Probability Abstracts 33
This document contains abstracts 669-710.
They have been mailed on May 29, 1996.
669. ON ASYMPTOTIC NORMALITY OF THE HILL ESTIMATOR
Laurens de Haan and Sidney Resnick
For iid observations $X_1,\dots,X_n$ from a common distribution
$F$ with regularly varying tail $1-F(x) \sim x^{-\alpha}L(x),$ $x\to
\infty$, the most popular estimator of $\alpha$ is the Hill estimator.
Regular variation of the distribution tail is equivalent to weak consistency
of the Hill estimator in a manner made precise in Mason (1983) but
necessary and sufficient conditions for asymptotic normality of this
estimator are still somewhat shrouded in confusion. This is in part due to
the different possibilities for a centering in the asymptotic normality
statement. We clarify the roles played by smoothness conditions such as
Von Mises conditions for the asymptotic normality and give a minimal
condition under which a non constant centering can be used.
Compressed postscript file TR1155.ps.Z available at
http://www.orie.cornell.edu/trlist/trlist.html
sid@orie.cornell.edu dehaan@cs.few.eur.nl
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information provided by the author
click here.
670. TRANSFORMATIONS ET EQUATIONS ANTICIPANTES POUR LES PROCESSUS DE POISSON
Jean Picard
We prove the existence and uniqueness of a solution for stochastic
anticipative equations driven by a point Poisson process. For a particular
class of linear equations, the solution may be interpreted as the
Radon-Nikodym density of an anticipative transformation of the Poisson process.
DVI file available in:
ftp://ucfma.univ-bpclermont.fr/pub/PUBLI/lma/1996/03-96.dvi
picard@ucfma.univ-bpclermont.fr
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information provided by the author
click here.
671. INTERFACE FOR ONE-DIMENSIONAL RANDOM KAC POTENTIALS
Thierry Bodineau
We consider a ferromagnetic Ising system with impurities where the
interaction is given by a Kac potential of scaling parameter $\gamma>0$.
The random position of the magnetic atoms is decribed by a quenched
variable $y$. In the Lebowitz Penrose limit $\gamma \to 0^+$, we prove
that the quenched Gibbs measure obeys a large deviation principle with
rate function depending on $y$.
We then show that for almost all $y$ the magnetization is locally approximately
constant. However, interfaces occur and magnetization change for almost
all $y$ at a distance of the order of $\exp(\frac{\Phi}{\gamma})$,
where $\Phi$ is a constant given by a variational formula.
bodineau@dmi.ens.fr (latex file available)
672. RELAXED CRITERIA OF THE DOBRUSHIN-SHLOSMAN MIXING CONDITION
Nobuo Yoshida
An interacting particle system (Glauber dynamics),
which evolves on a finite subset
in the $d$-dimensional integer lattice is considered.
It is known that
a mixing property of the Gibbs state in the sense of
Dobrushin and Shlosman is equivalent to
several very strong estimates in terms of the Glauber
dynamics.
We show that similar, but seemingly much milder
estimates are again equivalent to the Dobrushin-Shlosman
mixing condition, hence to the original ones.
This may be understood as the absence of intermediate speed
of convergence to equilibrium.
nobuo@math.mit.edu nobuo@kusm.kyoto-u.ac.jp (latex file available)
673. A CONSTRUCTION OF SUPERPROCESSES WITH MASS CREATION
Luis G. Gorostiza and Eliane Rodrigues
Superprocesses with mass creation (and killing) are constructed as
high-density limits of branching particle systems with a change of mass
mechanism attached to the branching. In the approach presented by E. B.
Dynkin ("An Introduction to Branching Measure-Valued Processes", CRM
Monograph Series, Vol.6, AMS, 1994), the single particle process already
contains the mass creation and the superprocess is built over it. In our
approach the mass creation in the superprocess limit is a consequence of
the change of mass connected with the branching in the approximating
particle system. This approach gives new insight into superprocesses with
mass creation. We make some assumptions on the model in order to present
the main ideas with simplicity. The model where the mass of a particle is
distributed uniformly among its offspring behaves differently. In this
case there is a superprocess limit which is deterministic, and the
fluctuaction limit process is a generalized Ornstein-Uhlenbeck process.
gortega@servidor.dgsca.unam.mx
674. SOME WIDTH FUNCTION ASYMPTOTICS FOR WEIGHTED TREES
Mina Ossiander, Ed Waymire, Qing Zhang
Consider a rooted labelled tree graph $\tau_n$ having a total of $n$
vertices. The {\it width function} counts the number of vertices as a
function of the distance to the root $\phi$. In this paper we compute
large $n$ asymptotic behavior of the width functions for two classes of
tree graphs (both random and deterministic) of the following types:
(i) Galton-Watson random trees $\tau_n$ conditioned or total progeny
and (ii) a class of deterministic self-similar trees which include an
\lq\lq expected\rq\rq \ Galton-Watson tree in a sense to be made precise.
The main results include:
(i) an extension of Aldous's theorem \cite {2} on \lq\lq search-depth\rq\rq
approximations by Brownian excursion to the case of weighted Galton-Watson
trees
(ii) a probabilistic derivation which generalizes previous results by
Troutman and Karlinger on the asymptotic behavior of the expected width
function and provides the fluctuation law;
and
(iii) width function asymptotics for a class of deterministic
self-similar trees of interest in the study of river network data.
ossiand@math.orst.edu, waymire@math.orst.edu
675. MODERATE DEVIATIONS FOR ITERATES OF EXPANDING MAPS
Amir Dembo and Ofer Zeitouni
We provide a mild mixing condition that carries the C.L.T.
for empirical means of centered stationary sequence of bounded random
variables to the whole range of moderate deviations. It is also key for
the exponential convergence to zero of these empirical means.
The motivating example for this work are iterates
of expanding maps, equipped with their unique invariant measure.
Postscript file available at
http://www-ee.technion.ac.il/~dembo/tent.ps
amir@playfair.stanford.edu (LaTeX file available)
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information provided by the author
click here.
676. THE SHERRINGTON KIRKPATRICK MODEL: A CHALLENGE FOR THE MATHEMATICIAN
Michel Talagrand
This is an effort to introduce a more unified methodology to the
study of the famous Sherrington-Kirkpatrick model of spin glasses, and
to point out some of the numerous questions that remain open, even in
the high temperature region. The main techniques are elementary, and
based on the computation of 2-spins correlation by induction over the
number of sites. Some use is also made of abstract tools, such as
concentration of measure.
Talagran@math.ohio-state.edu (Postcript file available)
677. SHORT TIME ASYMPTOTIC BEHAVIOUR AND LARGE DEVIATION
FOR BROWNIAN MOTION ON SOME AFFINE NESTED FRACTALS
Takashi Kumagai
We study so called Varadhan type short time asymptotic
estimates of heat kernels and Shilder type large deviation for
Brownian motion on some affine nested fractals.
For the Brownian motion on the Sierpinski gasket, we show that
there is a fluctuation so that the Varadhan type estimate does not hold.
We also show that when the ratio of the time scale of each cell is
irrational, these estimates holds using an intrinsic metric of the process.
As a corollary to our approach, we obtain sharper estimates of heat
kernels for a class of one dimensional diffusion processes studied in
[T.Fujita: Publ. RIMS, 26, 819--840, 1990].
Kumagai@math.nagoya-u.ac.jp (LaTex-file available)
678. SELF-NORMALIZED MODERATE AND LARGE DEVIATIONS
Amir Dembo and Qi-Man Shao
We prove Partial Large Deviation Principles for self-normalized partial sums
of i.i.d. R^d valued random vectors with minimal or no moment assumptions,
for both moderate and large deviation scales.
Applications to the self-normalized law of the iterated logarithm are
also discussed.
amir@playfair.stanford.edu (LaTeX file)
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information provided by the author
click here.
679. GROMOV'S HYPERBOLICITY AND PICARD'S LITTLE THEOREM FOR HARMONIC MAPS
Mike Cranston, Wilfrid Kendall, Yuri Kifer
This paper extends some previous results concerning limiting direction
of Brownian motion on manifolds satisfying Gromov's hyperbolicity
criterion and other conditions, by generalizing from Brownian motion to
the case of quasi-conformal $\Gamma$-martingales (at the price of
making some conditions more restrictive). The extended results are used
to give a probabilistic proof for a new generalization of Picard's
little theorem for harmonic maps of bounded quasi-conformality.
Preprint (hard copy) available as Warwick Statistics Research Report 280
from Secretary, Department of Statistics, University of Warwick,
Coventry CV4 7AL, UK.
W.S.Kendall@warwick.ac.uk
680. SPATIAL ESTIMATES FOR STOCHASTIC FLOWS IN EUCLIDEAN SPACE
Salah-Eldin A. Mohammed and Michael K. R. Scheutzow
We study the behavior for large $|x|$ of Kunita-type stochastic flows
$\phi (t, x)$ on $\R^d$, driven by continuous spatial semimartingales.
For this class of flows we prove new spatial estimates for large $|x|$,
under very mild regularity conditions on the driving semimartingale
random field. In particular we show that the stochastic flow $\phi_{s,t} (x)$
grows slower than $|x| (\log |x|)^{\epsilon}$ as $|x|$ goes to infinity,
for arbitrarily small positive $\epsilon$. We show by example that this
bound is sharp.
We give an example of a one-dimensional s.d.e. with sublinear coefficients
but with the underlying stochastic flow growing superlinearly for
large $|x|$. In this example the stochastic flow has a.s. unbounded spatial
derivatives, even though the driving martingale has local characteristics
with all derivatives globally bounded. It is interesting to note that in
this example the driving noise is infinite-dimensional. However
the infinite-dimensionality of the driving noise is not the crucial factor.
To illustrate this point we provide an example of a s.d.e. driven
by one-dimensional Brownian motion, has coefficients with globally bounded
derivatives, while its stochastic flow has a.s. unbounded derivatives.
This result is surprising since it is in sharp contrast with well-known
behavior of deterministic flows driven by vector fields whose
derivatives are globally bounded.
For one-dimensional s.d.e.'s, sufficient conditions on the coefficients
are given in order for the stochastic flow to have sublinear growth
and a.s. bounded derivatives.
It is expected that the results would be of interest for the theory of
stochastic flows on non-compact manifolds as well as in the study of
non-linear filtering, stochastic functional and partial differential equations.
salah@math.siu.edu ms@math.tu-berlin.de
(AMS TeX file or hard copy available)
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information provided by the author
click here.
681. INFINITE CLUSTERS IN DEPENDENT AUTOMORPHISM INVARIANT
PERCOLATION ON TREES
Olle H\"aggstr\"om
We study dependent bond percolation on the homogeneous tree ${\bf T}_n$ of
order $n\geq 2$, under the assumption of automorphism invariance. Excluding
a trivial case, we find that the number of infinite clusters a.s.\ is
either 0 or $\infty$. Furthermore, each infinite cluster a.s.\ has either
1, 2 or infinitely many topological ends, and under the additional assumption
of finite energy the only possible number of topological ends is $\infty$.
We also show that if the marginal probability that a single edge is open
is at least ${2/(n+1)}$, then the existence of infinite clusters has to have
positive probability. Several concrete examples are considered, and all
results have analogues for site percolation.
olleh@math.chalmers.se
682. UNIFORM AND MINIMAL ESSENTIAL SPANNING FORESTS ON TREES
Olle H\"aggstr\"om
Uniform and minimal random spanning trees for finite graphs are
well-known objects. Analogues of these for the nearest-neighbour graph
on ${\bf Z}^d$ have been studied by Pemantle and Alexander. Here we
propose analogous definitions of uniform resp.\ minimal essential
spanning forests for an infinite tree $\Gamma$. We study these random
objects in some detail in the case when $\Gamma$ is a
regular tree of order $n\geq 2$, where they share
the properties that each edge is present with probability $2/(n+1)$, and
every vertex a.s.\ has exactly one self-avoiding path to infinity. Despite
these similarities, the distributions are different. The uniform essential
spanning forest admits a simple description in terms of a Galton--Watson
process, and also arises as a limit of random-cluster measures.
olleh@math.chalmers.se
683. HYDRODYNAMIC SCALING, CONVEX DUALITY, AND ASYMPTOTIC
SHAPES OF GROWTH MODELS
Timo Seppalainen
We present a technique for simultaneously deriving two related results:
Hydrodynamic scaling limits for one-dimensional asymmetric particle
systems and asymptotic shapes for growth models. The idea is to specify
the particle dynamics in terms of a microscopic Lax-Oleinik formula
which leads directly to the macroscopic description in terms of a
nonlinear conservation law. The law of large numbers required for
this link comes from the growth model that is embedded in the particle
system. In the limit, the asymptotic shape of the growth model becomes
the convex conjugate of the flux of the conservation law, and the latter
is computable from the particle system in equilibrium. The asymptotic shape
is then obtained from the duality relation. The method is illustrated
with four applications.
seppalai\@iastate.edu (Hardcopy and AMS-TeX file available)
684. DIFFEOMORHISM OF THE CIRCLE AND THE BASED STOCHASTIC LOOP SPACE
R. Leandre
We use the theory of left quasi-invariant measure over the diffeomorphism
of the circle of Shavgulidze and Malliavin and Malliavin in order to define
the action of the diffeomorphism of the circle as well as its infinitesimal
action over a stochastic loop space, which gives an absolutely continuous
transformation of the based loop space.
Remi.Leandre@iecn.u-nancy.fr
685. HILBERT SPACE OF SPINORS FIELDS OVER THE FREE LOOP SPACE
R. Leandre
We construct Hilbert space of spinor fields over the free loop space of a
Riemannian manifold whose the first Pontryaguin class is equal to 0. The
transition function are almost surely measurable, and takes their values in
a central extension of a loop group. Therefore we cannot define the
topological space of the infinite dimensional spin bundle, but we define
its associated bundle by means of its measurable sections or of it $L^2$
sections. We recall some basical facts about the spin representations in
infinite dimension.
Remi.Leandre@iecn.u-nancy.fr
686. STOCHASTIC GAUGE TRANSFORM OF THE STRING BUNDLE
R. Leandre
We consider the frame bundle of the loop space whose the fiber is formally
a loop group. We put a measure over the fiber which lives either over
continuous loop or finite energy loop (Bismut scheme). Over the total space
we get a measure and therefore Sobolev spaces. We can define the string
bundle when the first Pontryaguin class is equal to 0 by its space of $L^p$
functionals. The gauge transforms act over the associated functional spaces
and use deeply the quasi-invariant formula over loop groups of
Albeverio-Hoegh-Krohn.
Remi.Leandre@iecn.u-nancy.fr
687. COVER OF THE BROWNIAN BRIDGE AND STOCHASTIC SYMPLECTIC ACTION
R. Leandre
The symplectic stochastic action is defined over the universal cover of the
Brownian bridge. A diffusion process is defined over the universal cover
of the brownian bridge by lifting the diffusion process of Driver-Rockner
over the based loop space. A stochastic de Rham complex is defined by
lifting the stochastic de Rham complex over the brownian bridge defined by
the author, as well as the lift of stochastic Chen forms.
Remi.Leandre@iecn.u-nancy.fr
688. PERCOLATION BEYOND Z^d
Itai Benjamini and Oded Schramm
The critical probability for percolation, $p_c$, and uniqueness of the
infinite open cluster are discussed. When a graph $G$ covers a graph
$G'=G/\Gamma$, we prove that the probability for percolation in $G'$ is
not greater than in $G$. It is shown that graphs $G$ with positive
Cheeger constant $h=h(G)>0$ (strong isoperimetric inequality) satisfy
$p_c(G)\leq (1+h)^{-1}$. A sufficient condition for non-uniqueness of
the infinite open cluster in almost transitive graphs is given. These
three results are proved using simple coupling methods. Additionally,
an elementary 0-1 law is proved for the uniqueness of the infinite open
cluster in almost transitive graphs, and it is shown that there are
infinitely many infinite clusters when there is non-uniqueness. A
simple planar example exhibiting uniqueness and non-uniqueness for
different $p>p_c$ is analysed. Thirteen open problems are presented.
itai@wisdom.weizmann.ac.il or schramm@wisdom.weizmann.ac.il
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information provided by the author
click here.
689. EXPONENTIAL STABILITY FOR NONLINEAR FILTERING
Rami Atar and Ofer Zeitouni
We study the a.s. exponential stability of the optimal filter w.r.t. its
initial conditions. A bound is provided on the exponential rate
(equivalently, on the memory length of the filter) for a general setting
both in discrete and in continuous time, in terms of Birkhoff's
contraction coefficient. Criteria for stability and explicit bounds on
the rate are given in the specific cases of a diffusion process on a
compact manifold, and discrete time Markov chains on both continuous
and discrete-countable state spaces.
rami@aluf.technion.ac.il zeitouni@ee.technion.ac.il
690. ERGODICITY OF $L^2$--SEMIGROUPS AND EXTREMALITY OF GIBBS STATES
S. Albeverio, Y.G. Kondratiev, and M. R\"ockner
We extend classical results of Holley--Stroock on the characterization
of extreme Gibbs states for the Ising model in terms of the
irreducibility (resp\. ergodicity) of the corresponding Glauber
dynamics to the case of lattice systems with unbounded (linear) spin
spaces. We first develop a general framework to discuss questions of
this type
using classical Dirichlet forms on infinite dimensional state spaces
and their associated diffusions. We then describe concrete
applications to lattice models with polynomial interactions (i.e., the
discrete $P(\varphi)_d$--models of Euclidean quantum field theory). In
addition, we prove the equivalence of extremality and
shift--ergodicity for tempered Gibbs states of these models and also
discuss this question in the general framework.
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
691. CONVERGENCE OF OPERATOR SEMIGROUPS GENERATED BY ELLIPTIC OPERATORS
Michael R\"ockner and Tu--Sheng Zhang
Consider a sequence of operator semigroups $\(T^{(n)}_t\)_{t>0}$, $n \in
\Bbb N$, whose generators are elliptic and are (informally) of type
$L^{(n)} = \sum^d_{i,j=1} \partial_i\(a_{ij}^{(n)} \partial_j +
d^{(n)}_i\) - \sum^d_{i=1} b^{(n)}_i \partial_i - c^{(n)}$ on a
possibly unbounded open set $U \subset \Bbb R^d$ with measurable
coefficients. Under weak assumptions on the coefficients we prove
strong convergence of $\(T^{(n)}_t\)_{t>0}$, $n \in \Bbb N$, on
$L^2(U;dx)$ resp. the Sobolev space $H^{1,2}_0(U;dx)$. In
particular, this is done without assuming that $b^{(n)}_i,
d^{(n)}_i, c^{(n)}$, are bounded in $L^\infty(U;dx)$ uniformly in
$n$.
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
692. MISSPECIFIED ASSET PRICE MODELS AND ROBUST HEDGING STRATEGIES
Hyungsok Ahn, Adviti Muni and Glen Swindle
The Black-Scholes theory of option pricing requires a perfectly
specified model for the underlying price. Frequently this is
taken to be a geometric Brownian motion with a constant, known
volatility. In practice, parameters such as the volatility
are not known precisely, but are simply estimates from
either historical prices or implied volatilities.
This paper presents a method for constructing hedging
(trading) strategies which are ``robust'' to misspecifications
of the asset price model.
swindle@pstat.ucsb.edu (Postscript or hardcopy available)
693. WHY NON-LINEARITIES CAN RUIN THE HEAVY TAILED MODELER'S DAY
Sidney I. Resnick
A heavy tailed time series that can be expressed as an
infinite order moving average has the property that the sample
autocorrelation function (acf) at lag $h$, converges in probability
to a constant $\rho (h)$ despite the fact that the mathematical
correlation typically does not
exist. A simple bilinear model considered by Davis and Resnick (1996) has
the property that the sample autocorrelation function at lag $h$ converges in
distribution to a non-degenerate random variable. Examination of various
data sets exhibiting heavy tailed behavior reveals that the sample
correlation function typically does not
behave like a constant. Usually, the sample
acf of the first half of the data set looks considerably different than the
sample acf of the second half. A possible explanation for this acf behavior
is the presence of nonlinear components in the underlying model
and this seems to imply that infinite order moving
average models and in particular ARMA models
do not adequately capture dependency structure in the presence of heavy
tails. Some additional results
about the simple nonlinear model are discussed and in particular we consider
how to estimate coefficients.
Available as TR 1157 at the url below.
sid@orie.cornell.edu
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information provided by the author
click here.
694. A LIMIT THEOREM FOR OCCUPATION TIMES OF FRACTIONAL BROWNIAN MOTION
Yuji Kasahara and Nobuko Kosugi
Recently, N.K\^ono gave a limit theorem for occupation times of
fractional Brownian motion, which result generalizes the well-known
Kallianpur-Robbins law for 2-dimensional Brownian motion.
The present paper studies a functional limit theorem for K\^ono's result.
It is proved that, under a suitable normalization, the limiting process is
the inverse of an extremal process.
kasahara@is.ocha.ac.jp (amsTeX-file available)
695. MARTINGALE DECOMPOSITION OF DIRICHLET PROCESSES ON THE BANACH SPACE
$C[0,1]$
T. J. Lyons, M. R\"ockner, T. S. Zhang
We prove that for a given symmetric Dirichlet form of type $\cal E (u,v)=
\int_E \<A(z)\nabla u(z), \nabla v(z)\>_H \mu(dz)$ with $E=C_0[0,1]$ and $H=$
classical Cameron-Martin space the corresponding diffusion process (under
$P_\mu$) can be decomposed into a forward and a backward $E$-valued
martingale. The construction of the martingale is direct and explicit
since it is based on a modification of Levy's construction of Brownian motion.
Applications to prove tightness of laws of diffusions of the
above kind are given.
(SFB-343-Bielefeld-Preprint-95-048)
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
696. FINITE DIMENSIONAL APPROXIMATION OF DIFFUSION PROCESSES ON INFINITE
DIMENSIONAL SPACES
M.~R\"ockner, T.~S.~Zhang
We prove that the laws of diffusion processes $\bfM $ on $E$
associated with Dirichlet forms of type $\calE (u,v)=\int_E
\<A(z)\nabla u(z), \nabla v(z)\>_H \mu(dz)$, where $H$, $E$ are
separable Hilbert spaces, are the weak limits of laws of finite
dimensional diffusions. These are associated with the image Dirichlet
forms obtained from $\calE$ under projections from $E$ onto finite
dimensional subspaces in $H$. As a by-product we obtain Hoelder
continuity of the sample paths as well as a new existence proof for
the infinite dimensional diffusion $\bfM$.
(SFB-343-Bielefeld-Preprint-95-080)
roeckner@mathematik.uni-bielefeld.de
- To see a preprint or other
information provided by the author
click here.
697. STOCHASTIC QUANTIZATION OF THE TWO-DIMENSIONAL POLYMER MEASURE
Sergio Albeverio, Yao-Zhong Hu, Michael R\"ockner, Xian Yin Zhou
We prove that there exists a diffusion process whose invariant
measure is the two-dimensional polymer measure $\nu_g$.
The diffusion is constructed by means of the theory of Dirichlet forms on
infinite-dimensional state spaces. We prove the closability of the appropriate
pre-Dirichlet form which is of gradient type,
using a general closability result by two of the authors. This result does
not require an integration by parts formula (which does not hold for the
two-dimensional polymer measure $\nu_g$) but requires the quasi-invariance of
$\nu_g$ along a basis of vectors in the classical Cameron-Martin space such
that the Radon-Nikodym derivatives (have versions which) form a continuous
process. We also show the Dirichlet form to be irreducible
or equivalently that the diffusion process is ergodic under
time translations.
(SFB-343-Bielefeld-Preprint-96-014)
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
698. STOCHASTIC QUANTIZATION OF THE THREE-DIMENSIONAL POLYMER MEASURE
Sergio Albeverio, Michael R\"ockner, Xian Yin Zhou
We prove that there exists a diffusion process whose invariant
measure is the three-dimensional polymer measure $\nu_\lambda$.
The diffusion is constructed by means of the theory of Dirichlet forms on
infinite-dimensional state spaces. We prove the closability of the appropriate
pre-Dirichlet form which is of gradient type,
using a general closability result by two of the authors. This result does
not require an integration by parts formula (which does not even hold for the
two-dimensional polymer measure $\nu_\lambda$) but
requires the quasi-invariance of $\nu_\lambda$ along a basis of vectors
in the classical Cameron-Martin space such
that the Radon-Nikodym derivatives (have versions which) form a continuous
process.
(SFB-343-Bielefeld-Preprint-96-015)
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
699. DIRICHLET FORMS ON INFINITE-DIMENSIONAL "MANIFOLD-LIKE" STATE SPACES:
A SURVEY OF RECENT RESULTS AND SOME PROSPECTS FOR THE FUTURE
Michael R\"ockner
We give a (to some extent pedagogical) survey on recent results about
Dirichlet forms on infinite-dimensional ``manifold-like'' state spaces including
path and loop spaces as well as spaces of measures. The latter are associated
with interacting Fleming-Viot processes resp. infinite particle systems. Also
some new results, further developing the Dirichlet form approach to infinite
particle systems, are enclosed. Finally, a summary of other research activities
in the theory of Dirichlet forms is given and some prospects for the future are indicated.
(SFB-343-Bielefeld-Preprint-96-026)
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
700. THE MARTINGALE PROBLEM FOR PSEUDO-DIFFERENTIAL OPERATORS ON
INFINITE-DIMENSIONAL SPACES
V. Bogachev, P. Lescot, M.~R\"ockner
A martingale problem for pseudo-differential operators on infinite dimensional
spaces is formulated and the existence of a solution is proved.
(SFB-343-Bielefeld-Preprint-95-093)
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
701. UNIQUENESS OF THE SKOROKHOD EQUATION IN LIPSCHITZ DOMAINS
Richard F. Bass
We consider the Skorokhod equation
$$dX_t=dW_t+\frac12\nu(X_t)\, dL_t$$
in a domain $D$, where $W_t$ is Brownian motion in $\R^d$,
$\nu$ is the inward pointing normal vector on the boundary of $D$,
and $L_t$ is the local time on the boundary. The
solution to this equation is reflecting Brownian motion in $D$.
In this paper we show that in Lipschitz domains the solution to
the Skorokhod equation is unique in law.
bass@math.washington.edu
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information provided by the author
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702. HITTING ESTIMATES FOR GAUSSIAN RANDOM FIELDS
Davar Khoshnevisan and Zhan Shi
We provide estimates for the probability
that a continuous, stationary Gaussian random field hits
a small ball. Our estimates are used to study the
sample functions of (N,d) Brownian sheet and its
corresponding integrated process.
davar@math.utah.edu
703. UNIQUENESS OF GIBBS STATES FOR QUANTUM LATTICE SYSTEMS
S. Albeverio, Yu.G. Kondratiev, M. R"ockner and T.V. Tsikalenko
We prove uniqueness of Gibbs states for certain quantum lattice
systems with unbounded spins. We use Dobrushins's uniqueness
criterion. The necessary estimates for the Vasershtein distance
between the corresponding one-point conditional distributions with
boundary conditions differing only at one side, are obtained by
proving a Log-Sobolev inequality on the infinite dimensional single
spin ( = loop ) spaces. Some important classes of concrete examples to
which all this applies are discussed.
(SFB-343-Bielefeld-Preprint-96-022)
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
704. PARTITIONS OF UNITY IN SOBOLEV SPACES OVER INFINITE DIMENSIONAL STATE
SPACES
S. Albeverio, Zhi-Ming Ma, M. R"ockner
We prove some general results on the existence of partitions of unity
in Sobolev type spaces on various infinite dimensional manifolds. As
special cases we obtain in particular, (continuous) partitions of
unity a) in the Malliavin test functions on an abstract Wiener space;
b) in the first order Sobolev space on pinned and free loop spaces; c)
in the first order Sobolev spaces associated with reversible
Fleming-Viot processes.
(SFB-343-Bielefeld-Preprint-95-009)
roeckner@mathematik.uni-bielefeld.de
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705. PERTURBED BROWNIAN MOTIONS
Mihael Perman and Wendelin Werner
The paper deals with one-dimensional Brownian motion
perturbed when it hits its minimum and/or its maximum.
It first presents some features of perturbed
reflected Brownian motion
defined as $\vert B \vert -\mu\ell$ where $B$ is standard Brownian motion,
$\ell$ its local time at 0 and $\mu$ a positive constant;
in particular it is shown that the positive and negative excursions
in the sense of It\^o of this ``one--sided'' perturbed
Brownian motion are two independent point processes,
which in turn is used to construct two--sided perturbed Brownian motion.
It is then shown that the constructed process is almost surely
the unique solution of the implicit stochastic equation
studied by Le Gall, Carmona-Petit-Yor and Davis, even when no
restrictions are imposed on the partial reflections.
Finally, the Hausdorff dimension of sets of exceptional points
for perturbed Brownian motion such as points of monotonicity are computed.
mihael.perman@uni-lj.si, wwerner@dmi.ens.fr
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706. LATTICE TREES AND SUPER-BROWNIAN MOTION.
Gordon D. Slade and Eric Derbez
This article discusses our recent proof that above eight dimensions the scaling
limit of sufficiently spread-out lattice trees is the variant of super-Brownian
motion called integrated super-Brownian excursion (ISE), as conjectured by
Aldous. The same is true for nearest neighbour lattice trees in sufficiently
high dimensions. The proof, whose details will appear elsewhere, uses the lace
expansion to perturb around a corresponding argument, valid in all dimensions,
showing that the scaling limit of a mean-field theory is ISE. The calculation
for the mean-field theory is sketched here. A connection is drawn between ISE
and certain generating functions and critical exponents, which may be useful in the study of high-dimensional percolation models at the critical point.
slade@mcmaster.ca derbez@mcmaster.ca
707. NUMERAIRE INVARIANCE AND GENERALIZED RISK NEUTRAL VALUATION
Aleksandar Kocic
We study the pricing of contingent claims in complete markets with arbitrary
number of assets. We show that basic concepts like replicating portfolios
and self financing trading strategies follow from simple dimensional
analysis arguments and give mathematical and economic explanations of the
numeraire invariance. For each choice of the numeraire there is a measure
with respect to which all dimensionless processes are martingales. When all
the variables are expressed in dimensionless form with one component of the
asset vector as a numeraire, dimensionless Black-Scholes equation follows.
We derive a generalization of the risk neutral valuation formula for
complete markets with an arbitrary number of assets without specifying their
stochastic character and show how risk preferences disappear. The trimming
of the degrees of freedom in this setting is manifest -- the dimensionality
of the pricing problem is lower than dimensionality of the asset space. We
give a simple geometric interpretation of diffusion in an arbitrage free
economy and illustrate the method with several examples.
kocic@uiuc.edu
708. ERGODIC THEOREMS FOR STOCHASTIC OPERATORS AND DISCRETE
EVENT NETWORKS
Francois Baccelli and Jean Mairesse
We present a survey of the main ergodic theory techniques which are
used in the study of iterates of monotone and
homogeneous stochastic operators.
It is shown that ergodic theorems on discrete event networks
(queueing networks and/or Petri nets) are a generalization
of corresponding results for stochastic operators.
Kingman's subadditive ergodic Theorem is the key tool
for deriving what we call first order ergodic results.
We also show how to use backward constructions (also called
Loynes schemes in network theory) in order
to obtain second order ergodic results.
We will propose a review of systems entering the framework insisting on two
models, precedence constraints networks and Jackson type networks.
baccelli@sophia.inria.fr, mairesse@statslab.cam.ac.uk
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here.
709. GEOMETRIC ASPECTS OF FLEMING-VIOT AND SUPERPROCESSES
Alexander Schied
The main purpose of this paper is to show that the intrinsic metric
of the Fleming-Viot process is given by an "angular distance" on the
space of probability measures. It turns out that it is closely related
to the branching structure of a continuous superprocess, which itself
induces the Kakutani-Hellinger distance. The corresponding geometries
are studied in some detail. In particular, representation formulae for
geodesics and arc length functionals are obtained. As an application, a
functional limit theorem for super-Brownian motion conditioned to local
extinction is proved.
schied@mathematik.hu-berlin.de (TeX or Postscript file available)
710. SOME AFFINE RANDOM WALKS ON FINITE FIELDS
Martin Hildebrand
This paper considers random processes of the form
$X_{n+1}=TX_n+B_n$ where $T$, $B_n$ are in a finite field
expressed as matrices and $X_0=0$. This paper considers how large
$n$ should be if $X_n$ is close to uniformly distributed for a
particular choice of $B_n$. If $T$ is a constant $d$x$d$ integer
matrix appearing in infinitely many fields of order $p^d$, then
this time depends on the complex eigenvalues of $T$. If $T$ has
no eigenvalues of length $1$ or $0$, then this time is at most
order $(\log p)^2$ while if $T$ has an eigenvalue which is a root
of unity, then this time is at least order some power of $p$.
Techniques involving probability on finite groups are used to
prove these results.
mvh@math.utexas.edu (TeX file and hard copy available)
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