makoto Maejima

60F

Limit laws for trimmed sums of triangular arrays of i.i.d. Banach space 
valued random vectors are studied.  It is shown that if the array belongs 
to the domain of attraction of an infinitely divisible law on a Banach space 
of type 2 without Gaussian component and if the random vectors are symmetric 
or the Banach space is a separable Hilbert space, then the trimmed sum
converges weakly to a nondegenerate random vector.
 
maejima@math.keio.ac.jp

102. EIGENSTRUCTURE OF THE INFINITELY-MANY-NEUTRAL-ALLELES DIFFUSION MODEL.

S. N. Ethier

The complete set of eigenvalues is found for the (unlabeled)
infinitely-many-neutral-alleles diffusion model.  The
transition density for the process, originally derived by
Griffiths, is rederived as an eigenfunction expansion.

ethier@math.utah.edu  (plain TeX file available)

103. THE TRANSITION FUNCTION OF A FLEMING--VIOT PROCESS.

S. N. Ethier and R. C. Griffiths

Let $S$ be a compact metric space, let $\theta \ge 0$, and let 
$\nu_0$ be a Borel probability measure on $S$.  An explicit
formula is found for the transition function of the
Fleming--Viot process with type space $S$ and mutation
operator $(Af)(x) = {1 \over 2}\theta \int_S (f(\xi)-f(x))\,
\nu_0(d\xi)$.

ethier@math.utah.edu  (plain TeX file available)
apm466b@monu1.cc.monash.edu.au  (griffiths)

104. THE TRANSITION FUNCTION OF A MEASURE-VALUED BRANCHING DIFFUSION WITH IMMIGRATION.

S. N. Ethier and R. C. Griffiths

Let $S$ be a compact metric space, let $\theta \ge 0$,
let $\nu_0$ be a Borel probability measure on $S$, and let
$\lambda$ be real.  An explicit formula is found for the
transition function of the measure-valued branching diffusion
with type space $S$, immigration intensity $\theta/2$,
immigrant-type distribution  $\nu_0$, and criticality
parameter $\lambda$.  If $\lambda > 0$, the formula shows that
the process is strongly ergodic.

ethier@math.utah.edu  (plain TeX file available)
apm466b@monu1.cc.monash.edu.au  (Griffiths)

105. A HAUSDORFF MEASURE CLASSFICATION OF G-POLAR SETS FOR SUPERDIFFUSION

Yuan-Chung Sheu

Our purpose is to give Hausdorff measure criteria for
determining which subsets of d+1-dimensional space are G-polar relative to
superdiffusions.We find a critical dimension e such that if the restricted
Hausdorff dimension of E is smaller(larger)than e,then E is(not) a G-polar set.

The restricted Hausdorff measure R-h-m(E) associated with a positive monotone
increasing function h has been introduced by S.J.Taylor and
N.A.Watson[Math.Proc.Camb.Phil.Soc.(1985)97,325] to investigate the class of
sets in d+1-dimensional
space which are not hit,a.s.,by the graph of the d-dimensional Brownian motion.
The definition of the R-h-m(E) is similar to the definition of the Hausdorff
measure but the balls of radius r are replaced by the products of d intervals
of length r(in space variable) and one interval of legth square of r(in time
variable).We investigate analogous problems for superdiffusions with parameters
(L,a) where L is an elliptic differential operator and a=2 or a is between 1
and 2(See,e.g.,Dynkin,Probab.Th.and Related Fields,90,1-36(1991))
A set E is called G-polar if it is not hit,a.s.,by the graph G of super-
diffusion. We use an analytic characterization of the class of all G-polar sets
given by E.B.Dynkin (Superdiffusions and Parabolic Nonlinear Differential Equa-
tions,to appear in Annals of Probab.) and "a non-linear potential theory"
developed by Maz'ya and Havin (Russians Mathematical Surveys,27(1972)) to prove
the following
 Theorem1
     A. The statement " If E is G-polar,then R-h-m(E)=0" is true under the
        following conditions on h :
        (1) h(0)=0
        (2) h(2r)/h(r) is bounded on an interval (0,c)
        (3) (h(r) to the power a-1) times (r to the power 1-d(a-1))is integrable
            over (0,c) for some c smaller than 1.
     B. The statement "If R-h-m(E)=0 then E is G-polar" is true in the following
        two cases
        (1) d is larger than 2/(a-1) and h(r)= r to the power d-2/(a-1)
         or
        (2) d=2/(a-1) and h(r)=(1/(-log r)) to the power 1/(a-1).

The restricted Hausdorff dimension R-dim(E) is defineed in terms of the
restricted Hausdorff measures in the same way as the Hausdorff dimension
H-dim(E) is defined in terms of the Hasdorff measures.

   Theorem 2. Suppose that d is larger than k(a)=2/(a-1).Then a set E is G-polar
   if d-(R-dim(E)) is larger than k(a) and it is not G-polar if d-(R-dim(E)) is
   smaller than k(a). If d is smaller than k(a),then there exists no non-empty
   G-polar set.[If d=k(a) then G-polarity can be characterized in terms of the
   restricted Carleson logarithmic dimension].

sheu@mssun7.msi.cornell.edu

106. ON REFLECTING DIFFUSION PROCESSES

Zheng-Qing Chen

Let $G$ be a general bounded $d$-dimensional Euclidean domain,
$H^1 (G)$ the set of $L^2$
function whose gradient (defined in the distribution sense) is square
integrable,
and let
$\|f\|^2_1 = {1\over 2} \int_G \| \nabla f \|^2 dx +
\int_G \|f\|^2 dx$.
There is a  compact separable Hausdorff space $\widetilde G$ containing $G$ as
a dense open subset and with the subspace topology on $G$ agreeing with the
original Euclidean topology on $G$ such that $H^1(G) \cap C(\widetilde G)$ is
$\| \cdot \|_1$ dense in $H^1 (G)$ and uniformly dense in $C(\widetilde G )$.
Under the condition that there exists an increasing sequence of
smooth subdomains with union $G$ such that their surface measures are
uniformly bounded, the surface measure $\sigma$ and the unit inward normal
to the boundary $\partial \widetilde G =  \widetilde G - G$ are defined
and they coincide with the usual definitions when $G$ is a Lipschitz domain.
It is shown that $\sigma$ does not charge polar sets and that the Green's
formula is valid on $\widetilde G$.
Let $\widetilde X$ be the continuous strong Markov process associated with the
regular Dirichlet space $(H^1 (G),\ \cal E)$ on $\widetilde G$ with
symmetrizing measure $\rho dx$.
Here for $f, g$ in $H^ 1 (G), \  {\cal E} (f,  g)$ is given by
${1\over 2} \int_G \sum_{i,j=1}^d {{\partial f} \over {\partial x_i}} (x)
a^{ij} (x) {{\partial g} \over {\partial x_j}} (x) \rho (x) dx $, where
$(a^{ij} (x))$ is a symmetric $d  \times d$  matrix with a positive constant
$\lambda $ such that $\lambda^{-1} I \leq (a^{ij} (x)) \leq \lambda I,
\ a.e. $ on $G$ and where $\rho$ is a bounded positive function
which is bounded away from zero.
Let $X =i (\widetilde X)$ be the projection of $\widetilde X$ on $
\overline G$, the Euclidean closure of $G$. Under the condition that
$\{ a^{ij},1
\leq i,j \leq d \}$ and $\rho$ are functions in $H^1 (G)$, the Skorokhod
decomposition for the reflecting diffusion process $X$ is derived under each
$P_x$ except possibly for $x$ in a polar subset of $\widetilde G$.

c31801zc@wuvmd.bitnet (EXP (Ver 2) file available)

107. PSEUDO JORDAN DOMAINS

Zhen-Qing Chen

The manifold metric between two points in a planar domain is
the minimum length of curves in the domain connecting these two points.
We define a bounded simply connected planar domain to be a pseudo Jordan
domain if its boundary under the manifold metric is homeomorphic to the
unit circle. Then the boundary correspondence theorem for conformal mappings
can be shown to hold for pseudo Jordan domains.
It is shown that for any point $x$ in a
pseudo Jordan domain $G$ and for any point $x$ on the boundary modulo a polar
set, the reflecting Brownian motion starting at $x$
can be constructed and has the expected sample path behavior.

c31801zc@wuvmd.bitnet (EXP (Ver 2) file available)

108. ON REFLECTED DIRICHLET SPACES

Zhen-Qing Chen

Reflecting diffusion processes on smooth Euclidean domains
are well understood. M.L. Silverstein has developed two variant procedures
for constructing the reflected processes for a general class of symmetric
transient Hunt processes
from a Dirichlet space point of view. A direct approach is
given in this paper and these two variant procedures are shown to yield
the same result. Only the techniques of martingales and standard Markov
processes are used.

c31801zc@wuvmd.bitnet (EXP (Ver 2) file available)

109. SOME APPLICATIONS OF QUASI-BOUNDEDNESS FOR EXCESSIVE MEASURES

P. J. Fitzsimmons and R. K. Getoor

Let \xi and m be excessive measures for a right Markov process X and let
Q_\xi and Q_m be the associated stationary Kuznetsov processes. We show
that if \xi and m are harmonic, then Q_\xi is absolutely continuous with respect
to Q_m if and only if \xi is quasi-bounded by m in the sense that 
\xi=\sum_k\xi_k where each term in the sum is an excessive measure dominated by 
m. This result allows us to describe the Lebesgue decomposition of Q_\xi 
relative to Q_m and to give an explicit formula for the Radon-Nikodym derivative
dQ_\xi/dQ_m in case  Q_\xi is absolutely continuous with respect to Q_m.  As a 
second application of quasi-boundedness, we obtain a general form of a theorem 
of U. Kuran, in which regularity for the Dirichlet problem is characterized by 
the quasi-boundedness of a suitable excessive measure. 

pfitzsim@ucsd.edu (plain.TeX file available)

110. FOURIER INVERSION FOR MULTI-DIMENSIONAL CHARACTERISTIC FUNCTIONS

Mark A. Pinsky

A density function $f(x), x \in {\bf R}^n$ is said to be {\it piecewise 
smooth}   if for each $x \in {\bf R}^n$, the mean value function $r 
\rightarrow M_r\,f(x) :=\int_{{\bf S}^{n-1}} f(x +r \omega)d\omega  $ is piecewise
$C^{\infty}$ with compact support. ($d\omega$ is normalized surface measure 
on the unit sphere).The Fourier transform is $\hat f(\mu)
= \int_{{\bf R}^n} f(x) e^{i\langle \mu,x \rangle}dx$ with spherical partial sum
$f_R(x) = (2\pi)^{-n}\int_{|\mu|\le R}  \hat f(\mu)e^{-i \langle \mu, x 
\rangle}d\mu.$

{\bf Theorem}. For such $f$,
$\lim_{R\uparrow \infty} f_R(x) = M_{0^+}f(x) $ if and only 
if $r\rightarrow M_rf(x)$ has $k = [{(n-3) \over 2}] $ continuous derivatives.
($[\,]$ = integer part). Otherwise we have 
$\liminf_{R \uparrow \infty} R^{-\nu}[f_R(x) - M_{0^+}f(x)] <0<
\limsup_{R \uparrow \infty} R^{-\nu}[f_R(x) - M_{0^+}f(x)]$ 
where $\nu \ge 0$ is uniquely determined. Note that if $n=1,2$ the conditions
are automatically satisfied.

m_pinsky@math.nwu.edu

111. INTERSECTION LOCAL TIME AND TANAKA FORMULAS

Richard F. Bass and Davar Khoshnevisan

A new approach to intersection local times of Brownian motion
is given, using additive functionals of a single Markov process and 
stochastic calculus. Many of the known results are obtained in a 
systematic way.  New results include the Tanaka formula for the $k$--multiple 
points of self--intersection local time and the joint H\"older continuity in
all variables of renormalized self--intersection local time for $k$--multiple
points, $k \geq 4$. We also obtain analogous results for elliptic diffusions.

bass@math.washington.edu  (Plain Tex)

112. THE RADIAL PART OF A $Gamma$-MARTINGALE AND A NON-IMPLOSION THEOREM

Wilfrid S. Kendall

An upper bound is given for the behaviour of the radial part of a
$\Gamma$-martingale, generalizing previous work of the author on the
radial part of Riemannian Brownian motion. This upper bound is applied
to establish an integral curvature condition to determine when
$\Gamma$-martingales cannot ``implode'' in finite intrinsic time,
answering a question of Emery and generalizing work of Hsu on the
$C_0$-diffusion property of Brownian motion.

w.s.kendall@warwick.ac.uk

113. ONSAGER MACHLUP FUNCTIONALS FOR A CLASS OF NON TRACE CLASS SPDE'S

Eddy Mayer Wolf and Ofer Zeitouni

An Onsager Machlup functional limit is derived for a class of SPDE's
whose principal part is not trace class. Both nondegenerate and degenerate
limits are obtained, and are illustrated by examples. The proof uses FKG type
inequalities.

zeitouni@ee.technion.ac.il

114. ON THE COVERING TIME OF A DISC BY SIMPLE RANDOM WALK IN TWO DIMENSIONS

Gregory F. Lawler

Let $T_n$ be the number of steps needed by a simple random walk in two
dimensions to cover all points within the disc of radius $n$.  This
paper improves bounds on the distribution of $T_n$ be proving that
for all $t > 0$,
\[  e^{-4/t} \leq \liminf P\{\ln T_n \leq t(\ln n)^2 \}
    \leq \limsup P\{\ln T_n \leq t(\ln n)^2\} \leq e^{-2/t}. \]
The methods also improve the lower bound for the covering time for a
n x n torus by simple random walk by showing that the covering time
is bounded below by (2/\pi) n^2 (\ln n)^2 .  

jose@math.duke.edu

115. HARNACK INEQUALITIES AND DIFFERENCE ESTIMATES FOR RANDOM WALKS WITH INFINITE RANGE

Gregory F. Lawler
Thomas W. Polaski

Difference estimates and Harnack inequalities for mean zero, finite
variance random walks with infinite range are considered.  An example
is given to show that such estimates and inequalities do not hold for
all mean zero, finite variance walks.  Conditions are then given under
which such results can be proved.

jose@math.duke.edu

116. ON THE SPEED OF CONVERGENCE IN FIRST-PASSAGE PERCOLATIOM

Harry Kesten

\subjclass Primary 60K35, Secondary 60F05, 60F10

We consider the standard first--passage percolation problem on
Z^d: {t(e):e an edge of Z^d} is an i.i.d. family
of random variables with common distribution F.
a_{0,n} := inf{\sum^k_1 t(e_i):(e_1,...,e_k) a path on
Z^d from 0 to  n\xi_1}, where \xi_1 is
the first coordinate vector.  We show that \sigma^2(a_{0,n}) \leq C_1n
and that P{|a_{0,n} - Ea_{0,n}| \geq x\sqrt{n}} \leq C_2\exp(-C_3x)
for x \leq C_4n and for some constants 0 < C_i < \infty.  It is known
that \mu := \lim \frac 1n Ea_{0,n} exists.  We show also that C_5n^{-1}
\leq Ea_{0,n} - n\mu \leq C_6n^{5/6}(\log n)^{1/3}.  There are
corresponding statements for the roughness of the boundary of the set
\tilde B(t) := {v:v a vertex of Z^d which can be
reached from the origin by a path (e_1,...,e_k) with
\sum t(e_i) \leq t}.

hak@cornella.cit.cornell.edu

117. INEQUALITIES FOR THE TIME CONSTANT IN FIRST-PASSAGE PERCOLATION

J. van den Berg and H. Kesten

\subjclass 60K35, 82A42

Consider first--passage percolation on  Z^d.  A classical result says,
roughly speaking, that the shortest travel time from (0,0,...,0) to
(n,0,...,0) is asymptotically equal to n\mu, for some constant \mu,
which is called the time constant, and which depends on the distribution
of the time coordinates.  Except for very special cases, the value of
\mu is not known.  We show that certain changes of the time coordinate
distribution lead to a decrease of \mu; usually \mu will strictly decrease.
Two examples of our results are:

(i)  If F and G are distribution functions with F \leq G,
F \not\equiv G, then, under mild conditions, the time constant for
G is strictly smaller than that for F.

(ii)  For 0 < \varepsilon_1 \leq \varepsilon_2 \leq a < b, the time
constant for the uniform distribution on [a-\varepsilon_2,b+\varepsilon_1]
is strictly smaller than for the uniform distribution on [a,b].

We assume throughout that all our distributions have finite first moments.

hak@cornella.cit.cornell.edu