Probability Abstracts 30

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593. LADDER HEIGHTS, GAUSSIAN RANDOM WALKS, AND THE RIEMANN ZETA FUNCTION

Joseph T. Chang and Yuval Peres

Let $\{S_n:n\ge 0\}$ be a random walk having normally distributed 
increments with mean $\theta$ and variance 1, and let $\tau$ be 
the time at which the random walk first takes a positive value, 
so that $S_\tau$ is the first ladder height. Then the expected 
value $E_\theta S_\tau$, originally defined for  $\theta>0$, may 
be extended to be an analytic function of the complex variable 
$\theta$ throughout the complex plane, with the exception of 
certain branch point  singularities.  In particular, the coefficients 
in a Taylor expansion about $\theta=0$ may be written explicitly as 
simple expressions involving the Riemann zeta function. Previously 
only the first coefficient of the series developed here was known;
this term has been  used extensively in developing approximations
for boundary crossing problems for Gaussian random walks.  
 Knowledge of the complete series makes more refined results possible; 
we apply it to derive asymptotics for boundary crossing probabilities
and the limiting expected overshoot.

chang@stat.yale.edu           peres@stat.berkeley.edu

594. A LARGE DEVIATIONS PRINCIPLE FOR HIERARCHICAL SYSTEMS OF INTERACTING JUMP PROCESSES

Boualem Djehiche and Alexander Schied

A large deviations principle from the McKean--Vlasov limit
for a collection of jump processes obeying a two--level hierarchy
interaction is derived. It is shown that the associated rate 
function admits a Lagrangian representation as well as a 
non-variational one. Moreover, it is proved that the admissible paths for
the weak solution of the McKean--Vlasov equation enjoy certain strong
differentiability properties.

boualem@math.kth.se, schied@mathematik.hu-berlin.de
   (Tex file or hard copy of the paper is available)

595. CANONICAL TRANSFORMATIONS FOR DIFFUSIONS

Boualem Djehiche and Torbj\"orn Kolsrud

We show that Morphisms of backward heat equations preserve 
in an appropriate sense
the forward Lagrangian, as well as the forward Newton's equations of
motion for a general class of Riemannian manifold--valued diffusions. Time 
reflection yields similar results for forward heat equations, in particular
for Bernstein-Schr\"odinger diffusions of Euclidean Quantum mechanics.

boualem@math.kth.se, kolsrud@math.kth.se
  (Tex file or hard copy of the paper is available).

596. BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH REFLECTION AND DYNKIN GAMES

Jaksa Cvitanic and Ioannis Karatzas

We establish existence and uniqueness results for adapted solutions of
Backward Stochastic Differential Equations (BSDE's) with two reflecting
barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng
and Quenez (1995). Existence is proved first by solving a related pair
of coupled optimal stopping problems, and then, under different
conditions, via a penalization method. It is also shown that the
solution coincides with the value of a certain Dynkin game, a stochastic
game of optimal stopping. Moreover, the connection with the Backward SDE
enables us to provide a pathwise (deterministic) approach to the game.

cj@stat.columbia.edu (TeX or ps-file available on request)

597. NONLINEAR FINANCIAL MARKETS: HEDGING AND PORTFOLIO OPTIMIZATION

Jaksa Cvitanic

This is a survey paper on the techniques and results of the theory 
of optimal trading for a single agent with a nonlinear wealth process,
in a continuous-time, diffusion model. Examples include the case of
policy dependent 
prices, portfolio constraints and different interest rates for borrowing and 
lending. We study the hedging (super-replication) problem and the portfolio 
optimization problem for the investor in this market. Mathematical
tools involved are those of continuous-time martingales, convex duality
in the case of convex nonlinearities, and forward-backward SDE's and
PDE's in the case of more 
general, but Markovian nonlinearities. We often present only the main
ideas of the proofs, and refer the reader to the literature for details.

cj@stat.columbia.edu (TeX or ps-file available on request)

598. ON THE POINTWISE CENTRAL LIMIT THEOREM AND MIXTURES OF STABLE DISTRIBUTIONS

Istv\'an Berkes and Endre Cs\'aki
 
Let $X_1, X_2, \ldots$ be i.i.d. r.v.'s with $EX_1 =0$,
$EX_1^2 =1$ and put $S_n = X_1 + \ldots +X_n$. We investigate the  
a.s. limiting behavior of 
$$L_N = {1\over \log N} \sum_{k \leq N} {1\over k} I \left\{ {S_k\over a_k}
 < x \right\} $$
for general norming sequences $(a_k)$. The pointwise central limit theorem
shows that $L_N$ converges a.s. to the normal distribution if $a_k = 
\sqrt k$; in our paper we prove the surprising result that for suitably 
chosen $(a_k)$ the expression $L_N$ can converge also to nongaussian 
limits, in particular, any symmetric stable distribution is a possible limit 
of $L_N$. We shall determine the class of limit distributions of $L_N$ 
and extend the result to the case when $X_n$ belong to the domain of 
attraction of a stable law. 

csaki@novell.math-inst.hu (TeX file available)

599. SOME ASYMPTOTIC RESULTS FOR THE INTEGRAL OF BROWNIAN MOTION

Davar Khoshnevisan and Zhan Shi

Consider A(t) the integral process of 
a d-dimensional Brownian motion started from 0. 
Recently, Kolokoltsov (1995) has shown that A is 
transient in dimension d>1. We characterize its exact 
rate of escape via an integral test. The process under 
the L^2 norm is also studied, for which we establish 
both limsup and Chung liminf iterated logarithm laws.

davar@math.utah.edu    (Postscript file available.)

600. SAMPLE FUNCTIONS OF L\'EVY--CHENTSOV RANDOM FIELDS

Narn-Rueih Shieh

Let $X$ be a L\'evy--Chentsov random field which
is a $(1/\alpha)$-self-similar, strongly stationary increments,
$N$-dimensionally indexed $S \alpha S$ field, where $0 < \alpha < 2$ and
$N \ge 2$. We prove two fundamental properties for sample functions of
$X$. We prove firstly that $X$ has a separable measurable locally bounded
version $\widetilde X$ for which each projection process $t \rightarrow
\widetilde X (te), \|\be\|=1$, is a $S \alpha S$ L\'evy process. We then
prove that $X$ has jointly continuous local times when $1 \le \alpha < 2$.
These results are non--Gaussian aspects of L\'evy's isotropic Brownian fields.
We also prove that $X$ has continuously differentiable local times when
$\alpha N > 3$.

shiehnr@math.ntu.edu.tw

601. LIMIT THEOREMS FOR LOCAL TIMES OF FRACTIONAL BROWNIAN MOTIONS AND SOME OTHER SELF-SIMILAR PROCESSES

Narn-Rueih Shieh

Let $X(t)$ be fractional Brownian motion of exponent $H,
0<H<1$, and let $L(t,x)$ be its local time. We prove some limit theorems for
certain functionals of $L(t,x)$(evaluation at $x=0$, supremum in $x$, integral
in $x$ w.r.t. a measure or function), normalized by $t^{1-H}$; these extend
previous results of Darling-Kac, Yamada, and Brosamler for Brownian motion.
We actually prove our results for more general self-similar processes,
including stable L\'evy processes and fractional stable processes.
The multi-parameter and multi-dimensional extensions are also given.

shiehnr@math.ntu.edu.tw

602. CLUSTER FORMATION IN A STEPPING STONE MODEL WITH CONTINUOUS, HIERARCHICALLY STRUCTURED SITES

Steven N. Evans and Klaus Fleischmann

A stepping stone model with site space a continuous,
hierarchical group is constructed via duality
with a system of (delayed) coalescing ``stable'' L\'evy processes. This
model can be understood as
a continuum limit of discrete state-space, two allele, genetics models with
hierarchically structured
resampling and migration. The existence of a process rescaling limit 
on suitable
large space and time scales is established and interpreted in terms of
the dynamics of cluster formation. 
This paper was inspired by  recent work of
Klenke.

evans@stat.berkeley.edu (LaTeX file available)
Compressed PostScript available from:

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• Or here.
• Or here.

603. COALESCING MARKOV LABELLED PARTITIONS AND A CONTINUOUS SITES GENETICS MODEL

Steven N. Evans

Let $Z$ be a Markov process with state-space
$E$.  For any finite set $S$ it is possible to associate with
$Z$ a process $\zeta$
of coalescing partitions of $S$ with components labelled by
elements of $E$ that evolve as copies of $Z$ between coalescences.
Subject to very weak  hypotheses on $Z$,
there is a Feller process $X$ with state-space a certain space
of probability measure valued functions on $E$.  The
process $X$ has its
``moments'' defined in terms of expectations for $\zeta$ in a manner
suggested by various instances of martingale problem
duality between coalescing Markov processes
and voter model particle systems,
systems of interacting Fisher-Wright and
Fleming-Viot diffusions that arise in population genetics,
and stochastic partial differential equations with Fisher-Wright noise
that appear as rescaling limits of long-range voter models
as well as in population genetics.
Some sample path properties are
examined in the special
case where $Z$ is a symmetric stable process on $\bR$ with
index $1 < \alpha \le 2$.  In particular, we show that for fixed
$t>0$ the essential range of the random probability measure
valued function $X_t$ is almost surely a countable set of point masses.

evans@stat.berkeley.edu  (Plain TeX file available)
Compressed PostScript available from:

• To see some information provided by the author click here.
• Or here.
• Or here.

604. A GAMBLING SYSTEM AND A MARKOV CHAIN

S. N. Ethier

``Oscar's system'' is a gambling system in which the aim is to win one
betting unit, at least with high probability, and then start over again. 
The system can be modeled by an irreducible Markov chain in a subset of the
two-dimensional integer lattice.   We show that the Markov chain, which
depends on a parameter $p$ representing the single-trial win probability,
is transient if $p < {1 \over 2}$ and positive recurrent if
$p \ge {1 \over 2}$.

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

605. UNE DEMONSTRATION ELEMENTAIRE D'UNE IDENTITE DE BIANE ET YOR

Christophe Leuridan

Let $P^t_0$ be the law of a brownian motion (in R), staring at 0 and 
killed at time t, and $P^{\tau^a_r}_0$ the law of a brownian motion 
starting at 0 and killed when its local time at point a reaches value r. 
We prove that:
$\int_0^{\infty} P^t_0 dt = \int_R \int_0^\infty P^{\tau^a_r}_0 da dr$. 
Transforming this equality by a space-time reversal, we give an elemen-
tary proof of an identity obtained by Biane and Yor.

Christophe.Leuridan@ujf-grenoble.fr

606. INCREASING PROPAGATION OF CHAOS FOR MEAN FIELD MODELS

G. Ben Arous and O. Zeitouni

Let $L_N$ denote the empirical mean of an i.i.d.($\mu$) sample, and
let $\mu^{(N)}$ denote the mean-field measure with potential $F$ constructed
from  it. Resulting independence properties of the measure
$\mu^{(N)}$ are investigated,
both in the degenerate and non--degenerate situations. In particular,
with $H(\cdot|\mu)$ denoting relative entropy,
if there exists a unique non--degenerate minimum of
$H(\cdot|\mu)-F(\cdot)$,
then propagation of chaos holds for blocks of size $o(N)$.

zeitouni@ee.technion.ac.il

607. ON TALAGRAND'S DEVIATION INEQUALITIES FOR PRODUCT MEASURES

Michel Ledoux

    We present a new and simple approach to some of the deviation
inequalities for product measures deeply investigated by M. Talagrand
in the recent years. Our method is based on functional inequalities of
Poincar\'e and logarithmic Sobolev type and iteration of these inequalities.
In particular, we establish with these tools sharp Gaussian deviation
inequalities from the mean on norms of sums of independent random vectors
and empirical processes. Concentration for the Hamming distance may also be
deduced from this approach.

ledoux@cict.fr (hard copy available)

608. FREE ENERGY AND SOME SAMPLE PATH PROPERTIES OF A RANDOM WALK WITH A RANDOM POTENTIAL

Sergio Albeverio and  Xian Yin Zhou

We study the asymptotic behaviour of the free energy for a model of one
dimensional random walk with random potential defined by Sinai. In 
particular, we obtain a central limit theorem and a strong law of large 
numbers for this free energy. We use some results on the free energy to 
study some sample path properties of this random walk which are related
respectively to its recurrence and localization. Some exponents 
describing the recurrence and localization are found.(to appear in J.Stat.
Phys.)

Xian.zhou@rz.ruhr-uni-bochum.de  (Latex file is available)

609. A NEW CONVERGENT LATTICE APPROXIMATION FOR THE $\phi_2^4$-QUANTUM FIELD

Sergio Albeverio and  Xian Yin Zhou

We define new lattice $\phi_2^4$-field models in which the lattice cutoffs
in the free and interacting parts are different. We construct the new 
continuum $\phi_2^4$-fields (as subsequence limits) by approximating,
in the superrenormalizable cases, these new lattice $\phi_2^4$-fields 
with or without spatial cutoffs. Some properties of these new continuum
$\phi_2^4$-fields are also discussed.

Xian.zhou@rz.ruhr-uni-bochum.de   (Latex file is available)

610. A CENTRAL LIMIT THEOREM FOR THE FOURTH WICK POWER OF THE FREE LATTICE FIELD

Sergio Albeverio and  Xian Yin Zhou

Let $G_a$ be the free lattice field measure of mass $m_0$ on $aZ^d$, and 
$:\phi_x^4:$ be the corresponding fourth Wick power of the lattice field 
$\phi_x$. Let $g\in C_0(R^d), g\ge 0$ be a given function and 
$a'=a'(a)\ge a$ satisfy: $\lim_{a\to 0^+}a'=0$ and $a'Z^d\subset a Z^d$.
We prove that if $d\ge 3$, or $d=2$ and $\lim_{a\to 0^+}a'|\log a|^2=\infty$,
then $\{a'^d\sum_{x\in a'Z^d}g_x:\phi_x^4:\}$ satisfies the central limit 
theorem: there is $V(a,a')$ with $\lim_{a\to 0^+}V(a,a')=\infty$ such that
the distribution of $V(a,a')^{-1}a'^d\sum_{x\in a'Z^d}g_x:\phi_x^4:$ 
under $G_a$ is convergent to the standard normal distribution, as $a\to 0^+$.

Xian.zhou@rz.ruhr-uni-bochum.de  (Latex file is available)

611. BOUNDS FOR DISCONNECTION EXPONENTS

Wendelin Werner

We slightly improve the upper bounds of disconnection exponents
that we derived in an earlier paper. We also give a plain 
proof of the $1 / (2 \pi)$ lower bound for the disconnection exponent
for one walk.

wwerner@dmi.ens.fr

• To see some information provided by the author click here.

612. MONTE-CARLO TESTS AND CONJECTURES FOR DISCONNECTION EXPONENTS

Emily E. Puckette and Wendelin Werner

Using Monte-Carlo simulations, we estimate numerically disconnection
exponents for planar Brownian motions. These simulations tend to
confirm conjectures by Duplantier and Mandelbrot.

eep@oxy.edu, wwerner@dmi.ens.fr

• To see some information provided by the author click here.

613. ON THE APPROXIMATION OF QUEUEING NETWORKS IN HEAVY TRAFFIC

R. J. Williams

This paper reports on the current state of research concerning
approximations to the queue length or workload processes in 
stable but heavily loaded queueing networks. Most
of the approximations that have been considered to date  are justified
by heavy traffic limit theorems that have a certain
conventional  form, namely,  (i) the queue length
or workload processes in a sequence of heavily
loaded networks are normalized by a central limit 
theorem type of scaling, and (ii) the weak limit of these
normalized processes is a reflecting Brownian motion.
Such conventional limit theorems for both open and closed networks
are surveyed first, with a view to outlining their general form.  Whilst there 
is a reasonable collection of limit theorems for single class networks
and some multiclass networks (especially those with feedforward structure),
recent developments have indicated that not all multiclass
networks with feedback have conventional heavy traffic behavior.
These developments are surveyed here and in particular,  the Harrison-Williams 
example of an unconventional heavy traffic limit theorem for a closed analogue
of the Lu-Kumar/Rybko-Stolyar priority network is described.

williams@math.UCSD.EDU  (Hard-copy and Postscript file available)

614. ESCAPE RATES FOR LEVY PROCESSES

Davar Khoshnevisan

We prove a space--time estimate for a L\'evy
process to hit a small set. As an application, we
present escape rates for L\'evy processes with
strictly stable components.

davar@math.utah.edu  (Postscript file available)

• To see some information provided by the author click here.

615. A PROOF OF A CONJECTURE OF BOBKOV AND HOUDR\'E

S. Kwapien, M. Pycia and W. Schachermayer

S. G. Bobkov and C. Houdr\'e recently posed the following question on
the internet (Problem posed in Stochastic Analysis Digest no. 15
(9/15/1995)): Let X,Y be symmetric i.i.d. random variables such that

         | X+Y |                              
     P ( - - - -   >  t )   <   P ( | X |  >  t )
         \sqrt 2   =        =              =        

for each t > 0. Does it follow that X has finite second moment (which
then easily implies that X is Gaussian)?
In this note we give an affirmative answer to this problem and present a
proof. Using a different method K. Oleszkiewicz has found another proof of
this conjecture, as well as further related results.

wschach@stat1.bwl.univie.ac.at   (LaTeX-version available)

• To see some information provided by the author click here.

616. DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS AND HYPOELLIPTICITY

Denis R. Bell

    This book is due to be published by the Longman Group LTD (UK) and Wiley
(US) in the series Pitman Monographs and Surveys in Pure and Applied
Mathematics, Volume 79. The main theme is the study of degenerate stochastic
differential equations, considered as transformations of the Wiener measure,
and their relationship with partial differential equations. The book contains
an elementary derivation of Malliavin's integration by parts formula, a proof
of the probabilistic form of Hormander's theorem, an extension of Hormander's
theorem for infinitely degenerate differential operators, and criteria for the
regularity of measures induced by stochastic hereditary-delay equations. The
final chapter treats an abstract version of the measure transformation problem
from the point of view of generalized divergence operators.
    A background chapter is included containing a brief introduction to the
Wiener measure, and the theory of stochastic integration and its relationship
with partial differential equations. Thus, while some knowledge of measure
theory and functional analysis is assumed, the book contains enough background
to be accessible to a wide mathematical readership. In particular, we hope it
will be of interest to advanced graduate students and to researchers in many
areas of mathematics, especially PDE theorists.
    The contents are as follows:

1 Background material
   1.1 The Wiener Space
     1.1.1 Brownian motion
     1.1.2 Wiener's theorem
     1.1.3 Abstract Wiener spaces
   1.2 Stochastic integration
     1.2.1 Construction of the Ito integral
     1.2.2 Computation of a stochastic integral
     1.2.3 The multidimensional Ito integral and its properties
   1.3 Stochastic differential equations
     1.3.1 Existence and uniqueness of solutions
     1.3.2 The Kolmogorov equations
   1.4 Criteria for the regularity of a Borel measure
2 Regularity of measures induced by stochastic differential equations
   2.1 A finite-dimensional analogue of the regularity problem
   2.2 The Malliavin calculus
     2.2.1 A summary of Malliavin's paper
     2.2.2 Stroock's functional analytic formulation
   2.3 An elementary proof of the Malliavin-Stroock theorem
     2.3.1 Motivation
     2.3.2 A sequence of differentiable approximations to the Ito map
     2.3.3 Derivation of the covariance matrix
     2.3.4 The regularity theorems
3 The probabilistic form of Hormander's theorem
   3.1 Existence of densities under Hormander's condition
   3.2 Smoothness of densities under Hormander's condition
4 Infinitely degenerate hypoelliptic operators
   4.1 An extension of Hormander's theorem
   4.2 Regularity of the heat semigroup
   4.3 Proofs of the theorems
   4.4 Lower bounds for degenerate Ito processes
5 Smooth densities for a class of stochastic functional equations
   5.1 Stochastic hereditary-delay equations
   5.2 Statement of hypotheses and theorems
   5.3 Analysis of the conditioned equation
   5.4 Proof of the main theorem
6 Generalized divergence operators and absolutely continuous
  transformations
   6.1 Transformations of Gaussian measure
   6.2 A formula for a class of Radon-Nikodym derivatives
   6.3 Differentiable and quasi-invariant measures

dbell@unf1vm.bitnet

617. R{\'E}GULARISATION D'\'EQUATIONS DE STRATONOVITCH \`A SAUTS ENTRE VARI\'ETES

S. Cohen and A.Estrade

We investigate a way to regularize a general manifold valued
semimartingale, in order to approximate the solution of some
stochastic differential equation. A regularization of the stochastic
driving noise is used.
The regularization procedure is the following: a
sample path of the c{\`a}dl{\`a}g manifold valued semimartingale is fixed, and
the semimartingale position at time $t$ is approximated by a
``mean value'' of the sample path just before time $t$. 
The usual definition of vector valued barycenter has to be extended
to construct this intrinsic ``mean value'', and
the resulting regularized processes, which are continuous with finite
variation, produce an estimate of the original semimartingale. The
solutions of the ordinary differential equations 
driven by those finite variation processes are proved to converge 
to the so-called
 solutions of stochastic differential equations between manifolds. This
``Wong-Zaka{\"\i}'' type approximation of a stratonovich differential
equation  is a natural geometrical extension of a paper by Kurtz,
Pardoux and Protter. At last we would like to emphasize that the geometric
point of view allows us to deal with a discontinuous driving process
even if the regularization is not linear.

sc@newaphro.enpc.fr  (Latex file available in French)

618. INFORMATION INEQUALITIES AND CONCENTRATION OF MEASURE

Amir Dembo

We derive inequalities of the form d(P,Q) =< H(P|R)+H(Q|R) which
hold for every choice of probability measures P,Q,R, where H(P|R) denotes
the relative entropy of P with respect to R and d(P,Q) stands
for a coupling type ``distance'' between P and Q. Using the chain rule
for relative entropies and then specializing to Q of a given support
we recover some of Talagrand's concentration of measure inequalities
for product spaces.

amir@playfair.stanford.edu  (LaTeX or Postscript file available)

619. ON THE SMOOTHNESS OF THE LAWS OF RANDOM FLAGS AND OSELEDETS SPACES OF RANDOM DYNAMICAL SYSTEMS GENERATED BY LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

Peter Imkeller

For a random dynamical system generated by a linear stochastic differential
equation driven by an $m-$dimensional Wiener process the Oseledets spaces 
carry the random invariant measures crucial for their asymptotic properties.
They are obtained by intersecting the nested random linear spaces of a
forward (in time) and backward flag. We study the existence and smoothness
of densities of their laws considered as random elements taking their values
on appropriate Grassmann manifolds with respect to Riemannian volume. These
laws are seen to be closely related to the laws of certain flows of 
diffeomorphisms which decompose the linear flow coming from the sde. The
latter are investigated using the tools of Malliavin's calculus and
H\"ormander's hypoellipticity conditions for the vector fields generating
the flows. We prove that if the Lie algebras induced by the actions of
the matrices on the tangent bundles of the Grassmannians span the tangent
spaces at any point for appropriate dimensions, the laws and also the 
conditional laws of the random subspaces are $C^{\un}-$smooth. As an 
application we find that the semimartingale property is well preserved if
the Wiener filtration is enlarged by the information present in the flag
or Oseledets spaces. Hence the tools of stochastic analysis are applicable
in a number of problems arising in the study of the asymptotics of random
dynamical systems despite the fact that the invariant measures are not adapted
to the history of the Wiener process.

imkeller@vega.univ-fcomte.fr