Probability Abstracts 8

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139. REGULARIZATION OF DIRICHLET SPACES

S. Albeverio, Z.M. Ma and M. Roeckner

This paper is essentially a summary of the results in Chap.VI of the mono=
graph "An Introdiction to the theory of (non-symmetric Dirichlet forms"
by the two last-named authors(see abstract no.88).We prove that any
quasi-regular Dirichlet form (E,D(E))on a general topological space X is (up
to an exceptional set) a regular Dirichlet form on a locally compact separable
metric space X' which is a "local compactification" of X. X' is constructed
in such a way that the capacities associated with (E,D(E)),(E',D(E'))
respectively coincide.As an easy consequence of this and well-known results
of Fukushima,Silverstein and Carillo-Menendez one obtains the existence of
a pair of m-perfect standard processes associated with (E,D(E)).We then show
that these processes are special.Moreover,we prove that any right process
M properly associated with (E,D(E)) is(up to restriction and trivial extension)
a Hunt process on X'associated with (E',D(E')).Hence all the classical results
about Hunt processes associated with regular Dirichlet forms on locally compact
separable metric spaces are applicable to M.In particular,the one to one
correspondence between additive functionals of M and smooth measures of
(E,D(E)) on X and the Fukushima decomposition for M are immediate consequences.

UNM30B@ibm.rhrz.uni-bonn.de

140. APPROXIMATE SOLUTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH HOLDER CONTINUOUS COEFFICIENTS

Xuerong Mao

In Ito's classical theory of stochastic differential equations where
the coefficients are assumed to be Lipschitz continuous, the solutions
are constructed through successive approximation and the uniqueness
of the solutions is shown immediately by the construction.  On the 
other hand, there are many examples where the coefficients do not
satisfy the Lipschitz condition yet we can prove the existence and 
uniqueness of the solutions. In particular, Skorohod in 1965 showed
the existence of solutions under the condition that coefficients
are only continuous.  Skorohod and Tanaka & Hasegawa in 1965 and 1964
respectively proved the pathwise uniqueness of the solution of a
one-dimensional Ito equation dx(t) = g(x(t)) dw(t) if g is Holder
continuous of order a > 1/2. Yamada and Watanabe in 1971 strengthened
the case and proved the uniqueness for a >= 1/2. However, in these
cases, the existence and uniqueness problems are treated by other
methods rather than successive approximation.  As a matter of fact,
the successive approximation problem for the solutions of stochastic
differential equations with Holder continuous coefficients has
remained open for more than twenty five years. This is the aim
of this present paper to close this gap.

xm@maths.warwick.ac.uk

141. A NOTE ON FIRST PASSAGES IN BRANCHING BROWNIAN MOTIONS

Ingemar Kaj and Paavo Salminen

Consider a realization of a one-dimensional branching Brownian motion
generated by a single particle at the origin. Fix $x>0$ and count any particle
visiting $x$ as the first among the members of its line of descent from the
initial particle. The resulting first-passage process indexed by $x$ is a
Markov branching process which can be characterized in terms of the original
branching mechanism. In this note we study the first-passage process with
emphasis on supercritical properties.

ingemar.kaj@math.uu.se

142. SYMMETRIC REFLECTED DIFFUSIONS

E. Pardoux and R. J. Williams

Consider a Dirichlet form $(E, D(E))$:
$$E(f,f) = 1/2 \int_D \nabla f \cdot a \nabla f \, p \, dx, f \in D(E),$$
$$D(E)  = \{f\in L^2(D, p\,dx )\cap H_{loc}^1(D) : \EE(f,f)<\infty\}, $$
where $D$ is a $d$-dimensional domain, $a$ is a bounded, symmetric,
locally elliptic, $d\times d$ matrix-valued  function on $D$, $p$ is a
strictly positive probability density on $D$, and $a,p$ are locally Lipschitz 
continuous on $D$. We investigate two methods for approximating the stationary, 
symmetric Markov process $X$ associated with $(E,D(E))$, which in the case 
of smooth non-degenerate data is a diffusion process with infinitesimal
generator ${1 \over {2p}} \nabla \cdot (ap\nabla)$ in $D$ and 
conormal reflection at the boundary of $D$. 
The first (or exterior) approximation is a conventional penalty
approximation by diffusions defined on all of $R^d$. The second (or
interior) approximation uses diffusions confined to $D$ by 
singular drifts that tend to infinity at the boundary of $D$. The existence 
of such singular diffusions is established as a result of possible independent 
interest.  For both approximation methods, the
approximating sequences of processes are shown to be tight using a
decomposition of Lyons and Zheng. The conditions under which one can
identify any weak limit as a realization of $X$ are most general for 
the interior approximation scheme and are satisfied for example if for any 
compact set $K\subset R^d$, $a$ is
uniformly elliptic on $D \cap K$ and $p$ is strictly bounded away from 
zero there. Finally, we show under further
mild regularity and non-degeneracy conditions on $a$ and $p$ that if
$\partial D$ is locally of finite $(d-1)$--dimensional upper 
Minkowski content, then $X$ is a semimartingale.

rjwilliams@ucsd.bitnet   (Latex file or hard copy form )

143. SELF-AVOIDING PATHS ON THE THREE DIMENSIONAL SIERPINSKI GASKET.

Kumiko Hattori,  Tetsuya Hattori,  Shigeo Kusuoka

We study self-avoiding paths on the three-dimensional pre-Sierpinski gasket.
We prove the existence of the limit distribution of the scaled path length,
the exponent for the mean square displacement, and the continuum limit.
We also prove that the continuum-limit process is a self-avoiding 
process on the three-dimensional Sierpinski gasket, and that
a path almost surely has infinitely fine creases.

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

144. EXPONENTIAL STABILITY OF LARGE-SCALE STOCHASTIC DIFFERENTIAL EQUATIONS

Xuerong Mao

Given a large-scale stochastic system described by a number of subsystems,
we introduce the corresponding isolated subsystems. It is shown that the
exponential stability of the isolated systems implies the exponential
stability of the large-scale system under some hypotheses added on the
interconnected terms. We also study a slightly special case where the 
large-scale system is described in a hierarchical form, i.e., the system
consists of several subsystems and each subsystem interacts with
"lower" subsystems but not with "higher" subsystems. In this hierarchical
case, it is proved that the large-scale system is exponentially stable
if and only if so is each of the isolated subsystems.
 
xm@uk.ac.warwick.maths

145. STOCHASTIC DIFFERENTIAL EQUATIONS AND PARABOLIC PDE'S WITH POLYNOMIAL DEGENERACIES

Denis R. Bell, Salah-Eldin A. Mohammed

     Extending techniques developed in a previous paper, the authors obtain
regularity results for the solution x of a stochastic differential equation
        dx(t) = H(t,x)dt + g(t,x(t - r)dW(t)
where g denotes a (possibly degenerate) matrix-valued function, r is a
positive time delay, and H is a non-anticipating functional of the whole
history of x. It is shown that if g has only a finite number of points of
degeneracy of polynomial type, then x(t) admits a smooth density for all
positive times. These techniques are also used to derive similar results
for degenerate time-dependent diffusions
        dx(t) = h(t,x(t))dt + g(t,x(t))dW(t).
The latter results are shown to imply the existence of smooth fundamental
solutions for the associated class of degenerate time-dependent parabolic
partial differential equations. In the case where g is space-independent,
the existence of smooth fundamental solutions is established under a very
weak mean ellipticity condition on gg*.

dbell@unf1vm.bitnet

146. A CALCIUM MODEL WITH RANDOM ABSORPTION: A STOCHASTIC APPROACH

Pali Sen, Denis R. Bell, Donna Mohr

Absorption of calcium, or any mineral, by the body is subject to the
random fluctuations typical of diffusion through membranes. In this
paper we consider the absorption of calcium from the gut as a white
noise process added to the deterministic model of Sen & Mohr (J. Theor.
Biol. 142, 179-188, 1990). The first two moments for the amount of
calcium in the extracellular fluid (ECF) have been derived using the
Ito calculus. A confidence interval for the total amount of calcium in
the ECF is constructed. The equations for the first two moments of the
fraction of dose calcium in the ECF are also given. Suggestions are
made for the collection of experimental data in a form which should
be helpful in investigating the magnitude of the stochastic effect.

dbell@unf1vm.bitnet

147. LEVY PROCESSES WITH NO POSITIVE JUMPS AT AN INCREASE TIME

Jean Bertoin

The purpose of this paper is to study the behaviour
of a L\'evy process with no positive jumps near its increase
times (in the sense of Dvoretzky, Erdos and Kakutani). 
Specifically, we construct a local time on the set of increase
times. Then we pick an increase time at random according to
the local time and we split the path of the L\'evy process
at this instant. The main result is a description of the two
resulting processes. As an application, we evaluate the rate
of escape before and after a 'typical' increase time.

jbertoin@euclid.ucsd.edu

148. THREE-DIMENSIONAL BROWNIAN PATH REFLECTED ON BROWNIAN PATH IS A FREE BROWNIAN PATH

Krzysztof Burdzy

The result announced in the title is an easy corollary
of recent results of Chen on reflected Brownian motion.
It supports a conjecture by physicists that self-reflecting
random walk (which they call "true" self-avoiding random walk)
converges to Brownian motion.

burdzy@math.washington.edu (AmSTeX file available)

149. DOOB-MEYER DECOMPOSITION FOR SET-INDEXED SUBMARTINGALES.

M. Dozzi, B.G. Ivanoff, E. Merzbach

Set-indexed martingales and submartingales are defined and studied. 
The admissible function of a submartingale is defined and some class $(D)$
conditions are given which allow the extension of the function to a
$\sigma$-additive measure on the predictable $\sigma$-algebra.  Then, we prove
a Doob-Meyer decomposition:  A set-indexed submartingale can be decomposed
into the sum of a weak martingale and an increasing process.  A hypothesis of
predictability ensures the uniqueness of this decomposition.  An explicit
construction of the increasing process associated with a submartingale is
given.   Finally, some remarks about quasimartingales are discussed.

merzbach@bimacs.cs.biu.ac.il

150. ON MULTIPLE PHASE TRANSITIONS FOR BRANCHING MARKOV CHAINS

Rinaldo Schinazi

We consider branching Markov chains on a countable set.
We give a necessary and sufficient condition in terms of the kernel of
the underlying Markov chain  to have two phase transitions. We compute
the critical values. We apply this result to prove that asymmetric
branching random walks on Z have two phase transitions.

schinazi@zeppo.colorado.edu

151. STRONG APPROXIMATIONS TO BROWNIAN LOCAL TIME

Richard Bass and Davar Khoshnevisan

Suppose we have mean 0 finite variance random walk(s) on the line
and suppose the random walk is either lattice valued or strongly nonlattice.
If the normalized random walk(s) converge a.s. to Brownian motion(s) with rate
$a_n$, then provided $a_n < (\log n)^{-10}$, the local times of the
random walk(s) will converge a.s. to the local times of the Brownian 
motion(s) at rate $a_n^{1/10}$. This provides a strong invariance
principle for the local times for the Hungarian embedding and a 
strong invariance principle for local times of random walks with $2+\epsilon$
moments. We also obtain weak invariance principles in the above cases.
Our methods also provide a strong invariance principle for intersection
local times of random walks.

bass@math.washington.edu

152. REFLECTING BROWNIAN MOTIONS: QUASIMARTINGALES AND STRONG CACCIOPOLI SETS

Z. Q. Chen, P. J. Fitzsimmons and R. J. Williams

A notion of strong Caccioppoli set is defined for bounded
Euclidean domains. It is shown that  
stationary (normally) reflecting Brownian motion 
on  the closure of a bounded Euclidean 
domain is a quasimartingale on each compact time interval
if and only if the domain is a strong Caccioppoli set. 
A similar result
is shown to  hold for symmetric reflecting diffusion processes.

rjwilliams@ucsd.bitnet (TEX file available)