Probability Abstracts 15
This document contains abstracts 247-256.
They have been mailed on July 1, 1993.
Click here to see the
list of all abstract titles.
247. SOLVING FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATION
EXPLICITLY --- A FOUR STEP SCHEME
Jin Ma, Philip Protter and Jiongmin Yong
We investigate the nature of the adapted solutions to a class of forward-
backward SDEs in which the forward equation is non-degenerate. We prove
that in this case the adapted solution can always be sought in an "ordinary"
sense over an arbitrarily prescribed time duration, via a direct "Four Step
Scheme". Using this scheme, we further prove that the backward componets of
the adapted solution are determined explicitly be the forward component via
the solution of a certain quasi-linear parabolic PDE system. Moreover the
uniqueness of the adapted solutions (over an arbitrary time duration), as
well as the continuous dependence of the solutions on the parameters, can
all be proved within this unified framework. Some special cases are studied
separately. In particular, we derive a new form of the integral representation
of the Clark-Haussmann-Ocone type for functionals (or functions) of diffusion,
in which the conditional expectation is no longer needed.
majin@math.purdue.edu
248. MULTIPLICATIVE CASCADES: DIMENSION SPECTRA AND DEPENDENCE
Ed Waymire and S.C. Williams
The study of spatial distributions which arise from independent
multiplicative cascades can be traced back at least to the efforts
by the Russian school led by Andrei N. Kolmogorov on the statistical
theory of turbulence. However, owing largely to the intriguing
statistical/geometric scaling properties emphasised by Benoit
Mandelbrot, and to the rich mathematical foundations begun by J.P.
Kahane and J. Peyriere in the middle 1970's, there has been a growing
interest in random cascades both in the physical sciences and
mathematics. This paper provides both a summary of some of the
principal aspects of the theory of independent cascades as well
as some recent extensions.
waymire@math.orst.edu
249. TWO-PARAMETER STRONG MARTINGALES
Ferenc Weisz
Two-parameter strong martingale inequalities and duality results
are proved. An atomic decomposition of the strong martingale Hardy
space $H_p^\sigma$ generated by the conditional quadratic variation
is given. A new Davis decompositipon is formulated with the help of
which a simple proof of Davis's inequality is demonstrated. For
previsible martingales the usually considered Hardy norms are all
equivalent. Amongst others it is proved that the dual of the
strong martingale Hardy space $H_1^*$ generated by the maximal
function is $BMO_2^-$, the dual of $H_1^\sigma$ is $BMO_2$ and the
dual of $H_p^\sigma$ is $H_q^\sigma$ (1<q<\infty, 1/p+1/q=1).
The equivalence of the $BMO_q^-$ spaces (2\leq q <\infty) and,
for regular stochastic basis, the equivalence between $BMO_q$ and
$BMO_2^-$ (2\leq q <\infty) are also verified.
Ferenc.Weisz@Mathematik.Uni-Muenchen.DBP.DE
250. MARKOV CASCADES
Edward C. Waymire and Stanley C. Williams
The Kahane-Peyriere theory for independent multiplicative cascades
is extended to a class of Markov cascades. Unlike the case of homogeneous
independent cascades, this class is shown to have a natural nontrivial
Kahane-Katznelson dimension spectral disintegration.
waymire@math.orst.edu
251. SINGULAR INITIAL CONDITIONS FOR THE HEAT EQUATION WITH A NOISE TERM
Carl Mueller
We consider the equation
u_t=u_{xx}+u^\gamma\dot W, t>0,\, 0\le x\le J,
u(0,x)=u_0(x)
u(t,0)=u(t,J)=0
where $\dot W =\dot W(t,x)$ is 2-parameter white noise. Assume that $u(0,x)$
is continuous and nonnegative. We show local existence and uniqueness
for unbounded initial functions satisfying certain conditions. For
$\gamma<3/2$, $u(0,x)$ may be a bounded measure. If $\gamma\ge 3/2$, we
assume that $u(0,x)$ lies in $L^p$ for some $p>2\gamma$. Our results are
motivated by work of Mueller and Sowers (1993), who showed that for large
$\gamma$, solutions of this equation can blow up. One would wish to show
that solutions can be extended beyond blow-up, and our results
can be viewed as a step in that direction.
cmlr@troi.cc.rochester.edu (AMStex file available)
252. THE SUPPORT OF MEASURE VALUED BRANCHING PROCESSES IN A RANDOM ENVIRONMENT
D. A. Dawson, Yi Li, and Carl Mueller
We consider the one dimensional catalytic process introduced by Dawson and
Fleischmann, which is a modification of the super-Brownian motion. The
catalysts are given by a nonnegative Levy random measure. We
give sufficient conditions for the support of the process to be compact,
and sufficient conditions for noncompact support. Since the catalytic
process is related to the heat equation, compact support may be surprising. On the other hand, the super-Brownian motion
has compact support. We find that all nonnegative stable Levy measures lead to
compact support, and we give an example of a very rarified Levy process which
leads to noncompact support.
cmlr@troi.cc.rochester.edu (AMStex file available)
253. A STOCHASTIC PDE ARISING AS THE LIMIT OF A LONG-RANGE CONTACT PROCESS, AND
ITS PHASE TRANSITIONS
Carl Mueller and Roger Tribe
We consider the equation
u_t=\frac16 u_{xx} + \theta u-u^2+u^{\frac12}\dot W,\, t>0,
x\in\R, \theta>0
u(0,x)=u_0(x) \geq 0
where $\dot W=\dot W(t,x)$ is spacetime white noise. We show that this equation
arises as a limit of the long-range contact process studied by Bramson,
Durrett, and Swindle. In addition, we show that solutions undergo a phase
transition. For large values of $\theta$, we show that solutions have a
positive probability of survival. By survival, we mean that for all
values of $t$, $u(t,x)$ is never
identically 0. We also show that for small values of $\theta$, the probability
of survival is 0. This work answers some questions of R. Durrett, and can by
viewed as a hydrodynamic limit theorem for stochastic PDE. For the convergence
theorem, we use a detailed analysis of the martingale problems satisfied by the
particle system and its limit. For the phase transition, we compare solutions
to a subcritical branching process, and a supercritical oriented percolation
process.
cmlr@troi.cc.rochester.edu (latex file available)
254. STOCHASTIC DIFFERENTIAL GEOMETRY WITH JUMPS 2 .
Serge Cohen
We have presented in a previous article an intrinsic second order calculus for
manifold valued cadlag semimartingales. This formalism allows us to exhibit
a discrete time approximation for solutions of Stochastic Differential
Equations with jumps. First we are proving a uniform convergence
in probability result as an application of stability theory for SDE with
random measure (cf Jacod79). Then we show how this intrinsic
stochastic calculus unifies the continuous calculus of Meyer-Schwartz
and various definitions of the non continuous calculus of Stratonovich
and Ito . The stochastic exponential presented by Estrade and the
stochastic development when the driving process has jumps are two
examples of this situation.
sc@newaphro.enpc.fr (Latex file available written in French)
255. BARYCENTRES ET MARTINGALES SUR UNE VARIETE
Jean Picard
On a manifold $V$, we consider a family of
$V$-valued maps, called barycentres, defined on the set of probability
measures on $V$. For each barycentre, we define the class of
martingales on $V$ as the set of $V$-valued semimartingales with jumps
which satisfy a condition involving the barycentre; this notion
extends the notion of continuous martingales defined by means of the
second order differential geometry. We give several equivalent
characterizations, and prove that under some assumptions, there exists
one and only one martingale with prescribed final value; this result
can be applied to the probabilistic study of $V$-valued harmonic maps.
picard@ucfma.univ-bpclermont.fr
256. SPANNING FORESTS, INVASION PERCOLATION, AND OPTIMAL
PATHS TO INFINITY IN PERCOLATION MODELS
Kenneth S. Alexander
Spanning forests which are analogs of the minimal spanning tree are
considered for certain infinite graphs with a value f(b) attached to each
bond b. The forest consists of infinite trees and contains one, or possibly
two,optimal (locally minimax for f) paths to infinity from each site. Of
particular interest are a lattice with iid uniform values of f(b), and the
Voronoi graph on the sites of a Poisson process, with f(b) the length of b.
The corresponding percolation models are Bernoulli bond percolation and
the "lily pad" model of continuous percolation, respectively. Using
percolation methods it is shown that with at most one exception, all trees
of the forest have finite branches, i.e. have no doubly infinite paths. If
there is a tree in the forest, necessarily unique, without finite branches,
it must contain all sites of an infinite cluster at the critical point in
the corresponding percolation model. In two dimensions the forest has
finite branches, and at least for lattice models it consists of only one
tree. Applications to invasion percolation are given.
alexandr@mtha.usc.edu
