Probability Abstracts 23

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376. OPTIMUM BOUNDS ON MOMENTS OF SUMS OF INDEPENDENT RANDOM VECTORS

Iosif Pinelis

For  the sum $S$ of independent zero-mean random vectors  
$X_1,\ldots,X_n$, optimum order upper bounds on  $\|\,\|S\|\,\|_p$ in  
terms of $\|\sup_i\|X_i\|\,\|_p$ and $\|\,\|S\|\,\|_1$ are given.}$

ipinelis@math.mtu.edu (hard copy or an AMS TeX file available)

377. A CONCEPTUAL PROOF OF THE KESTEN-STIGUM THEOREM FOR MULTI-TYPE BRANCHING PROCESSES

Thomas G. Kurtz, Russell Lyons, Robin Pemantle, and Yuval Peres

We give complete proofs of the theorem of convergence of types
and the Kesten-Stigum theorem for multi-type branching processes.  Very
little analysis is used beyond the strong law of large numbers and some
basic measure theory. The proof uses the ideas of the preprint,
"Conceptual Proofs of $L \log L$ Criteria for Mean Behavior of Branching
Processes" by Lyons, Pemantle and Peres.

rdlyons@silver.ucs.indiana.edu     (PlainTeX or hard copy available)

378. A SIMPLE PATH TO BIGGINS' MARTINGALE CONVERGENCE FOR BRANCHING RANDOM WALK

Russell Lyons

We give a simple non-analytic proof of Biggins' theorem on
martingale convergence for branching random walks ({\it J. Appl. Prob.}
{\bf 14} (1977), 25--37) inspired by the preprint,
"Conceptual Proofs of $L \log L$ Criteria for Mean Behavior of Branching
Processes" by Lyons, R. Pemantle and Y. Peres.

rdlyons@silver.ucs.indiana.edu     (PlainTeX or hard copy available)

379. ON INVARIANT MEASURES OF DIFFUSIONS ON BANACH SPACES

D.Gatarek and B.Goldys

We consider a stochastic differential equation on a Banach 
space with strongly nonlinear 
drift. Sufficient conditions are given for existence of a unique 
martingale solution. It is shown also that there exists a unique 
invariant measure for this equation and moreover it is 
equivalent to an invariant measure of the equation without 
drift. The main tool to show these results is Girsanov 
Transformation.

beng@unsw.edu.au (plain tex file available)

380. TWISTED SHEETS

J. R. Norris

We give a simplified presentation of two-parameter stochastic calculus
in terms of a symbolic calculus of differentials. We establish a new
existence, uniqueness and non-explosion theorem for certain hyperbolic
two-parameter stochastic differential equations driven by the Brownian
sheet. The theorem is applied to construct some natural path-valued
processes in a Riemannian manifold, where the twisting effect of a
connection makes the existing two-parameter theory inapplicable. An
analysis of these path-valued processes is made using the
two-parameter stochastic calculus.

J.R.Norris@statslab.cam.ac.uk

381. A COMPLETE CONVERGENCE THEOREM FOR AN EPIDEMIC MODEL

Enrique Andjel and Rinaldo Schinazi

We use an interacting particle system on $Z$ to model
an epidemic. Each site of $Z$ can be in either one of three states: empty, 
healthy or infected. An empty site $x$ gets occupied by a healthy individual
at a rate $\beta n_1(x)$ where $n_1(x)$ is the number of healthy nearest 
neighbors
of $x$. A healthy individual at $x$ gets infected at rate $\alpha  n_2(x)$
where $n_2(x)$ is the number of infected nearest neighbors of $x$. An 
infected individual dies at rate $\delta$ independently of everything else.
We show that for all $\alpha $, $\beta $ and $\delta>0$ and all initial
configurations, all the sites of a fixed finite set
remain either all empty or all healthy after an almost surely finite time.
Moreover if the initial configuration has infinitely many healthy individuals
then the process converges almost surely (in the sense described above) to the
all healthy state.
We also consider a model introduced by Durrett and Neuhauser where healthy
individuals appear spontaneously at rate $\beta>0$ and for which coexistence
of 1's and 2's was proved in dimension 2 for some values of $\alpha$ and
$\beta$. We prove that coexistence may occur in any dimension. 

schinazi@vision.uccs.edu

382. NO TRIPLE POINT OF PLANAR BROWNIAN MOTION IS ACCESSIBLE

Krzysztof Burdzy and Wendelin Werner

The boundary of a connected component 
of the complement of a planar Brownian path 
on a fixed time-interval contains almost surely no triple
point of this Brownian path.

W.J.J.Werner@statslab.cam.ac.uk or burdzy@math.washington.edu 
(plain TeX file available)

383. ITO FORMULA FOR AN ASYMPTOTICALLY 4-STABLE PROCESS

Krzysztof Burdzy and Andrzej Madrecki

An Ito-type formula is given for an asymptotically 
4-stable process which has been constructed
in an earlier paper (PAS 275). 

burdzy@math.washington.edu (plain TeX file available)

384. COMPLETE ANALYTICITY FOR 2D ISING COMPLETED

R.H.Schonmann and S.B.Shlosman

We study the behavior of the two-dimensional nearest neighbor ferromagnetic
Ising model under an external magnetic field $h$. 
We extend to every subcritical value of the temperature a result previously
proven by Martirosyan at low enough temperature, and which roughly states
that for finite systems with $-$ boundary conditions under a positive 
external field, the boundary effect dominates in the bulk if the linear size
of the system is of order $B/h$ with $B$ small enough, while if $B$ is large
enough, then the external field dominates in the bulk. As a consequence we
are able to complete the proof that ``complete analyticity for nice sets'' 
holds for every value of
the temperature and external field in the interior of the uniqueness region
in the phase diagram of the model.

The main tools used are the results and techniques developed to study
large deviations for the block magnetization in the absence of the 
magnetic field, and recently extended to all temperatures below the 
critical one by Ioffe.  

rhs@math.ucla.edu or shlosman@math.uci.edu
(AMSTEX file available from rhs@math.ucla.edu)

385. LEVY CLASSES AND SELF-NORMALIZATION

Davar Khoshnevisan

Among other results, we prove a Chung's
law of the iterated logarithm for essentially all
one-dimenional Markov processes which are recurrent.
In order to attain this level of generality, our 
normalization is random. In particular, when the 
Markov process in question is a diffusion, we obtain 
the integral test corresponding to a law of the iterated 
logarithm due to F. Knight.

davar@math.utah.edu

386. ASYMPTOTIC PROPERTIES OF LEVY PROCESSES IN LIE GROUPS

Ming Liao

Let $G$ be a semisimple Lie group of noncompact type and let 
$G = NAK$ be the Iwasawa decomposition, where $K$ is a maximal compact 
subgroup, $A$ is an abelian subgroup and $N$ is a nilpotent subgroup. 
Let $g_t$ be a Levy process on $G$. We obtain the following results.

(A) Let $g_t = x_t \exp(H_t) y_t$ be the Cartan decomposition with 
$x_t, y_t$ in $K$ and $H_t$ in the Lie algebra of $A$ (a Euclidean 
space). Assume that the support of the process $g_t$ generates $G$ and 
contains a contracting sequence in $G$ (see paper for the second 
assumption). Then almost surely, as $t$ tends to $\infty$, 
$\alpha(H_t)$ tends to $\infty$ for any positive root $\alpha$ 
and the projection of $x_t$ to the maximal Furstenberg boundary 
of $G$ converges. 

(B) Let $g_t = n_t a_t k_t$ be the Iwasawa decomposition $G = NAK$ with 
$a_t = \exp(H_t)$. Assume the hypotheses of (A). Then almost surely, 
as $t$ tends to $\infty$, $\alpha(H_t)$ tends to $-\infty$ for any 
positive root $\alpha$ and $n_t$ converges in $N$. 

(C) Let $H_t$ be the process in (B). Assume $g_t$ has a finite Levy
measure which satisfies a certain integrability condition. 
Then almost surely, as $t$ tends to $\infty$, $H_t/t$ 
converges to some vector $H$ contained in the closure of the negative 
Weyl chamber. Moreover, we obtain an explicit integral expression for 
$H$ in terms of the Levy measure of $g_t$ and the stationary measure 
of $k_t$. 

(D) If we drop the assumption that the support of the process $g_t$ 
contains a contracting sequence, then essentially all the above 
results still hold, but we have to replace the use of the maximal 
boundary of $G$ by the general boundary. 

The basic result (A) is proved by adapting the method used by Guivar'h 
and Raugi for Lie group-valued random walks. Our results generalize 
Malliavin and Malliavin's result for horizontal diffusions on 
semisimple Lie groups. We can apply these results to study the stability 
of stochastic flows generated by SDE driven by Brownian motion and 
Poisson point processes. 

mliao@mail.auburn.edu 

387. ALL TEMPERATURE BOUNDS FOR THE SHERRINGTON-KIRKPATRICK SPIN GLASS MODEL AND SIZE-BIASED T-MARTINGALES

Mina Ossiander, Edward C. Waymire, Stanley C. Williams

We give a representation of the Sherrington-Kirkpatrick (SK) 
model in terms of a sequence of positive T-martingales introduced by
Jean-Pierre Kahane (1988). From this we apply size-bias techniques 
described in Waymire and Williams (1993) to compute rigorous  
bounds on the free energy of the SK model with Gaussian couplings 
valid at all temperatures. For inverse temperatures down to 
$\beta _c = 2\sqrt{\log 2}$ the upper bound  coincides with
the (quadratic) annealed free energy. The upper bound for all 
higher inverse temperatures $\beta$ is linear $\sqrt{\log 2}\beta,$ 
as is the lower bound $.76\sqrt{\log 2}\beta.$ Waymire and Williams (1994) 
have shown that a size-bias estimate yields the exact free energies 
at all temperatures in a large class of correlated random polymer 
models for spin glasses. While it has been widely believed that 
the quenching transition occurs at unit temperature,
the results here suggest that that the temperature may be slightly
lower (i.e. ${1\over 2\sqrt{\log 2}}\approx 0.6$) if the free energy exists.

waymire@math.orst.edu             (PlainTex file or hardcopy available)

388. CLUSTERING BEHAVIOR OF FINITE VARIANCE PARTIAL SUM PROCESSES

U. Einmahl and V. Goodman

Clustering rates in Strassen's functional law of the iterated logarithm
are determined for finite variance partial sum processes in one dimension.
A general characterization of these rates, similar to one recently obtained
for one-dimensional Brownian motion, shows that relatively mild moment 
conditions on a partial sum process lead to high order clustering rates at
certain points of the Strassen set.

ueinmahl@ucs.indiana.edu, goodmanv@ucs.indiana.edu  

389. PERFECT COCYCLES THROUGH STOCHASTIC DIFFERENTIAL EQUATIONS

Ludwig Arnold and Michael Scheutzow

Smooth deterministic dynamical systems T(t) and differential equations 
dx/dt=f(x) (generating T through their solution flows) are basically the 
same class of objects.  Here we investigate the situation in the random 
case: When is a random dynamical system (cocycle) T(t,w) generated by a 
stochastic differential equation, and which stochastic differential 
equation generates a random dynamical system through its solution?

The situation will turn out to be roughly as follows: if T is a cocycle for
which t -> T(t,w)x is a semimartingale, then it is generated by a stochastic
differential equation driven by a vector field valued semimartingale with
stationary increments (helix), and conversely. The result is a basically
one-to-one correspondence, symbolically written as
        semimartingale cocycle = exp(semimartingale helix).
To implement this relation we lift stochastic calculus from the traditional
one-sided time to two-sided time and make this consistent with ergodic theory
by introducing the notion of a filtered dynamical system. We also prove a
general theorem on the perfection of a crude cocycle, thus solving a 
problem which was open for more than ten years.

arnold@mathematik.uni-bremen.de

390. NOISE-INDUCED TRANSITIONS FOR ONE-DIMENSIONAL DIFFUSIONS

Michael Scheutzow

We consider a one-dimensional diffusion X on a finite or infinite interval (l,r)
satisfying a SDE dX(t)=b(X(t))dt + c(X(t))dW(t), where W is standard Brownian
motion. For fixed drift b we investigate noise-induced transitions of the
boundary behavior i.e. the dependence of the behavior of X near r when c varies.
We are particularly interested in the question whether increasing c for all x
may cause or prevent explosion in finite time. A negative result is that if c
is bounded and bounded away from zero then multiplying c by a factor greater
than one can never prevent explosions. Various examples show that under 
different assumptions increasing the noise can have a drastic stabilising or 
destabilising effect (depending on b and c). In the final section we study the 
relation between the stochastic and deterministic (c=0) case.

ms@math.tu-berlin.de

391. CURVATURE CONDITIONS ON RIEMANNIAN MANIFOLDS WITH BROWNIAN HARMONICITY PROPERTIES

H. R. Hughes

The time and place that Brownian motion on a Riemannian manifold
first exits a normal ball of radius $\epsilon$ is considered and a
general procedure is given for computing asymptotic expansions, as
$\epsilon$ decreases to zero, for joint moments of the first exit
time and place random variables.  It is proven that asymptotic
versions of exit time and place distribution properties that hold
on harmonic spaces are equivalent to certain curvature conditions
for harmonic spaces.  In particular, it is proven that an asymptotic
mean value condition involving first exit place is equivalent to
certain levels of curvature conditions for harmonic spaces depending
on the order of the asymptotics.  Also, it is proven that an
asymptotic uncorrelated condition for first exit time and place is 
equivalent to weaker curvature conditions at corresponding orders of 
asymptotics.

hrhughes@c-math1.siu.edu

392. REFLECTING DIFFUSIONS ON LIPSCHITZ DOMAINS WITH CUSPS - ANALYTIC CONSTRUCTION AND SKOROHOD REPRESENTATION-

M. Fukushima and M. Tomisaki

For a general (not necessarily bounded) Lipschitz domain D of R^d posessing 
(possibly infinite number of ) cusp boundary points, a conservative diffusion 
process is constructed on the closure of D, which is associated with a Dirichletform expressed by uniform elliptic measurable functions a_{ij} and admits a
resolvent density jointly continuous off diagonal icluding the boundary. 
The Holder exponents at cusp points are assumed to be uniformly greater than 
(d-1)/d. The method of the construction is purely analytic and we use a standardPDE method due to Stampacchia and Moser. We need the above restriction on the 
exponents in proving a modified Sobolev inequality of Moser's type. 
The constructed process admits a semimartingale decomposition of Skorohod type 
involving a martingale associated with a_{ij} and a local time on the boundary 
(without exceptional starting point). This is based on a slight extension of a 
decomposition theorem of additive functionals stated in Theorem 5.5.5 in
a recent book (M.Fukushima,Y.Oshima and M.Takeda, Dirichlet forms and
symmetric Markov processes, Walter de Gruyter,1994). This type of
construction and decomposition were initiated by Bass and P.Hsu for
a bounded Lipschitz domain.  Recently Debrassie and Toby proved for a 
two dimensional standard cusp domain that a reflecting Brownian motion 
is semimartingale if and only if the Holder exponent is greater than
1/2.

fuku@sigmath.osaka-u.ac.jp
tomisaki@ccyi.ccy.yamaguchi-u.ac.jp

393. ON A STRICT DECOMPOSITION OF ADDITIVE FUNCTIONALS FOR SYMMETRIC DIFFUSION PROCESSES

M. Fukushima

This is a continuation of the previously announced preprint 'On a
decomposition of additive functionals in the strict sense for a symmetric 
Markov processes' which will appear in the Proc. Beijing Conf. of GFSP
eds Ma.Rockner,Yan, from Walter de Gruyter. Under the absolute continuity
assumption for the transition function of a symmetric diffusion process,
some basic properties of the energy zero part( the part of zero P_m quadratic
variation) N_t in the decomposition of additive functionals in the ordinary 
sense (admitting no exceptional starting point set of zero capacity) of 
u(X_t) - u(X_0) are studied.  Especially its support is related to the spectrum of u and further a necessary and sufficient condition for N to be of bounded
variation on each finite time interval is given in terms of the function 
u belonging locally to the Dirichlet space.

fuku@sigmath.osaka-u.ac.jp

394. PASSAGE-TIME MOMENTS FOR CONTINUOUS NON-NEGATIVE STOCHASTIC PROCESSES AND APPLICATIONS

M. Menshikov and R. J. Williams

We give  general criteria for 
the finiteness or not of the
passage-time moments for continuous 
non-negative stochastic processes in terms of 
sub/supermartingale inequalities for powers of these processes.
We apply these results to 
one-dimensional diffusions
and also  to reflected Brownian motion in a wedge.
The discrete-time analogue of this problem was
studied previously by Lamperti  and  more recently 
by Aspandiiarov, Iasnogorodski and  Menshikov.
Our results are continuous analogues  of
those in  the paper by Aspandiiarov et al.

williams@math.ucsd.edu (Plain tex file available)     

395. POINTS OF INCREASE FOR RANDOM WALKS

Yuval Peres

Say that a sequence S_0,...,S_n  has a (global) point of 
increase at k if S_k is maximal among S_0,...,S_k and
minimal among S_k,...,S_n. We show that an n-step symmetric
random walk on the line has a (global) point of increase
with probability comparable to 1/(log n). This implies  the
classical fact that Brownian motion has no points of increase.

peres@stat.berkeley.edu  (LaTeX file available)

396. FURSTENBERG-KHASMINSKII FORMULAS FOR LYAPUNOV EXPONENTS VIA ANTICIPATIVE CALCULUS

L. Arnold and P. Imkeller

ccording to a classical formula due to Furstenberg and Khasminskii the top
Lyapunov exponent of a random linear cocycle can be expressed as a phase
average of a function over projective space. Using a description of the
Oseledets spaces as intersections of two flags of random subspaces of the
phase space, one forward and one backward in time, we derive similar formulas
for the remaining Lyapunov exponents of the cocycle generated by a linear
stochastic differential equation. This approach makes elements of
anticipative calculus enter the scene quite naturally. Indeed,
the correction terms
appearing in the analogues of the Furstenberg-Khasminskii formula, averages
of the Malliavin gradient of the orthogonal projectors on the Oseledets spaces
come into play explicitly.

imkeller@grenet.fr

397. ON THE LAWS OF ORTHOGONAL PROJECTORS ON EIGENSPACES OF LYAPUNOV EXPONENTS OF LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

P. Imkeller

Consider a system of linear stochastic differential equations driven by
Brownian motions. Using an approach via Malliavin's calculus, we show that
the laws of the orthogonal projectors on the Oseledets spaces regarded as
objects of projective space are absolutely continuous with respect to surface
measure, provided H\"ormander's hypoellipticity criterion is satisfied by
the matrices governing the system. In case they are skew symmetric, even
the one dimensional orthogonal projectors appearing in the spectral 
decomposition
of the forward (backward) Ruelle's matrix possess smooth densities.

imkeller@grenet.fr

398. ON A PROBLEM OF ERDOS AND TAYLOR

D. Khoshnevisan, T.M. Lewis and Z. Shi

Let S(n) be a centered d-dimensional random 
walk with d>2. Consider  the so--called future 
infima process, J(n)= inf_{k geq n} |S(k)|.
This paper is concerned with obtaining precise integral
criteria for a function to be in the
Levy upper class of J. This solves an old
problem of Erdos and Taylor (1960), who
posed the problem for the simple symmetric random
walk on Z^d, d>2. These results are obtained
by a careful analysis of the future infima of transient
Bessel processes and using strong approximations.
Our results belong to a class of Ciesielski--Taylor theorems
which relate d and (d-2) dimensional Bessel processes.


davar@math.utah.edu
 

399. SPECTRA OF GRAPHS AND FRACTAL DIMENSIONS II

Andras  Telcs

This paper continues the study of exponents $ d(x), d_{\Omega}, d_{R} $
and $d_{\mu} for locally finite connected infinite graph $G$ and the
nearest neighbour random walk $\{X_{n}\}_{n \in N} $ on it with a fixed
starting point $X_{0}=x \in G $. The main result of the paper derives
for strongly recurrent graphs ($d_{\Omega}<2 $ ) from some technical
condition  that if $$ d_{R} = d + 2 - d_{\Omega} $$ (which is true if
the graph is smooth,( i.e. nice in some sense, see the authors earlier works)
then then $$  P_{n}(x,x)  \approx n^{-d/d_{R}} $$ and $$
d_{s}=2d/d_{R} $ where $d_{s}$ is the exponent of the spectral density or
density of states of the Laplacian given by the absorbing boundary condition.
This result conjectured already by Rammal and Toulouse but rigourously proven
only for particular very symmetric fractals like dendrits and some
nested recurrent fractals by Kigami and for the Sierpinsky gasket by Barlow
and Bass. Systematic study of the other exponents are provided in the paper
as well.

H197Tel@HUELLA.BITNET
 

400. PASSAGE-TIME MOMENTS FOR NON-NEGATIVE STOCHASTIC PROCESSES AND AN APPLICATION

S. Aspandiiarov, R. Iasnogorodski, M. Menshikov

We get some sufficient conditions for the finiteness or not of the 
first passage-times moments for non-negative discrete parameter processes
(here first passage-time means 
the first hitting time of a compact set containing the origin). 
The developed criteria are closely connected with the 
well-known results of Foster for the ergodicity of 
Markov chains and are given in terms of sub(super)-martingales.
Then, as an application of the obtained results, we get explicit 
conditions for the finiteness or not of the first passage-times moments 
for reflected random walks in a quadrant with zero drift in the interior. 

aspandij@oscar.proba.jussieu.fr, iasno@solaris.inria.fr,
    menshikov@llrs.math.msu.su (Latex file available)
 

401. TAILS OF PASSAGE-TIMES AND AN APPLICATION TO STOCHASTIC PROCESSES WITH BOUNDARY REFLECTION IN WEDGES

S. Aspandiiarov, R. Iasnogorodski

In this paper we obtain
lower bounds for the tails of the distributions
of first passage-times for some stochastic processes.
We consider first discrete parameter 
processes with asymptotically small drifts 
taking values in $ R_{+}$ and prove for them a general result
giving lower bounds for these tails. As an application of the obtained 
results, we obtain lower bounds for the  tails of the distributions
of the first passage-times for reflected random walks
in a quadrant with zero-drift in the interior.
This completes the results of our previous paper 
``Passage-time moments for non-negative stochastic 
processes and an application to reflected random walks 
in a quadrant'' giving a negative answer to the question of 
existence of the moment of the critical order 
$\alpha /2$ of the first passage-times for 2-dimensional reflected 
random walks ($\alpha$ here is a parameter depending on the
geometrical data corresponding to the reflected random walk).
Finally, the results on passage-times for reflected random walks
are used to get explicit conditions for the finiteness or not of 
the moments of the passage-time to the origin
for a Brownian motion with oblique reflection in a wedge.
Recently we have learned that using an approach based on
the continuous-time analogues of the results in
the above-mentioned paper, Menshikov and Ruth Williams
gave a direct proof of the results of concerning the first-passage times
for a Brownian motion with oblique reflection in all except the critical case
$p=\alpha /2$. Moreover, they are able to prove finiteness of the p-th
moments in the case $0<p<\alpha /2 <1$ which is not covered by our results.

aspandij@oscar.proba.jussieu.fr, iasno@solaris.inria.fr (Latex file available)

402. MARKOVIAN EXTENSIONS OF SYMMETRIC DIFFUSION OPERATORS

P. J. Fitzsimmons

Let $A$ be the $L^2$-infinitesimal generator of  a symmetric Markov diffusion
process $X$ with symmetry measure $m$, and let $\AA$ be the restriction of $A$
to a suitable core $\CC\subset D(A)$. Call a (non-positive) self-adjoint
operator $(L,D(L))$ in $L^2(m)$ a {\it Markovian extension\/} of
$(\AA,\CC)$ if $L|_\CC=\AA$ and if the semigroup $(e^{tL})_{t>0}$ generated
by $L$ is sub-Markov. The class $E_M(\AA)$ of all such extensions admits a
natural partial ordering, with respect to which $(A,D(A))$ is the minimum
element. With each element $L$ of $E_M(\AA)$ we can associate  a Dirichlet
form $(\EE_L,\DD_L)$, where $\DD_L=D(\sqrt{-L})$ and
$\EE_L(u,v)=(\sqrt{-L}u,\sqrt{-L}v)_m$. We say that $L\in E_M(\AA)$ is a
{\it Silverstein extension\/} of $\AA$ (and write $L\in E_S(\AA)$) provided
$\DD_A\cap L^\infty(m)$ is an algebra ideal in $\DD_L\cap L^\infty(m)$. The
importance of the class $E_S(\AA)$ stems from the fact, due to Silverstein,
that the Markov process corresponding to an infinitesimal generator $L\in
E_S(\AA)$ is a path-wise extension of $X$. We show that under very broad
conditions, these two classes of extensions coincide:  $E_M(\AA)=E_S(\AA)$.
In particular, if the diffusion $X$ is non-explosive ({\it i.e.\/} has
infinite lifetime), then $E_M(\AA)$ has precisely one element. That is, the
operator $\AA$ has a unique Markovian self-adjoint extension.  This result
turns on a convenient description of the maximum element of $E_M(\AA)$. The
paper provides a unified treatment of uniqueness problems studied by many
authors, in finite and infinite dimensional contexts.

pfitzsim@ucsd.edu  (plain TeX file available)

403. SOBOLEV INEQUALITIES AND MYERS' DIAMETER THEOREM FOR AN ABSTRACT MARKOV GENERATOR

D. Bakry and M. Ledoux 

We show that if a Markov generator $L $ satisfies the Sobolev inequality
$${\| f\| }_{2n/n-2}^2 \leq  {\| f\| }_2^2 
                + { 4\over {n(n-2)} } \int f (-L f) d\mu  $$
for some $n>2$, then its diameter (in a sense to be described) is less than
or equal to $\pi $, that is the diameter of the sphere that satisfies the
same
Sobolev inequality (for the conformal Laplacian). An abstract version of
Myers' theorem under curvature and dimension hypotheses on $L $ follows.
We also show that if the diameter is equal to $\pi $, there are
non-constant
extremal functions of the preceding Sobolev inequality, and we recover in
this way the Topogonov-Cheng sphere theorem.

bakry@cict.fr or ledoux@cict.fr  (plain tex file available)

404. STOCHASTIC P.D.E.'S ARISING FROM THE LONG RANGE CONTACT AND LONG RANGE VOTER PROCESSES

Carl Mueller and Roger Tribe

In one spatial dimension, a long-range contact process and a 
long-range voter process are scaled so that the distance between 
sites decreases and the number of neighbors of each site increases. 
The approximate densities of occupied sites, under suitable time 
scaling, converge in law to a continuous space time densities which
solve stochastic p.d.e.'s.  For the contact  process the limiting 
equation is the Kolmogorov-Petrovskii-Piscuinov equation driven by 
branching white noise:
$$
\partial_t u = (1/6) \Delta u + \theta u - u^2 + |2u|^{1/2} \dot{W} 
$$
For the voter process the limiting equaion is the heat equation 
driven by Fisher-Wright white noise:
$$
\partial_t u = (1/6) \Delta u + 2 \theta u(1-u) + |4u(1-u)|^{1/2} \dot{W} 
$$
Here, $\dot{W}$ is 2-parameter white noise.

To prove these results, the models are shown to satisfy a 
martingale problem that approximates the martingale problem for
the limiting processes. Tightness of the approximate densities is 
established. Passing to the limit in the approximate martingale 
problems, all limit points are shown to satisfy the limiting equations. 
The proof is then concluded by the uniqueness for solutions to the 
limiting equations. 

An earlier and weaker version of our limiting result for the contact 
process was contained in the preprint "A Stochastic PDE arising as the 
limit of a long-range contact process and its phase transition".  We 
have split up this preprint into 2 papers, the second of which is 
described in this abstract.

cmlr@troi.cc.rochester.edu	(latex file available)