Probability Abstracts 13

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215. OPTIMAL INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

H.M. Soner and S.E. Shreve

	A complete solution is provided to the infinite-horizon, discounted
problem of optimal consumption and investment in a market with one stock,
one money market (sometimes called a ``bond"), and proportional transaction
costs.  The utility function may be of the form $c^{p}/p$ where $p<0$ or
$0<p<1$, or may be $log c$.  It is assumed that the interest rate for the
money market is positive, the mean rate of return for the stock is larger
than this interest rate, the stock volatility is positive, and all these
parameters are constant.  The only other assumption is that the value
function is finite; necessary conditions for this are given.
	The analysis relies on the concept of viscosity solutions to
Hamilton-Jacobi-Bellman equations.  A self-contained treatment of this
subject, adequate for the present application, is provided.

shreve@galley.ece.cmu.edu

216. DIFFERENTIABLE FAMILIES OF MEASURES

O.G. Smolyanov and H. v. Weizscker


 The paper studies differential and related properties of 
functions of a real variable with values in the space of signed measures.
In particular the connections between different definitions of 
differentiability are described corresponding to different topologies on 
the measures. Some conditions are given for the equivalence of the measures 
in the range of such a function. These conditions are in terms of socalled 
logarithmic derivatives and yield a generalization of the 
Cameron-Martin-Maruyama-Girsanov formula. Questions of this kind appear 
both in the theory of differentiable measures on infinite-dimensional spaces
and in the theory of statistical experiments. 

hvw@mathematik.uni-kl.de

217. RENORMALIZATION OF FINITELY RAMIFIED FRACTALS

Volker Metz

Transition probabilities are calculated which make the construction
of diffusions on fractals relatively easy. Physically this is done 
by "coarse-graining-renormalization". Mathematically it is a 
convergence problem for quotients of Dirichlet forms. The main
result is that renormalization automatically ends up in nice
transition probabilities.

metz@mathematik.uni-bielefeld.de  (AMSTeX file available)

218. DIFFUSIVE CLUSTERING IN AN INFINITE SYSTEM OF HIERARCHICALLY INTERACTING DIFFUSIONS

Klaus Fleischmann and Andreas Greven                   

We study a system of linearly interacting diffusions on the  interval  [0,1], 
indexed by an infinite hierarchical group with parameter N, modeling the evo- 
lution of gene frequencies accounting for migration and  resampling.  Fisher- 
Wright diffusions are the typical example for the diffusion term. A  particu- 
lar choice of the migration term guaranties that we are in the so-called dif- 
fusive clustering regime. The processes in the  whole  class  considered  and 
starting with a shift-ergodic initial law have qualitative properties just as 
in the Fisher-Wright case (universality). Also, the initial law plays here no 
role. Clusters of components with values either close to 0 or close to 1 grow 
on various different scales (diffusive clustering). More precisely,  at  time  
N^t >> 1  (spatial) cluster sizes measured in a  hierarchical  distance  grow 
like at (i.e. consist of N^at components) where 1-a is the  hitting  time  of 
the traps {0,1} for a time transformed Fisher-Wright diffusion. Nevertheless, 
the single components oscillate infinitely often between values  close  to  0 
and close to 1, but in such a way that they spend fraction one of their  time 
together close to the boundary. Diffusive clustering phenomena we believe are 
in fact common to many interacting systems and were first  discussed  by  Cox 
and Griffeath (1986) for the planar simple voter model. On the way, we  prove 
some scaling results on a specific coalescing random walk with delay  on  the 
hierarchical group.

fleischmann@iaas-berlin.dbp.de

219. ON BOUNDS OF HEAT KERNELS UNDER LOCAL ELLIPTICITY CONDITIONS

Weian Zheng

We consider the estimates of the upper and lower bounds of heat kernel on
a non-smooth Lipschitz Riemannian manifold. Under the condition that
the volume growth is less fast than the exponential of squared radius and
some mild conditions for the local ordinate systems,
we proved that Aronson's inequality holds for small time. This result is
new in geometry even when the Riemannian metric is smooth. From the analysis 
point of view, we got an upper bound estimate for small time under the a weaker
condition than the uniform ellipticity which was essential in many
known results.

wzheng@math.uci.edu   (LaTeX file available) 

220. SUBDIFFUSIVE FLUCTUATIONS FOR INTERNAL DIFFUSION LIMITED AGGREGATION

Gregory F. Lawler

Internal diffusion limited aggregation (internal DLA) is a cluster model
in $Z^d$ where new points are added by starting random walkers at the
origin and letting them run until they have found a new point to add to
the cluster.  It has been shown that the limiting shape of internal DLA
clusters is spherical.  Here we show that the fluctuations are
subdiffusive, in fact, they are of order at most $n^{1/3}$, at least
up to logarithmic corrections.  More precisely, we show that for all
$n$ sufficiently large the cluster after $m =[\omega_d n^d]$ steps
covers all points in tbe ball of radius $n - n^{1/3}(\ln n)^2$ and is
contained in the ball of radius $n + n^{1/3} (\ln n)^4$.

jose@math.duke.edu

221. EXACT RATES OF CONVERGENCE TO BROWNIAN LOCAL TIME.

Davar Khoshnevisan 

In this paper we are mainly concerned with proving lower
bounds on some uniform approximation schemes for the local times of
one-dimensional Brownian motion. Consequently, this leads to many 
exact limit theorems for Brownian local times. The latter results
are Paul Levy's occupation time approximation, the downcrossing theorem
and an intrinsic construction.

davar@math.washington.edu     (TeX file available)

222. ON EXACT TAILS FOR LIMITING DISTRIBUTIONS OF U-STATISTICS IN THE SECOND GAUSSIAN CHAOS

Peter Imkeller

Random variables in the second Gaussian chaos appear as limits in CLT for
U-statistics of order 2. We study the exact tail behaviour of their
distributions. Let $A$ be the symmetric Hilbert-Schmidt operator
associated in a canonical way with a given random variable. If $a^+
(a^-)$ is the largest positive (smallest negative) eigenvalue of $A$,
positive (negative) tails are asymptotically equivalent with those of
a $\chi^2$ with the multiplicities of $a^+ (a^-)$ as numbers of degree of
freedom, and depend essentially only on $a^+ (a^-)$. If $A$ is positive
definite, negative tails depend sensitively on the tail behaviour of the
series associated with the sequence of eigenvalues of $A$.

Peter.Imkeller@mathematik.uni-muenchen.dbp.de

223. CHAOS EXPANSIONS OF DOUBLE INTERSECTION LOCAL TIME OF BROWNIAN MOTION IN $R^d$ AND RENORMALIZATION

Peter Imkeller, Victor Perez-Abreu, Josep Vives

Double intersection local times $\alpha (x,.)$ of Brownian motion $W$ in
${\bf R}^d$ which measure the integral closeness of $W_t$ and $W_s + x$
for $s\not= t$ and $x \in {\bf R}^d$ can be developed into series of
multiple Wiener-Ito integrals. These series representations reveal on the
one hand the degree of smoothness of $\alpha (x,.)$ in terms of eventually
negative order Sobolev spaces with respect to the canonical Dirichlet
structure on Wiener space. On the other hand, they offer an easy access
to renormalization of $\alpha (x,.)$ as $|x| \to 0$. The results, valid for
any dimension $d$, describe a pattern in which the well known cases
$d=2,3$ are naturally embedded.

Peter.Imkeller@mathematik.uni-muenchen.dbp.de

224. THE ASYMPTOTIC BEHAVIOUR OF LOCAL TIMES AND OCCUPATION INTEGRALS OF THE N_PARAMETER WIENER PROCESS IN $R^d$

Peter Imkeller, Ferenc Weisz

Let $L(x,T), x \in {\bf R}^d, T \in {\bf R}_+^N$, be the local time of the
$N-$ parameter Wiener process $W$ taking values in ${\bf R}^d$. Even in
the distribution valued cases $d \ge 2N$, $L$ can be described in a series
representation by means of multiple Wiener-Ito integrals. This setting
proves to be a good starting point for the investigation of the asymptotic
behaviour of $L(x,T)$ as $|x| \to 0$ and/or $T \to \infty$ and of related
occupation integrals $X_T (f) = \int_{[0,T]} f(W_s) ds$ as $T \to \infty$.
We obtain the rates of explosion in laws of the first order, i.e.
normalized convergence laws for $L(x,T)$ resp. $X_T (f)$, and of the second
order, i.e. normalized convergence laws for $L(x,T) - E(L(x,T))$
resp. $X_T(f) - E(X_T(f))$.

Peter.Imkeller@mathematik.uni-muenchen.dbp.de

225. L\'EVY MEASURES OF INFINITELY DIVISIBLE RANDOM VECTORS AND SLEPIAN INEQUALITIES

Gennady Samorodnitsky and Murad S. Taqqu

  We study Slepian inequalities for general non-Gaussian infinitely divisible
random vectors. Conditions for such inequalities are expressed in terms of
the corresponding L\'evy measures of these vectors. These conditions are
shown to be nearly  best possible, and for a large subfamily of
infinitely divisible random vectors, these conditions are necessary
and sufficient. 
  As an application we consider symmetric $\alpha$ stable
Ornstein-Uhlenbeck processes and a family of infinitely divisible
random vectors introduced by Brown and Rinott. 

gennady@orie.cornell.edu     murad@math.bu.edu

226. SHOCK FLUCTUATIONS IN THE ASYMMETRIC SIMPLE EXCLUSION PROCESS

P. A. Ferrari, L. R. G. Fontes

We consider the one dimensional nearest neighbors
asymmetric simple exclusion process with rates $q$ and $p$ for left and right
jumps respectively; $q<p$.  Ferrari, Kipnis and Saada
(1991) have shown that if the initial measure is $\nu_{\rho,\lambda}$, 
a product
measure with densities $\rho$ and $\lambda$ to the left and right of the origin
respectively, $\rho<\lambda$, then there exists a (microscopic) shock for 
the system. 
A shock is a random
position $X_t$ such that the system as seen from this position at time $t$ has
asymptotic product distributions with densities $\rho$ and $\lambda$ to the left
and right of the origin respectively, uniformly in $t$.  We compute the
diffusion coefficient of the shock 
$D=\lim_{t\to\infty} t^{-1}(E(X_t)^2 - (EX_t)^2)$ and
find $D=(p-q)(\lambda-\rho)^{-1} (\rho(1-\rho)+\lambda(1-\lambda))$ as 
conjectured by
Spohn (1991). We show that in the scale $\sqrt t$ the position of $X_t$ is
determined by the initial distribution of particles in a region of lenght
proportional to $t$. We prove that the distribution of the process at the
average position of the shock converges to a fair mixture of the product
measures with densities $\rho$ and $\lambda$. This is the so called 
dynamical phase
transition. Under shock initial conditions we show how the density fluctuation
fields depend on the initial configuration. 

pablo@ime.usp.br

227. ENTROPY, LIMIT THEOREMS, AND VARIATIONAL PRINCIPLES FOR DISORDERED LATTICE SYSTEMS

Timo Seppalainen
 
Abstract: We study infinite volume limits and equilibrium states 
of disordered lattice systems with bounded and continuous
potentials. Our main tools are a generalization of
the relative entropy functional for random reference
measures and a large deviation theory for
nonstationary independent processes. We find that
many familiar results of invariant potentials,
such as large deviation theorems, variational
principles, and equivalence of ensembles, continue to
hold for disordered models, with  suitably modified
statements.

timosepp@function.mps.ohio-state.edu (AMS-TeX file available)

228. EXISTENCE OF MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Ludger Overbeck and Michael Roeckner 

We prove that (as for a Dirichlet form) the quasi-regularity of a Semi-
Dirichlet form implies the existence of an associated special standard
Markov process.Such bilinear forms,by definition,have the contraction
(or Dirichlet) property only with respect to one of its arguments.For
the construction of the process because of lack of duality some new
results on the analytic potential theory of Semi-Dirichlets form
are required.

UNM30B@IBM.rhrz.uni-bonn.de

229. EXPONENTIAL STABILITY OF LARGE-SCALE STOCHASTIC DIFFERENTIAL- FUNCTIONAL EQUATIONS

X. Mao

One of the effective methods developed for large-scalesystems is the 
decomposition technique which has been studied by many authors e.g.
Michel, Miller, Tang, Siljak and Sontag. The idea is as follows:
Decompose a given large-scale equation into several sub-equations
(rearrage the order of variables if necessary) and introduce the 
isolated sub-equations.  Under certain hypotheses added on the 
interconneted terms it can be shown that the given large-scale
equation is exponentially stable if every isolated sub-equation 
is exponentially stable. The aim of this paper is to investigate
the exponential stability of large-scale stochastic differential-
functional equations by the decomposition technique. We obtain
several sufficient conditions under which the exponential stability
of every isolated sub-equation implies the exponential stability
of the large-scale stochastic differential-functional equations.

xuerong@stams.strath.ac.uk 

230. RESOLVENT ESTIMATES FOR FLEMING-VIOT OPERATORS AND UNIQUENESS OF SOLUTIONS TO RELATED MARTINGALE PROBLEMS

Donald A. Dawson and Peter March 

We define a two-parameter scale of Banach spaces contained in
the continuous functions on the space of probability 
measures on a compact topological space X, and show that the
resolvent of the Fleming-Viot process is a bounded operator in the
scale. We use this result to prove uniqueness of solutions
to the martingale problem for Fleming-Viot operators whose coefficients,
(mutation operator  and sampling rate)
may depend on the current state, provided they are
sufficiently close to constant in a sense made precise by the norms 
defining the scale. We also prove an existence result with no such
restriction on the coefficients.

march@math.ohio-state.edu (AMS TeX file available)

231. HEAT KERNEL ON A LIPSCHITZ MANIFOLD COMPOSED OF TWO HALFSPACES WITH DIFFERENT METRICS

W. Zheng

When we consider a Lipschitz Riemannian manifold M 
there are many difficulties due to the non-smoothness of the 
geometric structure. In this paper, we will just consider the simplest 
case where the Riemannian matrix as a function may only assume 
two positive definite matrices as its possible values, namely
it is equal to A on the left half-space and equal to B on the 
right half-space. We give a precise
expression of the heat kernel H(x,t,y) on that manifold. As its
direct consequences, we show the following two results:
1) H(x,t,y) is Lipschitz continuous in (x,y) for fixed t>0;
2) For each a>0, there are positive constants C and T
such that
${1\over Ct^{d\over 2}}\exp\{ -{d^2(x,y)\over 2t(1-a )}\}
\leq H(x,t,y)$
for all t<T. Both results are unknown yet in the general
case.

wzheng@math.uci.edu (LaTeX file available)

232. ASYMPTOTICALLY ONE-DIMENSIONAL DIFFUSIONS ON THE SIERPINSKI GASKET AND THE ABC-GASKETS.

Kumiko Hattori,  Tetsuya Hattori,  Hiroshi Watanabe 

Diffusion processes on the Sierpinski gasket and 
the abc-gaskets are constructed as limits of random walks.
In terms of the associated renormalization group,  
the present method uses the inverse trajectories which converge to 
unstable fixed points corresponding to the random walks on 
one-dimensional chains.
In particular, non-degenerate fixed points are unnecessary for 
the construction.
A limit theorem related to the discrete-time 
multi-type non-stationary branching processes is applied.

kumiko@tansei.cc.u-tokyo.ac.jp
hattori@tansei.cc.u-tokyo.ac.jp
(LaTex file available.)