Probability Abstracts 44

978. ENERGY AND CUTSETS IN INFINITE PERCOLATION CLUSTERS

David Levin  and  Yuval Peres

Grimmett, Kesten and Zhang (1993) showed that for $d>2$, simple random
walk on the infinite cluster of supercritical percolation on $Z^d$ is 
a.s. transient. Their result is equivalent to the existence of a
nonzero flow $f$ on the infinite  cluster such that the energy 
$\sum_{e}f(e)^2$ is finite. Here we improve this result, and show that
if $d>2$ and $p>p_c$, then the infinite cluster supports a nonzero
flow $f$ such that the  $q$--energy $\sum_{e}|f(e)|^{q}$ 
is finite for all $q>d/(d-1)$.   As a corollary, we obtain that for any
sequence ${A_n}$ of disjoint cutsets in the infinite cluster that
separate a fixed vertex from infinity, the sum $\sum_n |A_n|^r$  is
finite for all $r > 1/(d-1)$. Our proofs are based on the method of
``unpredictable paths'', developed by Benjamini, Pemantle and Peres (1998) 
and refined by Haggstrom and Mossel (1998).

levin@stat.berkeley.edu  peres@stat.berkeley.edu

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979. AN EXPLICIT DESCRIPTION OF THE LYAPUNOV EXPONENTS OF THE NOISY DAMPED HARMONIC OSCILLATOR

Peter Imkeller  and  Christian Lederer

The Lyapunov exponents of the linearization
$$ dv_t \ =\ \left(\begin{array}{cc} 0 & 1 \\ -1 & \beta\end{array} \right)
  v_t ~ dt + \left(\begin{array}{cc} 0 & 0 \\ \sigma & 0\end{array} \right) 
  v_t \circ dW_t $$
of a two-dimensional noisy Duffing-van der Pol oscillator
are key quantities in the investigation of the stochastic Hopf
bifurcation of this system. 
We derive a simple equation exhibiting them explicitly as  functions
of the second moment of the invariant measure of an associated diffusion
with drift given by a potential function and additive noise, and, 
consequently, in terms of hypergeometric functions.
This representation leads to different kinds of complete and explicit
asymptotic expansions, as well as a rather complete account of global 
properties of the Lyapunov exponents as functions of $\beta$ and $\sigma$.

imkeller@mathematik.hu-berlin.de    lederer@mathematik.hu-berlin.de 

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980. STOCHASTIC BIFURCATION MODELS

Richard F. Bass and Krzysztof Burdzy

We study an ordinary differential equation controlled by 
a stochastic process. We present results on existence
and uniqueness of solutions, on associated local
times (Trotter and Ray-Knight theorems), and on time
and direction of bifurcation. A relationship with Lipschitz
approximations to Brownian paths is also discussed.

bass@math.washington.edu,   burdzy@math.washington.edu

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981. BIFURCATION OF RANDOM DYNAMICAL SYSTEMS INDUCED BY ONE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS.

Hans Crauel    and    Peter Imkeller    and    Marcus Steinkamp  

In this paper we consider families of random dynamical systems induced by 
parametrized one dimensional stochastic differential equations. 
We give necessary and sufficient conditions on the invariant measures of the 
associated Markov semigroups which ensure a stochastic bifurcation. 
This leads to sufficient conditions on drift and diffusion coefficients for a
stochastic pitchfork and transcritical bifurcation of the family of random 
dynamical systems.

steink@math.tu-berlin.de

982. THE 2-STAGE CONTACT PROCESS

Stephen M. Krone

We introduce a multi-type interacting particle system which is a 
natural generalization of the contact process.  Here, individuals in
the population have 2 life stages, young and adult.  Only adults can
give birth and each new offspring is young.  Transition from young to 
adult occurs at constant rate, and individuals die at rates that depend
on their life stage.  Important to the analysis of this process is the
construction of a multi-type dual process.

krone@uidaho.edu  (hardcopy available)

983. DIRICHLET FORMS ON TOTALLY DISCONNECTED SPACES AND BIPARTITE MARKOV CHAINS

David Aldous and Steven N. Evans

We use Dirichlet form methods to construct
and analyse a general class of reversible Markov processes with
totally disconnected state spaces.
We study in detail the special case of bipartite
Markov chains.  The latter processes have a state space
consisting of an ``interior'' with a countable number
of isolated points and a, typically uncountable, ``boundary''.
The equilibrium measure assigns all of its mass to the interior.
When the chain is started at a state in the interior, it holds
for an exponentially distributed amount of time and then
jumps to the boundary.  It then instantaneously re-enters the
interior.  There is a ``local time on the boundary''.
That is, the set of
times the process is on the boundary is uncountable and
coincides with the points of increase of a continuous
additive functional.
Certain processes with values in the space of
trees and the space of vertices of a fixed tree provide
natural examples of bipartite chains.  Moreover,
time--changing a bipartite chain by its
local time on the boundary leads to  interesting processes,
including particular L\'evy processes on local fields
(for example, the $p$-adic numbers) that have been considered
elsewhere in the literature.

aldous@stat.Berkeley.EDU
evans@stat.Berkeley.EDU

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984. SEQUENTIAL SELECTION OF AN INCREASING SEQUENCE FROM A MULTIDIMENSIONAL RANDOM SAMPLE

Yuliy Baryshnikov and Alexander Gnedin

We consider selection of a long increasing subsequence as a sequential
decision problem. Random points $X_1,\ldots,X_n$ are sampled 
in strict sequence from a continuous product distribution on
Euclidean $d-$space. At the time $X_j$ is observed it must be
accepted or rejected. If a point is accepted it cannot be 
discarded later and if rejected cannot be recalled.
The subsequence of accepted points must increase in
each coordinate. We show that the maximum expected length
of a subsequence selected is asymptotic to $\gamma n^{1/(d+1)}$
with $\gamma=\frac{d+1}{ \left\{(d+1)!\right\}^{1/(d+1)} }.$
This extends the $\sqrt{2n}$ result by Samuels and Steele for $d=1$.
We also give simple selection policies which approach optimality
as $n\to\infty$.

gnedin@math.uni-goettingen.de

985. INVARIANT MEASURES AND SYMMETRY PROPERTY OF LEVY TYPE OPERATORS

Sergio Albeverio, Barbara R"udiger and Jiang-Lun Wu

Second order elliptic integro-differential operators 
(Levy type operaotrs) are investigated. The regularity 
of invariant probability measures for such operators 
is proved. Sufficient conditions for the existence of 
invariant measures are obtained and the symmetrization
problem is discussed.

sergio.albeverio@ruhr-uni-bochum.de, rudiger@mat.utovrm.it
jiang-lun.wu@ruhr-uni-bochum.de
(Plain Tex file, PS file and hard copy are available)

986. ENTROPIE METRIQUE ET CONVERGENCE PRESQUE PARTOUT

Michel WEBER

Entropy numbers (associated to a given metric space) are classical tools
of analysis, and their role in the study of the regularity properties 
of stochastic processes is already old, since it goes back to the seminal 
works of Kolmogorov. Their role in ergodic theory, in the study of problems of
convergence almost everywhere, was discovered much more recently by Bourgain 
in a remarkable paper. This work as well as its developments by other 
contributors, constitute the basis and the justification of this monograph. 
In order to do easier its reading, it is written in a self-contained way. Thus,
for instance, Chapters 1 and 2 are devoted to
classical tools in ergodic theory that are Banach's Principle and the
Continuity Principle. We took this opportunity to include an extension by 
Bellow-Jones of Banach Principle for $L^\infty$, which is crucial to prove the
second entropy criterion of Bourgain. Besides, a direct proof of the
Continuity Principle is purposed, by means of a Gaussian randomization 
technique. Chapter 3 collects Gaussian tools necessary to the reading of 
Chapters 4 and 5. Chapter 4 is the central part of this monograph, various 
metric entropy criteria are presented, commented and proved in detail. In
Chapter 5, we treat some remarkable applications of these criterias to: the
study of Riemann sums, a conjecture of Khintchine and the regularity of 
Gaussian processes indexed by product sets. In Chapter 6, a method of spectral
regularization introduced by Talagrand in the study of entropy numbers
associated to averages of contractions, is presented. Developments and 
simplifications of this method due to the works of Lifshits-Weber take place 
in this Chapter.  An application to the study of oscillations functions 
of ergodic averages parallel to the recent works of Jones,
Kaufman, Ostrowskii, Rosenblatt and Wierdl is included. Finally, Chapter 7
concerns the theory of orthogonal sums, and present an approach jointly
developped by Talagrand and the author, issued from the theory of
regularity of stochastic processes.


  Chapitre 1: LE PRINCIPE DE BANACH  

1.1 Un theoreme de Banach    
1.2 Un theoreme de Bellow-Jones   
 
  Chapitre 2: LE PRINCIPE DE CONTINUITE   

2.1 Le principe de continuite de Stein   
2.2 L'extension de Sawyer   
2.3 Un principe de domination   
 
  Chapitre 3: PROCESSUS GAUSSIENS-PROPRIETES FONDAMENTALES   

3.1 Definitions   
3.2 Lois de 0-1. Integrabilite forte  
3.3 Lemme de comparaison de Slepian   
3.4 Regularite des processus gaussiens  
3.5 Geometrie des espaces de Hilbert  
 
  Chapitre 4: CRITERES D'ENTROPIE METRIQUE 

4.1 Le critere d'entropie dans $L^p$, $2\le p<\infty$  
4.2 Le critere d'entropie dans $L^\infty$    
4.3 Le critere d'entropie dans $L^p$, $1<p\le 2$   
4.4 Cas des procedes de sommation  
 
  Chapitre 5: QUELQUES APPLICATIONS   

5.1 Applications en theorie ergodique  
5.1.1 Les sommes de Riemann  
5.1.2 Une conjecture de Khintchine   
5.2 Applications en theorie des processus gaussiens   
5.2.1 De nouvelles classes de GB ensembles  
5.2.2 De nouvelles classes de GC ensembles  
 
  Chapitre 6: UN PRINCIPE DE REGULARISATION SPECTRALE  
 
6.1 L'estimation de Talagrand  
6.2 L'inegalite de regularisation spectrale   
6.3 Extensions a la transformee de Hilbert    
6.4 Regularisation spectrale relative aux fonctions d'oscillations 
  
  Chapitre 7: SERIES ORTHOGONALES 

7.1 Introduction   
7.2 Criteres de convergence presque sure   
7.3 Convergence presque sure des series orthogonales   
7.4 Systemes quasi-orthogonaux  

weber@math.u-strasbg.fr        

987. A SELF-SIMILAR INVARIANCE OF CRITICAL BINARY GALTON-WATSON TREES

Gregory A. Burd, Ed Waymire, Ronald D. Winn

A coding of heirarchical structure in finite directed tree graphs
was introduced by Robert Horton in 1945 which, after a modification
by Strahler in 1952, has become a standard river network code
defined as follows:  Edges along a path
adjoining a source (valence 1 vertex) to a junction (valence 3 or higher
vertex) are  coded as order 1.  Such a path of order 1 edges is also
referred to as an order 1 stream. Having defined edges and streams of
order $i$, one now recursively defines edges and
streams of order $i+1$ by the rule that the \lq\lq order 1\rq\rq
streams of the
tree obtained by pruning the streams of order $i$ (and lower) are assigned
order $i+1$.  This code has helped to identify a number of naturally
occuring patterns in river network structure as well as in other 
naturally occuring dendritic structures.  Let $T_{i,j}(s)$ denote
the number of order $j <  i$ junctions in a stream $s$ of  order $i$
and let $\overline{T}_{i,j}$ denote the sample average over all
such streams $s$ of order $i$. 
Empirically it has been observed that 
$\overline{T}_{i,j}$ is, allowing small sample fluctuations,
approximately (Toeplitz) a function of $i-j$ for large
classes of river networks.  A calculation of Ronald Shreve 
published in 1967 revealed that $\Bbb{E}T_{i,j} = {1\over 2}2^{i-j}$ for the 
critical binary Galton-Watson distribution.  This is our starting point.
In particular, we introduce a 
more general notion of {\it stochastic self-similarity}
and show that within a
class of Galton-Watson trees both this and the consequent mean 
Toeplitz property are characteristic of the critical 
binary offspring distribution.
In addition we obtain  interesting conditioned limit theorems corresponding to
the invariance of the critical binary branching process under a pruning
dynamic in the space of rooted trees.

waymire@math.orst.edu

988. SOME CENTRAL LIMIT THEOREMS FOR SUPER BROWNIAN MOTION

Zeng-Hu Li

A well-known result of Dawson (1977) is that, if the underlying motion is
a transient symmetric stable process, the critical continuous
Dawson-Watanabe superprocess started with the Lebesgue measure converges
to a non-trivial steady state. It was also shown by Dawson (1977) that
that the steady random measure has an interesting spatial central limit
theorem which leads to a Gaussian random field. Some central limit
theorems for the weighted occupation time of the super stable process
were proved by Iscoe (1986). In the recurrent underlying motion case, the
superprocess exhibits local extinction and there is no central limit
theorem on the lines of Dawson (1977) and Iscoe (1986).

In this paper we prove a central limit theorem for the super Brownian
motion which covers both the transient and recurrent underlying motions.
We show that the renormalized super Brownian motion converges to a
limiting Gaussian random field if it is properly started. In the
transient case, the covariance kernel of the Gaussian field is given by
the potential kernel of the underlying motion as in Dawson (1977). In
the recurrent case, the Gaussian field is spatially uniform. We also
give a central limit theorem for the weighted occupation time of the
super Brownian motion with underlying dimension number < or = 3,
completing the results of Iscoe (1986).

lizh@bnu.edu.cn (AmS-TeX file and hard copy available)

989. ABSOLUTE CONTINUITY OF MEASURE BRANCHING PROCESSES WITH INTERACTION

Zeng-Hu Li

A continuous measure branching process, or Dawson-Watanabe superprocess,
$X$ over $R^d$ depends on three parameters $(A,c,b)$, where $A$ is the
generator of a Markov process on $R^d$, and $c\ge 0$ and $b$ are bounded
continuous functions on $R^d$. The processes may be characterized either by
its Laplace functionals via a non-linear equation or by a martingale
problem. The measure branching process with (mean field) interaction is a
solution to a similar martingale problem where the parameters $(A,c,b)$ are
all dependent on the present state of the process $X$. The existence of
this solution was proved by M\'el\'eard and Roelly (1992, 1993) and
M\'etivier (1987), but its uniqueness is still unknown. Several special
cases of the process have been considered by others.

It is well-known that when $d=1$, the DW superprocess is absolutely
continuous with respect to the Lebesgue measure for a large class of
admissible generators $A$. The same result for the measure branching
process with interaction was conjectured by M\'el\'eard and Roelly (1992).
In the special case where $A$ is independent on $X$, a measurable density
of the process was obtained by Zhao (1998).

In this paper we establish the absolute continuity of the measure
branching process with interaction in a typical situation where $(A,c,b)$
are all dependent on $X$. When $A$ is independent of $X$, we show that the
density field has a continuous version and satisfies a stochastic partial
differential equation. One essential step of the proof is some moment
estimates for the process. In the non-interacting case, Konno and
Shiga (1988) proved those by using the log-Laplace equation, which is not
available in our situation. We obtain the estimates from the martingale
characterization by an induction argument.

lizh@bnu.edu.cn (AmS-TeX file and hard copy available)

990. LIMIT THEOREMS FOR SUMS OF p-ADIC RANDOM VARIABLES

Anatoly N. Kochubei

We study $p$-adic counterparts of stable distributions, that is
limit distributions for sequences of normalized sums of
independent identically distributed $p$-adic-valued random
variables. In contrast to the classical case, non-degenerate
limit distributions can be obtained only under certain
assumptions on the asymptotic behaviour of the number of
summands in the approximating sums. This asymptotics determines
the ``exponent of stability''.

ank@ank.kiev.ua

991. STRICT INEQUALITY FOR CRITICAL PERCOLATION VALUES IN FRUSTRATED RANDOM CLUSTER MODELS

Massimo Campanino

We establish an inequality between the critical percolation value
of frustraded random cluster models and that of an unfrustrated model
where the free occupation probabilities of some bonds are strictly
decreased. Using the FK representation this gives  an inequality
between the critical temperatures
of the corresponding spin models. In this way
we can prove a strict inequality
between critical temperatures for symmetry breaking of some frustrated
Ising models and the corresponding ferromagnetic models.

campanin@dm.unibo.it

992. A COUNTEREXAMPLE TO THE ``HOT SPOTS'' CONJECTURE

Krzysztof Burdzy and Wendelin Werner

We construct a counterexample to the ``hot spots'' conjecture; 
there exists a bounded connected planar domain such that the 
second eigenvalue of the Laplacian in that
domain with Neumann boundary conditions is simple
and such that the corresponding eigenfunction
attains its strict maximum at an interior point of
that domain.

burdzy@math.washington.edu

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993. SMOOTHNESS OF PROJECTIONS, BERNOULLI CONVOLUTIONS AND THE DIMENSION OF EXCEPTIONS

Yuval Peres and Wilhelm Schlag

Erdos (1939, 1940) studied the distribution $\nu_s$ of the random series
$\sum r_n s^n$, where $r_n$ are i.i.d. uniform random signs. He showed
that the resulting measure is singular for infinitely many $s \in (1/2,1)$,
and absolutely continuous for a.e. $s$ in a small interval below 1. 
Solomyak (1995) proved a conjecture made by Garsia (1962), that $\nu_s$ 
is absolutely continuous for a.e. $s \in (1/2,1)$. In order to sharpen
this result, we have developed a general method to bound the Hausdorff
dimension of exceptional parameters in several settings. We prove:

1. For any $t>1/2$, the set of $s \in [t,1)$ such that $\nu_s$ 
   is singular, has Hausdorff dimension strictly less than 1.
2. For any Borel set $A$ in $\R^d$ with Hausdorff dimension $\dim A>(d+1)/2$, 
   there are points $x$ in $A$ such that the set of distances 
   ${|x-y| : y \in A}$ has positive Lebesgue measure. Moreover, the set of 
   $x$ where this fails has Hausdorff dimension at most $d+1-\dim A$.
3. Let $K_s$ denote the middle-$a$ Cantor set for $a=1-2s$ and let $K$ 
   be any compact set of reals. Peres and Solomyak (1998) showed that for 
   a.e. $s$ in $(t, 1/2)$ such that $\dim(K)+\dim(K_s)>1$, the sumset
   $K+K_s$ has positive length; we show that the set of exceptional $s$ 
   in this statement has Hausdorff dimension at most $2-\dim(K)-\dim(K_t)$.
4. For any Borel set $E$ in $\R^d$ with $\dim E>2$, for a.e. line $L$ through 
   the origin, the orthogonal projection of $E$ to $L$ has nonempty interior, 
   and the set of lines where this fails has dimension at most $d+1-\dim E$. 
5. If $\mu$ is a measure on $\R^d$ with correlation dimension greater than 
   $m+2a$, then for a ``prevalent'' set of $C^1$ maps $f: \R^d \to \R^m$  
   (in the sense studied by Hunt, Sauer and Yorke 1992), the image of $\mu$ 
   under $f$ has a density with at least $a$ fractional derivatives in $L^2$.

peres@stat.berkeley.edu      schlag@math.princeton.edu

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994. HOW MISLEADING CAN SAMPLE ACF'S OF STABLE MA'S BE? (VERY!)

Fong Xue, Sidney Resnick and Gennady Samorodnitsky,

For the stable moving average process 
\[ X_t = \int_{-\infty}^{\infty}f(t+x) M(dx), t=1,2,... \]
we find the weak limit of its sample autocorrelation function as the sample size $n$ increases to $\infty$. It turns out that, as a rule, this limit is random! This shows how dangerous it is to rely on sample correlation as a model fitting tool in the heavy tailed case. We discuss for what functions $f$ this 
limit is non-random for all (or only some -- this can be the case, too!) lags.

Available as ma.ps at www.cornell.edu/~gennady/techreports.html

sid@orie.cornell.edu gennady@orie.cornell.edu xue@cam.cornell.edu

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995. A NOTE ON SUMS OF INDEPENDENT RANDOM VARIABLES

P. Hitczenko and S. Montgomery-Smith

In this note a two sided bound on the tail probability of sums 
of independent, and either symmetric or nonnegative, random variables 
is obtained. We utilize  a recent result by Latala
on bounds on moments of such sums. We also give a new proof of Latala's
result for nonnegative random variables, and improve one of the constants
in his inequality.

stephen@math.missouri.edu        pawel@math.ncsu.edu

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996. SUR LE CARACTERE GAUSSIEN DE LA CONVERGENCE PRESQUE PARTOUT

Michel Weber 

We establish functional type inequalities linking the regularity properties
of sequences of operators $S=(S_n)$ acting on $L^2$-spaces, with those of
the restriction of the canonical Gaussian process on the associated subsets
$(S_n(f))$ of $L^2$. These inequalities allow to easily deduce as
corollaries famous Bourgain's entropy criterias in the theory of almost
everywhere convergence. These also provide a better understanding of the
role of the Gaussian processes in the study of phenomenon of almost
everywhere convergence. 

weber@math.u-strasbg.fr 

997. EXISTENCE OF HYDRODYNAMICS FOR THE TOTALLY ASYMMETRIC SIMPLE $K$-EXCLUSION PROCESS

Timo  Seppalainen

In a totally asymmetric simple $K$-exclusion process particles 
take nearest-neighbor steps to the right on the lattice $\mmZ$, 
under the constraint that each site contain at most $K$ particles. 
We prove that such processes satisfy hydrodynamic limits under 
Euler scaling, and that the limit of the empirical particle 
profile is the entropy solution of a scalar conservation law 
with a concave flux function. Our technique requires no knowledge 
of the invariant measures of the process, which is essential here 
because the equilibria of asymmetric $K$-exclusion are unknown. 
But we cannot calculate the flux function precisely without some
further information about the process. The proof proceeds via a 
coupling with a growth model on the 2-dimensional lattice. In 
addition to the basic $K$-exclusion with constant exponential jump
rates, we treat the site-disordered case where each site has its 
own jump rate, randomly chosen but frozen for all time. The 
hydrodynamic limit under site disorder is new even for the 
simple exclusion process (the case $K=1$). Our proof makes no use 
of the Markov property, so at the end of the paper we indicate how 
to treat the case where the waiting times are not exponential. 

seppalai@iastate.edu   (AMS-TeX, postscript, hardcopy available) 

998. A LOWER BOUND ON THE GROWTH EXPONENT FOR LOOP-ERASED RANDOM WALK IN TWO DIMENSIONS

Gregory F. Lawler

  The growth exponent $\alpha$ for loop-erased or Laplacian random walk on the
integer lattice is defined by saying that the expected time to reach the sphere
of radius $n$ is of order $n^\alpha$. We prove that in two dimensions, the
growth exponent is strictly greater than one. The proof uses a known estimate
on the third moment of the escape probability and an improvement on the
discrete Beurling projection theorem.

jose@math.duke.edu

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999. ON INCREASING SUBSEQUENCES OF IID SAMPLES

J. D. Deuschel and O. Zeitouni

  We study the fluctuations, in the large deviations regime, of the longest
increasing subsequence of a random i.i.d. sample on the unit square. In
particular, our results yield the precise upper and lower exponential tails for
the length of the longest increasing subsequence of a random permutation.

zeitouni@eecs.berkeley.edu

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1000. LOCAL FIELDS, GAUSSIAN MEASURES, AND BROWNIAN MOTIONS

Steven N. Evans

  These are lecture notes from a course given at the CRM in Montreal in 1992.
They survey the author's attempts to find and understand canonical
probabilistic entities in a local field (e.g. p-adic) setting. We propose
answers to the related questions: ``What are the analogues for Gaussian
measures?'' and ``What are the analogues for Brownian motion and its
multiparameter relatives?''
  There is a suitable concept of orthogonality in the local field setting, and
one can mimic a classical abstract definition of Euclidean Gaussian measures.
The resulting theory is, in some sense, an L^\infty rather than an L^2 one; and
so although there are many similarities to the Euclidean case there are also
some striking differences.
  Local field Brownian motion is the most natural local field Gaussian process
that takes values in a local field vector space and is indexed by another local
field vector space. Characterising the polar sets for such processes lead us to
study the notion of additive functionals or homogeneous random measures and a
related Riesz-like potential theory.
  We study local times, show they can be recovered by counterparts of classical
intrinsic contstructions using dilation and Hausdorff measures, and establish
an analogue of Trotter's theorem.
  We finish with a discussion of random series with local field Gaussian
coefficients. Although we show that there are broad classes of random series
which are stationary, we also find that there is no obvious counterpart of the
representation of a general stationary Gaussian process on the circle as a
random Fourier series.

evans@stat.berkeley.edu

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1001. MARKOV PROCESSES WITH IDENTICAL BRIDGES

P. J. Fitzsimmons

  Let X and Y be time-homogeneous Markov processes with common state space E,
and assume that the transition kernels of X and Y admit densities with respect
to suitable reference measures. We show that if there is a time t>0 such that,
for each x\in E, the conditional distribution of (X_s)_{0 < s < t}, given X_0 =
x = X_t, coincides with the conditional distribution of (Y_s)_{0 < s < t},
given Y_0 = x = Y_t, then the infinitesimal generators of X and Y are related
by [L^Y]f = \psi^{-1}[L^X](\psi f)-\lambda f, where \psi is an eigenfunction of
L^X with eigenvalue \lambda. Under an additional continuity hypothesis, the
same conclusion obtains assuming merely that X and Y share a ``bridge'' law for
one triple (x,t,y). Our work entends and clarifies a recent result of I.
Benjamini and S. Lee.

pfitz@euclid.ucsd.edu

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1002. DISTRIBUTION OF ZEROS OF RANDOM AND QUANTUM CHAOTIC SECTIONS OF POSITIVE LINE BUNDLES

Bernard Shiffman and Steve Zelditch

  We study the limit distribution of zeros of certain sequences of holomorphic
sections of high powers $L^N$ of a positive holomorphic Hermitian line bundle
$L$ over a compact complex manifold $M$. Our first result concerns `random'
sequences of sections. Using the natural probability measure on the space of
sequences of orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$, we show that for
almost every sequence $\{S^N_j\}$, the associated sequence of zero currents
$1/N Z_{S^N_j}$ tends to the curvature form $\omega$ of $L$. Thus, the zeros of
a sequence of sections $s_N \in H^0(M, L^N)$ chosen independently and at random
become uniformly distributed. Our second result concerns the zeros of quantum
ergodic eigenfunctions, where the relevant orthonormal bases $\{S^N_j\}$ of
$H^0(M, L^N)$ consist of eigensections of a quantum ergodic map. We show that
also in this case the zeros become uniformly distributed.

shiffman@chow.mat.jhu.edu

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