Probability Abstracts 42
This document contains abstracts 908-938.
They have been mailed on December 1, 1997.
908. HOW SYSTEM PERFORMANCE IS AFFECTED BY THE INTERPLAY OF AVERAGES IN A FLUID
QUEUE WITH LONG RANGE DEPENDENCE INDUCED BY HEAVY TAILS
David Heath, Sidney Resnick and Gennady Samorodnitsky
We consider a
fluid queue with sessions arriving according to a Poisson
process. A long--tailed distribution of session lengths induces
long range dependence in the system and
causes its performance to deteriorate.
The deterioration is due to occurrence of load
regimes far from average ones. Nonetheless, the
{\it extent} of this performance deterioration is shown to
depend crucially on the average values of the system parameters.
davidh@orie.cornell.edu, sid@orie.cornell.edu, gennady@orie.cornell.edu
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information provided by the author
click here.
909. MEASURE-VALUED IMMIGRATON PROCESSES AND KUZNETSOV PROCESSES
Zeng-Hu Li
An immigration process associated with the Borel right
Dawson-Watanabe superprocess does not always have a right
continuous realization. We construct the immigration process by
adding up measure-valued paths in the Kuznetsov process determined
by an entrance rule for the superprocess. The path behavior of the
Kuznetsov process is studied, which gives insights into trajectory
structures of the immigration process. Some known results on
excessive measures are interpreted probabilistically in terms of
stationary immigration processes. It is also shown that a class of
infinite dimensional Ornstein-Uhlenbeck processes may arise as
fluctuation limits of the stationary immigration processes.
lizh@bnu.edu.cn (AMS-TeX file and hardcopy available)
910. THE ASYMPTOTICS OF WAITING TIMES BETWEEN STATIONARY PROCESSES,
ALLOWING DISTORTION
Amir Dembo and Ioannis Kontoyiannis
Given two independent realizations of the stationary processes
$\{X_n ; n\geq 1\}$ and $\{Y_n ; n\geq 1\}$, our main quantity
of interest is the waiting time $W_n(D)$ until a $D$-close version
of the initial string $(X_1,X_2,\ldots,X_n)$ first appears as a
contiguous substring in $(Y_1,Y_2,Y_3,\ldots)$, where closeness is
measured with respect to some ``average distortion'' criterion.
We study the asymptotics of $W_n(D)$ for large $n$ under various
mixing conditions on $\{X_n\}$ and $\{Y_n\}$. We first prove a
strong approximation theorem between $\log W_n(D)$ and the
logarithm of the probability of a $D$-ball around $(X_1,X_2,\ldots,X_n)$.
Using large deviations techniques, we show that this probability can,
in turn, be strongly approximated by an associated random walk,
and we conclude that: (i) $n^{-1}\log W_n(D)$ converges almost surely
to a constant $R$ determined by an explicit variational problem;
(ii) $[\log W_n(D)-R]$, properly normalized, satisfies a central
limit theorem, a law of the iterated logarithm, and, more generally,
an almost sure invariance principle.
yiannis@isl.stanford.edu
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information provided by the author
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911. THE QUASI-SURE RATIO ERGODIC THEOREM
P. J. Fitzsimmons
Let (P_t)_{t\ge 0} be the transition semigroup of a right Markov process, and
let m be a conservative (P_t)-invariant measure. Let f and g be elements
of L^1(m) with g > 0. We show that, with the exception of an m-polar set of
starting points x, the ratio int_0^t P_s[f](x)ds / int_0^t P_s[g](x)ds
converges as t --> infty, and we identify the limit as a ratio of
conditional expectations with respect to the appropriate invariant
sigma-algebra. This improves upon earlier work of M. Fukushima and M.G.
Shur, in which the exceptional set was shown to be m-semipolar. The proof
is based on Neveu's presentation of the Chacon-Ornstein filling scheme,
adapted to continuous time. The method yields, as a by-product, a local
limit theorem for the ratio of the ``characteristics'' of two continuous
additive functionals, extending work of Mokobodzki.
pfitzsim@ucsd.edu
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information provided by the author
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912. BROWNIAN SPACE-TIME FUNCTIONS
OF ZERO QUADRATIC VARIATION DEPEND ONLY ON TIME
P. J. Fitzsimmons
Let B(t), t >= 0, be a d-dimensional Brownian motion and let f be a
continuous function mapping R^d x [0,infty[ into R. We show that if
t --> f(B(t),t) is locally of zero quadratic variation, then
f(x,t) = f(0,t) for all (x,t). This result extends recent work of F.B.
Knight, thereby confirming a conjecture of T. Salisbury.
pfitzsim@ucsd.edu
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information provided by the author
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913. RANDOM WALKS ON GRAPHICAL SIERPINSKI CARPETS
Martin T. Barlow and Richard F. Bass
We consider random walks on a class of graphs derived from
Sierpinski carpets. We obtain upper and lower bounds (which are non-Gaussian)
on the transition probabilities which are, up to constants, the best possible.
We also extend some classical Sobolev and Poincar\'e inequalities to this
setting.
bass@math.washington.edu
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information provided by the author
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914. CRITICAL PERCOLATION ON ANY NONAMENABLE GROUP HAS NO INFINITE CLUSTERS
Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm
We show that independent percolation on any Cayley graph of a
nonamenable group has no infinite components at the critical
parameter. We first obtained this result as a corollary of a general study
of group-invariant percolation. The goal here is to present a simpler
self-contained proof that easily extends to quasi-transitive graphs
with a unimodular automorphism group. The key tool is a ``mass-transport''
method, which is a technique of averaging in nonamenable settings.
itai@wisdom.weizmann.ac.il rdlyons@indiana.edu
peres@stat.berkeley.edu schramm@wisdom.weizmann.ac.il
915. BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH CONSTRAINTS ON
THE GAINS-PROCESS
Jaksa Cvitanic, Ioannis Karatzas and Mete Soner
We study Backward Stochastic Differential Equations with convex
constraints on the gains (or intensity-of-noise) process. Existence and
uniqueness of a minimal solution
are established in the case of a drift coefficient which is
Lipschitz-continuous in the state- and gains-processes,
and convex in the gains-process. It is also shown that
the minimal solution can be characterized as the unique solution
of a functional stochastic control-type equation. This representation
is related to the penalization method for constructing solutions of stochastic
differential equations, involves change of measure techniques, and
employs notions and results from convex analysis, such as the
support function of the convex set of constraints and its various properties.
cj@stat.columbia.edu
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information provided by the author
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916. THE MAXIMUM OF THE PERIODOGRAM FOR A HEAVY--TAILED SEQUENCE
Thomas Mikosch, Sidney Resnick, and Gennady Samorodnitsky
We consider the maximum of the periodogram based on an infinite variance
heavy--tailed sequence. For $\alpha<1$ we show that the maxima
constitute a weakly convergent sequence, and find its limiting distribution.
For $1\leq\alpha <2$ we show that the sequence of the maxima is not
tight, and find a normalization that makes it tight.
sid@orie.cornell.edu
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information provided by the author
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917. AMENABILITY AND PHASE TRANSITION IN THE ISING MODEL
Johan Jonasson and Jeffrey E. Steif
We consider the Ising model with external field $h$ and coupling constant
$J$ on an infinite connected graph $G$ with uniformly bounded degree.
We prove that if $G$ is nonamenable, then the Ising model exhibits phase
transition for some $h \neq 0$ and some $J<\infty$.
On the other hand, if $G$ is amenable and quasi--transitive, then phase
transition cannot occur for $h \neq 0$.
In particular, a group is nonamenable if and only if the Ising model on
one (all) of its Cayley graphs exhibits a phase transition for some $h \neq 0$
and some $J<\infty$.
expect@math.chalmers.se steif@math.chalmers.se
918. APPLICATIONS OF A REPRESENTATION OF LONG MEMORY GAUSSIAN PROCESSES
Philippe Carmona, Laure Coutin and Gerard Montseny
We study a class of Gaussian processes which have a moving average
representation with respect to a fixed driving Brownian motion, and can
be represented as a mixture of Markovian semimartingale Gaussian
processes with a numerically efficient approximation scheme (a typical
process in this class is fractional Brownian motion).
In particular, we try to answer the following questions:
-- When is this process a Brownian motion ? a semimartingale ?
-- Do we have an Ito formula ? Can we solve stochastic differential
equations ? What does this mean in terms of the underlying driving
Brownian motion ?
carmona@cict.fr, coutin@cict.fr, montseny@laas.fr
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information provided by the author
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919. TAILS AND MOMENTS ESTIMATES FOR SOME TYPES OF CHAOS
Rafa{\l} Lata{\l}a
Let $X_{i}$ be a sequence of independent symmetric real random variables
with logarithmically concave tails (that is such that the functions
$-\ln P(|X_{i}|\geq t)$ are convex on $(0,\infty)$). We consider a
random variable $X=\sum_{i\neq j}a_{i,j}X_{i}X_{j}$, where $a_{i,j}$
are real numbers. We derive approximate formulas for the tails and moments
of $X$ and its decoupled version, which are exact up to some universal
constants.
rlatala@math.gatech.edu (LaTex file available)
920. RATES OF CONVERGENCE FOR A NON-REVERSIBLE MARKOV CHAIN SAMPLER
Martin Hildebrand
Diaconis, Holmes, and Neal have described a non-reversible Markov chain
sampler. Like the Metropolis algorithm, this Markov chain sampler
depends on a stationary probability $\pi$. This Markov chain
involves a duplication of states; the transition probabilities will
depend on which duplicate the algorithm is on at the start of a step as
well as the ratio of probabilities at 2 states. The transition probabilities
also depend on a parameter $\theta$ which governs transitions between
the duplicate sets. This paper examines the behavior of this Markov chain
for certain $\theta$ and $\pi$ and finds examples where the
Markov chain converges to the stationary distribution faster than the
corresponding Metropolis algorithm's Markov chain does. The proofs of
these results use various facts from Markov chain and probability theory.
martinhi@math.albany.edu
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information provided by the author
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921. DIFFUSIONS AND ELLIPTIC OPERATORS
Richard F. Bass
This is a book about the interplay of diffusion processes and
elliptic PDEs. The best clue to what is in the book is the
table of contents.
I STOCHASTIC DIFFERENTIAL EQUATIONS
Preliminaries
Pathwise solutions
Lipschitz coefficients
Types of uniqueness
Markov properties
One-dimensional case
Examples
Some estimates
Stratonovich integrals
Flows
SDEs with reflection
SDEs with reflection: pathwise results
II REPRESENTATIONS OF SOLUTIONS
Poisson's equation
Dirichlet problem
Cauchy problem
Schr\"odinger operators
Girsanov transformation
The Neumann and oblique derivative problems
Fundamental solutions and Green functions
Adjoints
III REGULARITY OF SOLUTIONS
Variation of parameters
Weighted H\"older norms
Regularity of hitting distributions
Schauder estimates
Dirichlet problem
Extensions
Neumann and oblique derivative problem
Calder\'on-Zygmund estimates
Flows
IV ONE-DIMENSIONAL DIFFUSIONS
Natural scale
Speed measures
Diffusions as solutions of SDEs
Boundaries
Eigenvalue expansions
V NONDIVERGENCE FORM OPERATORS
Definitions
Some estimates
Examples
Convexity
Green functions
Resolvents
Harnack inequality
Equicontinuity and approximation
VI MARTINGALE PROBLEMS
Existence
The strong Markov property
Some useful techniques
Some uniqueness results
Consequences of uniqueness
Submartingale problems
VII DIVERGENCE FORM OPERATORS
Preliminaries
Inequalities
Moser's Harnack inequality
Upper bounds on heat kernels
Off-diagonal upper bounds
Lower bounds
Extensions
Path properties
VIII THE MALLIAVIN CALCULUS
Integration by parts formula
Smooth functionals
A criterion for smooth densities
Vector fields
H\"ormander's theorem
An alternative approach
bass@math.washington.edu
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information provided by the author
click here.
922. POLAR BOUNDARY SETS FOR SUPERDIFFUSIONS AND
REMOVABLE LATERAL SINGULARITIES FOR NONLINEAR PARABOLIC PDE'S
S. E. Kuznetsov
Suppose $L$ is a second order elliptic differential operator in $R^d$ and $D$
is a bounded smooth domain in $R^d$. Let $\alpha \in (1,2]$ and let
$\Gamma$ be a closed subset of $\partial D$. It is known (Dynkin and Kuznetsov,
1996) that the following three properties are equivalent:
(i) $\Gamma$ is d-polar, that is $\Gamma$ is not hit by the range of the
corresponding $(L,\alpha)$-superdiffusion in $D$;
(ii) The Poisson capacity of $\Gamma$ is equal to zero;
(iii) $\Gamma$ is a removable boundary singularity for the equation
$Lu=u^\alpha$ in $D$.
We investigate a similar problem for a parabolic operator in a smooth cylinder
$Q = [0,\infty)\times D$. Let $\Gamma$ be a compact set on the lateral
boundary of $Q$. We show that the following three properties are equivalent:
(a) $\Gamma$ is G-polar, that is $\Gamma$ is not hit by the graph of the
corresponding $(L,\alpha)$-superdiffusion in $Q$;
(b) The Poisson capacity of $\Gamma$ is equal to zero;
(c) $\Gamma$ is a removable lateral singularity for the equation
$\dot u + Lu = u^\alpha$ in $Q$,
that is if $u$ is a non-negative solution of the problem $\dot u + Lu=u^\alpha$
in $Q$, $u=0$ on $\partial Q \setminus \Gamma$ and $u \to 0$ if $t$ tends to
infinity, then $u=0$.
sk47@cornell.edu
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information provided by the author
click here.
923. REMOVABLE LATERAL SINGULARITIES OF SEMILINEAR PARABOLIC PDE'S AND
BESOV CAPACITIES
S. E. Kuznetsov
Suppose $\alpha$ belongs tp (1,2], $L$ is a second order elliptic differential
operator in $R^d$ and $D$ is a bounded smooth domain in $R^d$. Let $Q =
[0,\infty)\times D$ and let $\Gamma$ be a compact set on the lateral boundary
of $Q$. Refining previous result of the author, we prove that $\Gamma$ is a
removable lateral singularity for the equation $\dot u + Lu = u^\alpha$ in $Q$
if and only if
$$
Cap_{1/\alpha,2/\alpha,\alpha')(\Gamma) = 0
$$
where $Cap$ stands for the Besov capacity on the boundary and
$\alpha' = \alpha/(\alpha-1)$.
sk47@cornell.edu
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information provided by the author
click here.
924. $\sigma$-MODERATE SOLUTIONS of $Lu = u^\alpha$
AND FINE TRACE ON THE BOUNDARY
S. E. Kuznetsov
We investigate the class of all positive solutions of the equation
$Lu = u^\alpha$, $\alpha$ belongs to (1,2], in a smooth domain $E$ in $R^d$.
We define the fine trace $tr(u)$ of a solution $u$ as a pair $(\Gamma,\nu)$
where $\Gamma$ is a set of singular boundary points of $u$ and $\nu$ is a
certain $\sigma$-finite measure on the complement of $\Gamma$. We describe
all possible traces and we construct the minimal solution with the given trace.
sk47@cornell.edu
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information provided by the author
click here.
925. FINE TOPOLOGY AND FINE TRACE ON THE BOUNDARY ASSOCIATED WITH A CLASS
OF SEMILINEAR DIFFERENTIAL EQUATIONS
E. B. Dynkin and S. E. Kuznetsov
We investigate the set of all positive solutions of a semilinear equation
$Lu=\psi(u)$ where $L$ is a second order elliptic differential operator in a
domain $E$ of $R^d$ or, more generally, in a Riemannian manifold and $\psi$
belongs to a wide class of convex functions which contains $\psi(u)=u^\alpha$
for all $\alpha>1$. We define boundary singularities of a solution $u$ in
terms of points of rapid growth of the right derivative $\psi^+(u)$, we
introduce a fine topology and a fine trace of $u$ on the Martin boundary
and we construct the minimal solution for every possible value of this trace.
ebd1@cornell.edu, sk47@cornell.edu
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information provided by the author
click here.
926. ON THE EQUIVALENCE OF CERTAIN ERGODIC PROPERTIES FOR GIBBS STATES
F. den Hollander and J.E. Steif
We extend our previous work by proving that for translation invariant
Gibbs states on $\Z^d$ with a translation invariant interaction potential
$\Psi = (\Psi_A)$ satisfying $\sum_{A \ni 0} |A|^{-1} [\mbox{diam}(A)]^d
\|\Psi_A\| < \infty$ the following hold:\\
(1) the Kolmogorov-property implies a Trivial Full Tail;
(2) the Bernoulli-property implies F\o lner Independence.
The existence of bilaterally deterministic Bernoulli Shifts tells us that
neither (1) nor (2) is true for random fields in general without some
further assumption (even when $d=1$).
denholla@sci.kun.nl
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information provided by the author
click here.
927. A NEW INDUCTIVE APPROACH TO THE LACE EXPANSION FOR SELF-AVOIDING WALKS
R. van der Hofstad, F. den Hollander and G. Slade
We introduce a new inductive approach to the lace expansion, and apply
it to prove Gaussian behaviour for the weakly self-avoiding walk on
${\Bbb Z}^d$ where loops of length $m$ are penalised by a factor
$e^{-\beta/m^{p}}$ ($0<\beta \ll 1$) when:
(1) $d>4$, $p \geq 0$;
(2) $d \leq 4$, $p > \frac{4-d}{2}$.
In particular, we derive results first obtained by Brydges and Spencer
(and revisited by other authors) for the case $d>4$, $p=0$. In addition,
we prove a local central limit theorem, with the exception of the
case $d>4$, $p=0$.
denholla@sci.kun.nl
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information provided by the author
click here.
928. SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS ON A RIEMANNIAN
MANIFOLD AND THEIR TRACE ON THE MARTIN BOUNDARY
E.B.Dynkin and S.E.Kuznetsov
Let $L$ be a second order elliptic differential operator on a
Riemannian manifold $E$ with no zero order terms. We say that a
function $h$ is $L$-harmonic if $Lh=0$. Every
positive $L$-harmonic function has a unique representation
$$
h(x)=\int_{E'} k(x,y) \nu(dy)
$$
where $k$ is the Martin kernel, $E'$ is the Martin boundary and
$\nu$ is a finite measure on $E'$ concentrated on the minimal part
$E^*$ of $E'$. We call $\nu$ the trace of $h$ on $E'$.
Our objective is to investigate positive solutions of a nonlinear equation
$$
L u=u^\alpha\text{ on } E \tag *
$$
for $1<\alpha\le 2$ [the restriction $\alpha\le 2$ is imposed because our
main tool is the $(L,\alpha)$-superdiffusion which is not defined for
$\alpha>2$]. We associate with every solution $u$ of (*) a pair
$(\Gamma,\nu)$ where $\Gamma$ is a closed subset of $E'$ and $\nu$ is a
Radon measure on $O=E'\setminus \Gamma$. We
call $(\Gamma,\nu)$ the trace of $u$ on $E'$. $\Gamma$ is empty if
and only if $u$ is dominated by an $L$-harmonic function. We call such
solutions moderate. A moderate solution is determined
uniquely by its trace. In general, many solutions can have the same
trace.
In an earlier paper, we investigated the case when
$L$ is a second order elliptic
differential operator in $\rd$ and $E$ is a bounded smooth
domain in $\rd$. We obtained necessary and sufficient
conditions for
a pair $(\GG,\nu)$ to be a trace and we gave a probabilistic formula
for the maximal solution with a given trace.
General theory developed in the present paper is applicable, in
particular, to elliptic operators $L$ with bounded
coefficients in an arbitrary bounded domain of $R^d$ assuming only that the
Martin boundary and the geometric boundary coincide.
ebd1@cornell.edu, sk47@cornell.edu
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information provided by the author
click here.
929. STOCHASTIC BOUNDARY VALUES AND BOUNDARY SINGULARITIES
FOR THE EQUATION $Lu = u^\alpha$
E.B.Dynkin
We investigate positive solutions of a nonlinear equation $Lu=u^\alpha$ where
$L$ is a second order elliptic differential operator in a Riemannian manifold
$E$ and $1<\alpha\le 2$. The restriction $\alpha\le 2$ is imposed because our
main tool is $(L,\alpha)$-superdiffusion $X$ which is not defined for
$\alpha>2$. We establish a 1-1 correspondence between the set $U$ of positive
solutions and a class $\frak Z$ of functionals of $X$ which we call linear
boundary functionals (they depend only on the behavior of $X$ near the Martin
boundary $E'$).
The class $\frak Z$ is a closed convex cone and $u\in U$ is a subadditive
function of $Z \in \frak Z$. Special roles belong to moderate solutions
corresponding to $Z$ with finite mathematical expectations and to a family of
solutions determined by the range of $X$. A new formula is deduced connecting
$u, Z$ and $L$-diffusions conditioned to hit the boundary $E'$ at a given
point $y$. A concept of a singular boundary point for $u$ is introduced in
terms of the conditioned diffusion.
ebd1@cornell.edu
930. ON EXIT SPACES FOR THE DAWSON-WATANABE SUPERPROCESSES
E.B.Dynkin
Let $\xi$ be a Markov process in a space $E$ with transition function
$p(r,x;t,dy)$ and let $X$ be the corresponding Dawson-Watanabe superprocess
[i.e. the superprocess with the branching characteristic $\psi(u)=\gamma u^2$].
Denote by $\Cal P$ the transition function of $X$ and put
$$
p_n(r,x;t,dy)=
\prod\limits_{i=1}^n p(r,x_i;t,dy_i).
$$
To every $p_n$-exit law $\ell$ at time $\beta$ there corresponds a
$\cal P$-exit law $L_\ell$ at the same time such that, for every $t$,
$L_\ell^t(\mu)$ is a polynomial of degree $n$ in $\mu$ with the leading
term $\<\ell^t,\mu^n\>$.
Polynomial exit laws for $\xi$ form a convex cone. Under broad conditions on
$p$, we describe all extremal elements of this cone and we prove existence
and uniqueness theorem for decomposition into extremal elements. (If $p$ is
the transition function of a diffusion and if $\beta<\infty$, then the classes
of proportional extremal polynomial exit laws are in a natural correspondence
with finite integer-valued non-zero measures on $E$.)
ebd1@cornell.edu
931. ROTATION NUMBERS FOR LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS
Ludwig Arnold and Peter Imkeller
Let $dx=\sum_{i=0}^m A_ix\circ dW^i$ be a linear SDE
in ${\bf R}^d$, generating the flow $\Phi_t$ of
linear isomorphisms. The multiplicative ergodic
theorem asserts that every vector
$v\in{\bf R}^d\setminus\{0\}$ possesses a Lyapunov
exponent (exponential growth rate) $\lambda(v)$
under $\Phi_t$ which is a random variable taking
its values from a finite list of canonical
exponents $\lambda_i$ realized in the invariant
Oseledets spaces $E_i$. We prove that, in the
case of simple Lyapunov spectrum, every 2-plane
$p$ in ${\bf R}^d$ possesses a rotation number
$\rho(p)$ under $\Phi_t$ which is defined as the linear growth
rate of the cumulative infinitesimal rotations
of a vector $v_t$ inside $\Phi_t(p)$. Again, $\rho(p)$ is a random
variable taking its values from a finite list
of canonical rotation numbers $\rho_{ij}$
realized in span$(E_i,E_j)$. We give
rather explicit formulas for the $\rho_{ij}$.
This carries over results of Arnold and San
Martin from random to stochastic differential
equations which is made possible by utilizing
anticipative calculus.
imkeller@mathematik.hu-berlin.de
932. ADDITIONAL LOGARITHMIC UTILITY OF AN INSIDER
Juergen Amendinger, Peter Imkeller, and Martin Schweizer
In this paper, we consider a security market in
which two investors on different information
levels maximize their logarithmic utility from
terminal wealth. While the ordinary investor's
portfolio decisions are based on the public
information flow, the insider possesses from the
very beginning some extra information about the
outcome of some random variable $G$, e.g., the
future price of a stock. We solve the two
optimization problems and rewrite the insider's
additional expected logarithmic utility in terms
of a relative entropy. This enables us to provide
simple conditions on $G$ for the finiteness of
this additional utility and to show that it is
basically given by the entropy of $G$.
imkeller@mathematik.hu-berlin.de
933. COALESCENTS WITH MULTIPLE COLLISIONS
Jim Pitman
For each finite measure $\Lambda$ on $[0,1]$, a coalescent Markov process,
with state space the compact set of all partitions of the set $N$ of
positive integers, is constructed so the restriction of the partition
to each finite subset of $N$ is a Markov chain with the following transition
rates: when the partition has $b$ blocks, each $k$-tuple of blocks is merging
to form a single block at rate $\int_0^1 x^{k-2} (1 - x) ^{b-k} \Lambda(dx)$.
Call this process a {\em $\Lambda$-coalescent}. Discrete measure valued
processes derived from the $\Lambda$-coalescent model a system of masses
undergoing coalescent collisions. Kingman's coalescent, which has numerous
applications in population genetics, is the $\delta_0$-coalescent for
$\delta_0$ a unit mass at 0. The coalescent recently derived by Bolthausen and
Sznitman from Ruelle's probability cascades, in the context of the
Sherrington-Kirkpatrick spin glass model in mathematical physics, is the
$U$-coalescent for $U$ uniform on $[0,1]$. For $\Lambda=U$, and whenever an
infinite number of masses are present, each collision in a $\Lambda$-coalescent involves an infinite number of masses almost surely, and the proportion of
masses involved exists almost surely and is distributed proportionally to
$\Lambda$. The two-parameter Poisson-Dirichlet family of random discrete
distributions derived from a stable subordinator, and corresponding
exchangeable random partitions of $N$ governed by a generalization of the
Ewens sampling formula, are applied to describe transition mechanisms for
processes of coalescence and fragmentation, including the $U$-coalescent and
its time reversal.
pitman@stat.Berkeley.edu
Technical Report No. 495, Department of Statistics, U.C. Berkeley
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information provided by the author
click here.
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here.
934. EMPIRICAL PROCESSES BASED ON PSEUDO-OBSERVATIONS
Kilani Ghoudi and Bruno Remillard
Usually, empirical distribution functions are used to estimate
the theoretical distribution function of known functions
$\theta(X)$ of the observable random variable $X$.
In practice, many researchers are using empirical distribution functions
constructed from residuals, which are estimations of a non-observable
error terms in linear models. This falls under a class of more general
problems in which one is interested in the estimation of the
distribution function of a non-observable random variable $\theta(Q,X)$
depending on an observable random variable $X$ together with its unknown
law $Q$. When $Q$ is estimated by some $Q_n$, the quantities
$\theta(Q_n,X_i)$ are called pseudo-observations. Some work has been done
recently when the pseudo-observations are the so-called residuals of linear
models.
The aim of this paper is to provide some tools to study the asymptotic
behavior of empirical processes constructed from general
pseudo-observations. Examples of pseudo-observations will be
given together with applications to copulas, weighted symmetry, regression
and other statistical concepts.
ghoudi@uqtr.uquebec.ca or remillar@uqtr.uquebec.ca
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information provided by the author
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935. LARGE DEVIATIONS FOR NONLINEAR HYPERBOLIC SPDE'S IN THE
BESOV-ORLICZ SPACE
Boualem Djehich and M'Hamed Eddahbi
We investigate the regularity of the solutions of a class of nonlinear
hyperbolic stochastic partial differential equations in the anisotropic
Besov--Orlicz space $\Cal B_{\tau,\omega}^0$ modulated by the Young
function $\tau(t)=\exp(t^2) -1$ and the modulus of continuity
$\omega(t)=\left(t(1+\log(1/t))\right)^{1/2}$. Moreover, we derive in
the Besov--Orlicz norm a large deviation estimate of Freidlin--Wentzell
type for the solution.
boualem@math.kth.se
936. CONCENTRATION OF MEASURE AND LOGARITHMIC SOBOLEV INEQUALITIES
Michel Ledoux
These notes corresponds to a mini-course given at the Technische
Universitat Berlin in November 1997. The aim of these notes is to present a
complete account on the applications of logarithmic Sobolev inequalities
to the concentration of measure phenomenon. The basic argument we develop
is going back to I. Herbst and has been revived recently by several authors
including S. Aida, T. Masuda, I. Shigekawa, D. Stroock, S. Bobkov, F.
G\"otze, L. Gross, O. Rothaus. We exploit Herbst's original argument to
deduce from the logarithmic Sobolev inequalities some differential
inequalities on the Laplace transforms of Lipschitz functions. According to
the family of entropy-energy inequalities we are dealing with, these
differential inequalities yield various behaviors of the Laplace
transforms of Lipschitz functions and of their concentration properties.
Besides applications to Talagrand's concentration inequalities for
product measures, we present recent joint work with S. Bobkov on
modified logarithmic Sobolev inequalities and their applications to
non-Gaussian tails. In addition, we include some new tail estimate
for Brownian motion on a manifold with non-negative Ricci curvature as
well as upper bounds on the logarithmic Sobolev constant in compact
manifolds. In particular, we determine the behavior of the logarithmic
Sobolev constant of the Euclidean unit ball as a function of the dimension.
TABLE OF CONTENTS
1. ISOPERIMETRIC AND CONCENTRATION INEQUALITIES
1.1 Introduction
1.2 Isoperimetric inequalities for Gaussian and Boltzmann measures
1.3 Some general facts about concentration
2. SPECTRAL GAP AND LOGARITHMIC SOBOLEV INEQUALITIES
2.1 Abstract functional inequalities
2.2 Examples of logarithmic Sobolev inequalities
2.3 Herbst's argument
2.4 Entropy-energy inequalities and non-Gaussian tails
2.5 Poincar\'e inequalities and concentration
3. DEVIATION INEQUALITIES FOR PRODUCT MEASURES
3.1 Concentration with respect to the Hamming metric
3.2 Deviation inequalities for convex functions
3.3 Concentration inequalities for contracting Markov chains
3.4 Applications to bounds on empirical processes
4. MODIFIED LOGARITHMIC SOBOLEV INEQUALITIES FOR LOCAL GRADIENTS
4.1 The exponential measure
4.2 Modified logarithmic Sobolev inequalities
4.3 Poincar\'e inequalities and modified logarithmic Sobolev inequalities
5. MODIFIED LOGARITHMIC SOBOLEV INEQUALITIES IN DISCRETE SETTINGS
5.1 Logarithmic Sobolev inequality for Bernoulli and Poisson measures
5.2 Modified logarithmic Sobolev inequalities and Poisson tails
5.3 Sharp bounds
6. SOME APPLICATIONS TO LARGE DEVIATIONS AND TO BROWNIAN MOTION
ON A MANIFOLD
6.1 Logarithmic Sobolev inequalities and large deviation upper bounds
6.2 Some tail estimate for Brownian motion on a manifold
7. ON REVERSED HERBST'S INEQUALITIES AND BOUNDS ON THE
LOGARITHMIC SOBOLEV CONSTANT
7.1 Reversed Herbst's inequality
7.2 Dimension free lower bounds
7.3 Upper bounds on the logarithmic Sobolev constant
ledoux@cict.fr (postcript file only available)
http:/www-sv.cict.fr/lsp/Ledoux/index.html
937. ON MINIMAL PARABOLIC FUNCTIONS AND TIME-HOMOGENEOUS
PARABOLIC $h$-TRANSFORMS
Krzysztof Burdzy and Thomas S. Salisbury
Does a minimal harmonic function $h$ remain minimal
when it is viewed as a parabolic function?
The question is answered for a class of long thin
semi-infinite tubes $D\subset \R^d$ of variable width
and minimal harmonic functions $h$ corresponding to the
boundary point of $D$ ``at infinity.'' Suppose $f(u)$
is the width of the tube $u$ units away from its
endpoint and $f$ is a Lipschitz function.
The answer to the question is affirmative if and only if
$\int^\infty f^3(u)du = \infty$. If the test fails, there
exist parabolic $h$-transforms of space-time Brownian motion
in $D$ with infinite lifetime which are not time-homogenous.
burdzy@math.washington.edu or salt@nexus.yorku.ca
- To see a preprint or other
information provided by the author
click here.
938. IMPROVED BONFERRONI INEQUALITIES
Klaus Dohmen
We prove that the binomial moments appearing in classical Bonferroni
inequalities can be replaced by smaller quantities involving much
fewer terms. The new bounds, which are sharper and faster to compute,
are applied to reliability theory, chromatic graph theory and extreme
value statistics.
dohmen@informatik.hu-berlin.de
- To see a preprint or other
information provided by the author
click here.
