Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
Talk
Abstracts:
July
21-August 1, 2003
Reprints/Preprints
Materials
from Talks Photo Gallery
Hassan
Allouba
(Department of Mathematical Sciences, Kent State University)
allouba@mcs.kent.edu
http://www.mcs.kent.edu/~allouba
From
Brownian-Time Processes to Linearized Kuramoto-Sivashinsky
PDE
One
of the current "hot" areas of stochastic processes is the
study of stochastic processes in which time is replaced in
one way or another by a Brownian motion. An example of such
a process is the Iterated Brownian Motion (IBM) of Burdzy.
We introduce a family of processes, which we call Brownian-time
process (BTPs), which gives rise to new processes as well
as serve as a canonical family for other interesting processes
like IBM and the Markov Snake of Le Gall. We link BTPs to
some 4th order PDEs, and we finish the talk by solving a linearized
Kuramoto-Sivashinsky PDE (in any dimension) using an imaginary-Brownian-time-Brownian-angle
process. The theory of nonlinear Kuramoto-Sivashinsky PDEs
in $d\ge2$ is still being developed (even at the level of
existence/uniqueness) and we think that our probabilistic
approach is a good step towards resolving these questions
and more.
Krishna
B. Athreya (Department of Operations Research and
Industrial Engineering, Cornell University and Iowa State
University) athreya@orie.cornell.edu
Markov
Chains Generated by Iteration of iid Maps on R+
Markov
chains generated by iteration of random logistic maps where
the parameter of the map changes in an iid fashion has been
studied in the literature. The results for this case will
be reviewed and extended to maps on R+. A trichotomy for the
existence of nontrivial invariant probabilities will be presented.
More detailed studies of the three cases will be given. Recent
results on Harris irreducibilty of random s unimodal maps
will be discussed.Some open problems will be identified.
Siva
Athreya
(Indian Statistical Institute) athreya@isid.ac.in
http://www.isid.ac.in/~athreya
Hölder
Norm Estimates for Elliptic Operators on Finite and Infinite
Dimensional Spaces abstract.pdf
abstract.ps
Slides:
html
Michele
Lorenzo Baldini (Physics Department, New York
University) mlb257@nyu.edu
http://www.physics.nyu.edu/~mlb257
The
invariant measure of a infinite dimensional diffusion: how
can we compute it? (poster session)
I
will be presenting part of my Ph.D. thesis (work in progress)
that I am developing with my advisor Prof. Henry Mckean.
Randomly forced parabolic-type partial differential equations
are an increasingly interesting topic in mathematical physics.
They represent the evolution of a deterministic system subject
to some "wiggles" due to an external noise. They require
a different language and they can be interpreted as a diffusion,
namely a markov process with continuous path, in an infinite
dimensional space. The invariant measure is a probability
distribution that represents the statistical steady state
in which the solutions of the equation are going to stabilize.
In finite dimensions the invariant measure is solution of
a certain type of elliptic equation, but in infinite dimensions
it is a more elusive object. I will show a new interpretation
of the invariant measure in terms of the underlying diffusion
and also a new method to compute it.
Rabi
N. Bhattacharya
(Department of Mathematics, University of Arizona) bhattach@indiana.edu
Multiscale
Diffusions and a Transport Problem in Composite Media
Slides: html
We
study the effect of slowly evolving heterogeneities on the
transport of a substance from a point source in a composite
medium. In terms of dimensionless units, it is shown that
if new heterogeneities propagate at the rate 1/a for a large
spatial scale parameter a, then their effects are not manifested
until a time of the order o(a2/3). A more detailed
analysis for periodic media reveals that after an initial
Gaussian profile of the diffusing substance, which occurs
at times 1<< t << a2/3, a final Gaussian phase
shows up at times t>> a2. Examples for stratified
media show that different non-Gaussian phases may occur
between the two Gaussian phases mentioned above.
Dirk
Blömker (Mathematics Research Centre, University
of Warwick) bloemker@instmath.rwth-aachen.de
Structure of Invariant Measures Near Bifurcations
We
consider a general class of stochastic partial differential
equations (SPDEs) driven by additive noise, such that the
deterministic (unperturbed) PDE exhibits a change of stability.
We study the SPDE at a distance 
from this bifurcation, where
is supposed to be small.
The
dynamics of this problem exhibits a natural separation of
time-scales. Using multi-scale analysis, the transient dynamics
of the SPDE is well approximated, to the lowest order in ,
by a stochastic ODE called amplitude equation that describes
the dynamics of the (finite number of) unstable modes.
By
computing higher order corrections to the amplitude equation,
we give the first two terms in an -expansion
of the invariant Markov measure for the SPDE, and establish
rigorous error estimates. (joint work with Martin
Hairer, Warwick)
Marco
Cannone
(Laboratoire d'Analyse et de Mathématiques Appliquées, CNRS
UMR, Université de Marne-la-Vallée) cannone@math.univ-mlv.fr
Smooth
and Singular Solutions for the Navier-Stokes Equations
So
far, only two ways for attacking the Cauchy problem for the
Navier-Stokes equations are known: the first is due to J.
Leray (1933), and the second is due to T. Kato (1984). None
of them can be considered the ``golden rule'' for solving
the Navier-Stokes equations because they both leave open the
following celebrated question. In three dimensions, does the
velocity field of a fluid flow that starts with smooth initial
data (velocity and external force) remain smooth and unique
for all time? Based on a priori energy estimates, Leray's
theory gives the existence of global weak, possibly irregular
and possibly non-unique solutions to the Navier-Stokes equations,
whereas Kato's approach, based on the fixed point scheme,
imposes a priori a regularization effect on solutions we look
for. In other words, Kato's solutions are considered as fluctuations
around the solution of the heat equation with same initial
data, and are as such a priori regular. There exist however
two exceptions, more exactly two critical spaces where Kato's
method applies without imposing any a priori regularizing
condition : the Lorentz space L3, ,
considered from an analytical viewpoint by M. Yamazaki (1999)
and Y. Meyer (1999), and the pseudomeasure space, introduced
by Y. Le Jan and A.S. Sznitman (1997), and associated to a
probabilistic representation of solutions of the Navier-Stokes
equations.
In
this lecture, based on a series of joint works with P. Biler,
I. Guerra and G. Karch (2002), we will show how Kato's approach
gives existence and uniqueness of a (small) solution in a
larger space which, in our case, contains genuinely singular
solutions that are not smoothed out by the action of the nonlinear
semigroup associated. More exactly, using the pseudomeasure
space of Le Jan-Sznitman we can prove the following results.
The existence of singular solutions associated to singular
(e.g. the Dirac delta) external forces, thus allowing to describe
the solutions considered by L.D. Landau (1944) and by G. Tian
and Z. Xin (1998). The existence of regular solutions for
more regular external forces. The asymptotic stability of
small solutions including stationary ones. A pointwise loss
of smoothness for solutions for large data. Applying the same
techniques we will prove similar results for a model equation
of gravitating particles. Moreover, in the case of this particular
model, we will show that the loss of smoothness for large
data holds in the distributional sense as well.
Rene
Carmona
(Department of Research and Financial Engineering, Princeton
University) rcarmona@Princeton.EDU
Malliavin
Calculus for Stochastic Partial Differential Equations
Slides: pdf
First,
we review a couple of applications of the Malliavin calculus
to the analysis of stochastic PDE's. Then we concentrate on
a particular random Schroedinger equation introduced by Papanicolaou
et al. for the study of time reversal mirrors. We show how
one can compute the dependence of quantities of physical interest
with respect to the parameters of the equation.
Amit
Chakraborty
(Department Of Pure Mathematics, University Of Calcutta, 35,
Ballyganj Circular Road, Calcutta-19, West Bengal, India)
chakra_a@hotmail.com
A
Mathematical Model On Biodegradation
Joint
work with Dilip Kumar Bhattacharaya.
A
Mathematical model has been developed to describe the biodegradation
process, which generally occurs in Wetlands or rice fields.
Two types of Monod Kinetics are found to be highly suitable
for our purpose. Monod kinetics of type-I has been used to
describe the process through which complex materials like
biomass/Organic matters are converting to substrate/simple
compounds. Monod Kinetics of type-II has been used to describe
the last stage of the process in which substrate are converting
to biogases through some micro bacterial activity. The growth
of biomass has been taken into account by assuming its logistic
type growth. Maximum biomass utilization rate (Bmax)
and maximum substrate utilization rate(Smax) are
highly influenced by temperature and soil water content. The
condition for vulnerability of the system with respect to
the effect components (temperature and soil water) has been
defined using Liapunovs function. Using Nonlinear local and
global stability conditions of the system an Emission index
has been defined which is able to describe the biogass emission
trend.
Discontinuous
Markov Processes and Pseudodifferential Operators Slides:
pdf
It
is well known that there are rich interplay between Ito's
diffusions processes and second order partial differential
operators. It is also known that Markov processes with discontinuous
sample paths constitute an important family of stochastic
processes in probability theory, and that many physical and
economic systems should be and in fact have been successfully
modeled by discontinuous processes, such as stable processes.
However when a Markov process is discontinuous, its infinitesimal
generator is not local but a pseudo differential operator.
For example, the infinitesimal generator for symmetric stable
process Rn is a fractional Laplacian.
In
this talk, I will survey some recent progress in the study
of boundary potential theory for discontinuous Markov processes
which I have been involved. It includes Green function estimates,
Harnack and boundary Harnack inequalities, Martin boundary
and Martin kernel estimates, discontinuous Feynman-Kac transform,
gauge and conditional gauge theorems. While some of these
results have been established for a large class of discontinuous
Markov processes, I will use resurrected (or censored) stable
processes as a concrete model in the talk.
Part
I: Published Papers Relevant to the Talk:
-
Z.-Q. Chen and P. Kim: Green function estimate for censored
stable processes. Probab. Theory Relat. Fields, 124
(2002), 595-610.
- Z.-Q.
Chen and R. Song: General gauge and conditional gauge theorems.
Ann. Probab. 30 (2002), 1313-1339.
- Z.-Q.
Chen and R. Song: Conditional gauge theorem for non-local
Feynman-Kac transforms. Probab. Theory Relat. Fields, 125
(2003), 45-72.
- Z.-Q.
Chen: Gaugeability and conditional gaugeability. Trans.
Ameri. Math. Soc. 354 (2002), 4639-4679.
-
Z.-Q. Chen: Analytic characterization of conditional gaugeability
for non-local Feynman-Kac transforms. J. Funct. Anal. 202
(2003), 226-246.
- Z.-Q.
Chen and R. Song: Drift transforms and Green function estimates
for discontinuous processes. J. Funct. Anal. 201
(2003), 262-281.
Part
II: Preprints:
(psf files available at http://www.math.washington.edu/~zchen/paper.html)
- K.
Bogdan, K. Burdzy and Z.-Q. Chen: Censored stable processes.
To appear in Probab. Theory Relat. Fields.
- Z.-Q.
Chen and T. Kumagai: Heat kernel estimates for stable-like
processes on d-sets. To appear in Stoch. Proc. Appl.
-
Z.-Q. Chen and P. Kim: Stability of Martin Boundary under
Non-local Feynman-Kac Perturbations. Preprint 2003.
Erhan
Cinlar (Chairman
of the Department of Operations Research and Financial Engineering
(ORFE) and the Norman Sollenberger Professor of Engineering,
Princeton University) ecinlar@Princeton.EDU
Stochastic
Flows with Jumps
We
define Markovian flows of discontinuous transformations which,
further, have jumps in time. The aim is to model the development
of cracks in brittle solids. The flows are defined by random
differential equations driven by Poisson random measures whose
intensities are dependent on the flows themselves. A few properties
of such flows are explored.
Michael
Cranston
(Department of Mathematics, University of Rochester) cran@math.rochester.edu
Some
Results on the Parabolic Anderson Model
The
parabolic Anderson model in the lattice setting is a parabolic
pde du/dt =Hu, where H is the a small parameter,k, times the
discrete Laplacian plus a potential which is the stratonovich
differential of a Brownian motion at x. Here, for different
integer lattice points x, the Brownian motions are independent.
In the continuous spatial setting, the discrete Laplacian
is replaced by the ordinary Laplacian and the Brownian motion
field, indexed by x in d-dimensional Euclidean space, is no
longer independent but smoothly correlated. Another variant
in the lattice setting is to consider a potential which is
the stochastic differential of a field os independent Levy
processes. We consider the parabolic Anderson equation in
the above contexts and show how to use percolation arguments
to obtain existence of Lyapunov exponents. That is we exhibit
the exact exponential growth rate of positive solutions of
the Anderson pde. We also examine the behavior of this exponential
growth rate as k tends to zero.
Ian
M. Davies
(Department of Mathematics, University of Wales Swansea) I.M.Davies@swansea.ac.uk
Stochastic
Heat and Burgers Equations and their Singularities Slides:
pdf
The
Arnol'd-Thom classification of caustics for the Burgers equation
suggests that there should be an analogous one for the wavefronts
of the corresponding heat equation. We present a general theorem
for Hamiltonian systems characterizing how the level surfaces
of Hamilton's principal function meet the caustic surface
in both the deterministic and stochastic cases. Such a characterization
allows one to give a fairly detailed description of the behaviour
of the solution of the heat equation in the vicinity of the
wavefront and caustic. It allows one to propose some reasons
for the "blow-up" of the Burgers velocity field on the caustic.
In the case of small noise the shapes of the random wavefront
and random caustic may easily be obtained, and to first order
the caustic is merely displaced. In the stochastic case we
have the possibility of "rapid" changes in the caustic-wavefront
intersection. This will engender stochastic turbulence in
the Burgers velocity field and, due to its stochasticity,
be of an intermittent nature. There is no analogue of this
in the deterministic case. Throughout our studies much use
has been made of computer algebra packages in building an
understanding of the archetypal cases. Numerical simulations
and numerical solutions of the partial differential equations
involved have been immensely useful in clarifying conjectures
and determining apt characterizations.
Jinqiao
Duan (Department of Applied Mathematics Illinois
Institute of Technology) duan@iit.edu
Ergodicity,
Fluctuations and Stabilization in Fluid Flows
We consider a boundary feedback control problem for 3D Navier-Stokes
fluid flows, taking into account of small random fluctuations
arising from numerical simulations. Assuming the random fluctuations
form an independently identically distributed process, we
show that the real solution process to the linearized fluid
control problem is ergodic, i.e., it approaches a unique invariant
measure exponentially. Coupling techniques on invariant measures
are used in the proof. This is joint work with Andrei
V. Fursikov, Moscow State University, Russia.
William
G. Faris
(Department of Mathematics, University of Arizona) faris@math.arizona.edu
A
Gentle Introduction to Cluster Expansions Slides:
pdf
ps
A
cluster expansion is the representation of a set function
as a combinatorial exponential. That is, it represents the
contribution of a set as the sum over partitions of the set,
where the contribution of each partition is a product over
the subsets belonging to the partition. In the simplest case
of independence the only contribution is from the partition
into one point sets. The advantage of the representation is
that it gives a computationally effective way of estimating
dependence, by estimating the contributions of the other partitions.
This technique is well known in probability in the context
of the expansion of moments in terms of cumulants.
Cluster
expansions are used to analyze complex systems in many areas
of applied mathematics and physics, often non-rigorously.
However, it is possible to get rigorous estimates for cluster
expansions involving large numbers of variables. This is particularly
useful in controlling measures on infinite dimensional spaces,
where approximate independence is used as a replacement for
absolutely continuity. In particular, in rigorous renormalization
group analysis, each step involves a cluster expansion to
control irrelevant variables.
This
talk is an self-contained exposition of these ideas. It will
review the basic relation between the combinatorial exponential
and the ordinary exponential. This leads to an elementary
derivation of the Mayer equations for an equilibrium lattice
gas with two-particle interaction. These may be solved rigorously
under a cluster estimate of the Kotecky-Preiss type. One special
case of the lattice gas is a polymer system. Pairs of sites
are classified as compatible or incompatible, and the interaction
is that no two particles may occupy incompatible sites.
A
partition of a set is a collection of non-empty subsets that
do not overlap and that has union equal to the whole set.
The condition of no overlap is an incompatibility condition
on pairs of subsets. Thus it turns out that the special case
of a polymer system is the key to analyzing other cluster
expansions.
Mark
Freidlin
(Department of Mathematics, University of Maryland) mif@math.umd.edu
Multiparameter
Asymptotic Problems for Stochastic Differential Equations
and PDE's
When
various asymptotic problems for differential equations are
considered, one should keep in mind that the original equations
themselves, as a rule, are a result of neglect of some terms
which are considered as small. Therefore one, actually, must
consider a multiparameter asymptotic problem. The main terms
of the asymptotics can depend on the way how the parameters
approach zero. I will consider those questions for the Smoluchovski
- Kramers approximation of stochastic differential equations.
Problems related to stabilization as time goes to infinity,
homogenization, large deviations including exit problem and
stochastic resonance will be considered.
Victor
W. Goodman (Department of Mathematics, Indiana
University, Bloomington) goodmanv@lear.ucs.indiana.edu
Interest
Rate Explosions in HJM Bond Models
Slides: pdf
ps
HJM
Models allow a no-arbitrage family of bond prices to be specified
by setting the volatilities of forward interest rates. Some
natural choices for volatilities produce such strong positive
drift in SDE's for the rates that the rates explode in finite
time. We show how to remove this explosion problem by changing
the risk-neutral measure in a rather drastic way. Our new
measure, in effect, conditions the original model so that
the explosion event is deferred until after some specified
constant time. This is joint work with Kyounghee
Kim.
Priscilla
E. Greenwood
(Cindy) (Department of
Mathematics, Arizona State University) pgreenw@graph.la.asu.edu
Stochastic
Resonance Slides:
pdf
ps
If
a signal is below a threshold, no data about the signal is
obtained. If noise is added, signal plus noise is occasionally
above the threshold and the signal can be estimated. If the
noise variance is increased the information about the signal
first increases and then decreases. There is an optimal amount
of noise. This phenomenon, called stochastic resonance, is
of interest in, e.g., neuroscience and engineering, and most
work is experimental or simulation. This talk will be about
recent stochastic studies of stochastic resonance.
Martin
Greiner (Corporate
Technology Department, CT IC4, Siemens AG) martin.greiner@mchp.siemens.de
Data-driven stochastic processes in fully developed turbulence
In order to achieve a satisfactory understanding of fully developed
turbulence, two main routes can be taken in principle. Whereas
"top-down" attempts to derive almost everything directly from
the Navier-Stokes equation, "bottom-up" starts from data, brings
order into data phenomenology and achieves a consistent description
of turbulent statistics. The latter approach is advocated here.
At first, concentration is on some perfidies hidden in the processing
of measured time series and on the question, what are good observables.
Compared to the velocity field the energy dissipation field
turns out to be more fundamental. Simple hierarchical multiplicative
cascade processes yield a surprisingly robust stochastic modelling
of the latter. Beyond a consistent description of multifractal
scaling exponents, also observed scale correlations are quantitatively
reproduced. An analytical solution of the multivariate characteristic
function for such processes is given. Some model extensions
will be presented. At the end the focus will be on parallels
between turbulence and the modelling of financial markets and
communication networks.
Reprints:
greiner_EPL61_2003_756.pdf
greiner_PLA266_2000_276.pdf
greiner_PLA273_2000_104.pdf
greiner_PLA281_2001_249.pdf
greiner_PRE51_1995_1948.pdf
greiner_PRE58_1998_554.pdf
greiner_PRE59_1999_2451.pdf
greiner_PRL80_1998_5333.pdf
greiner_PhysicaA247_1997_41.pdf
greiner_PhysicaA325_2003_577.pdf
greiner_PhysicaD136_2000_125.pdf
Siwei
Jia
(Department of Statistics, Oregon State University) jia@stat.orst.edu>
A
Note on the Economic Management of Inventory or Resource under
Stochastic Prices (poster
session)
The
Markovian optimal policies are studied for the problem of economic
inventory control or resource management in a finite time horizon.
Under some conditions, in particular, when the prices are stochastic
and there is a positive fixed setup cost K, the existence of
{S, s}-type Markovian optimal management policies is proved.
When K=0, the optimal policies are of {S}-type, in which case
a comparison is made between the optimal policies under stochastic
and deterministic prices. It turns out that under stochastic
prices the optimal policies should be more conservative in order
to maximize the present value of expected revenue.
Kyounghee
Kim
(Mathematics Department, Indiana University, Bloomington)
kimkh@indiana.edu
Moment
Generating Function of the Reciprocal of Integral of Geometric
Brownian Motion (poster session)
In
this paper we obtain a simple, explicit integral form for the
moment generating function of the reciprocal of the random variable
defined by A( )t
:=  t
0 exp (2Bs+ 2 s)
ds , where {Bs: s>0} is a one dimensional Brownian
motion starting from 0. In case =
1, the moment generating function has a particularly simple
form.
Vassili
N. Kolokoltsov
(School of Computing and Mathematics, Nottingham Trent University)
vassili.kolokoltsov@ntu.ac.uk
Mathematics
of the Feynmann path integral applied to the Schrödinger
equation (Jump processes approach) (for
the workshop)
First
a short review is given of the basic approaches to the rigorous
construction of the path integral representation to the solutions
of the Schrödinger equation. The main part is devoted to
the development of an approach based on the jump Markov processes.
It will be shown that this approach allows the rigorous construction
for almost any reasonable Schrödinger equation including
singular (e.g. measure-valued) potentials and magnetic fields.
Various probabilistic interpretation will be given including
a lifting of the problem into a Fock space that allows, in particular,
a representation in terms of the standard Wiener measure. Connection
with semiclassical approximation will be also discussed. The
main new results of the talk are published in the author's book
V.N. Kolokoltsov. Semiclassical Analysis for Diffusions and
Stochastic Processes, Springer LNM 1724 (2000) and papers V.N.
Kolokoltsov, Math. Proc. Camb. Phil. Soc. 132 (2002), 353-375
and V.N. Kolokoltsov, Matem. Zbornik 194:6 (2003), 105-126.
Talk
Preprints:
serbia.pdf
serbia.ps
singular.pdf
singular.ps
Vassili
N. Kolokoltsov
(School of Computing and Mathematics, Nottingham Trent University)
vassili.kolokoltsov@ntu.ac.uk
Measure-valued
Limits of Interacting Particle Systems with k-nary Interaction
(poster session)
It
is shown that Markov processes describing the general k-nary
(in particular, usual binary) interacting particle systems under
a natural scaling converge to measure-valued Markov processes
with (generally speaking, infinite-dimensional) pseudo-differential
generators having symbols p(x,q) depending polynomially (of
order k) on x. In particular, our general scheme yields a unified
description for a large variety of models that are intensively
studied in different domains of natural and social studies including
(i) superprocesses, (ii) coagulation-fragmentation and collision
processes of statistical mechanics, (iii) birth and death processes
of mathematical biology, (iv) evolutionary games of evolution
biology.
Poster
Preprints:
fel.pdf
fel.ps
p2.pdf
p2.ps
p4nn.pdf
p4nn.ps
super.pdf
super.ps
Robert
Krasny (Department of Mathematics, University of
Michigan) krasny@umich.edu
Particle
Simulations of Vortex Sheet Roll-Up in Fluid Dynamics Slides:
pdf
A
vortex sheet is a moving surface in a fluid flow across which
the tangential component of fluid velocity has a jump discontinuity.
Vortex sheets are commonly used in fluid dynamics to model thin
shear layers in slightly viscous flow, for example the trailing
wake behind a airplane. The initial value problem for vortex
sheet motion is ill-posed in the sense of Hadamard due to Kelvin-Helmholtz
instability, and analytic solutions typically develop a curvature
singularity in finite time. Past the critical time, the sheet
rolls up into a tight spiral, although some form of regularization
is needed to capture this process. This talk will show how particle
simulations are being used to shed light on these issues. Recent
results indicating the onset of Hamiltonian chaos in vortex
sheet flow will be described (joint work with Monika Nitsche,
University of New Mexico).
Yves
Le jan
(Département de Mathématiques, Universite Paris Sud XI ) Yves.LeJan@math.u-psud.fr
Flows, Coalescence, Noise and Glue
Stochastic
flows of maps or kernels can be defined from SDE's or, in general,
from consistent systems of Markovian semi-groups. The detailed
study of models related to turbulent advection shows the interest
and the complexity of this theory: Non uniqueness of the solutions
and non linearity of the noise occur in certain cases.
Papers:
Bmatrix.pdf Bmatrix.ps
artsticky.pdf
artsticky.ps
coalflow.pdf coalflow.ps
Kening
Lu
(Department of Mathematics, Michigan State University) klu@math.msu.edu
Invariant
Manifolds for Stochastic PDE's
Invariant
manifolds are essential for describing and understanding dynamical
behavior of nonlinear and random systems. Stable, unstable and
center manifolds have been widely used in the investigation
of both finite and infinite dimensional deterministic dynamical
systems. In this talk, I will report some recent work on invariant
manifolds and foliations manifolds for a class of stochastic
partial differential equations. This talk is based on jointed
work with J. Duan and B.
Schmalfuss.
Mukul
Majumdar
(Department of Economics, Cornell University) mkm5@cornell.edu
Random
Dynamical Systems with Monotone Laws of Motion: Examples from
Economics Paper:
pdf
In
a number of contexts dealing with optimal allocation and management
of resources over time,one encounters dynamical systems with
monotone laws of motion. Examples from deterministic models
of growth and dynamic optimization will first be reviewed. Attempts
to capture repeated random shocks lead to the study of random
dynamical systems. Suppose that the state space S is an interval,
and the set of all possible laws of motion consists of monotone
maps from S to S from which a particular law is chosen independently
in each period according to the same distribution Q. If the
Markov process of states satisfies a "splitting" condition,
some strong results on the existence, uniqueness and stability
can be derived. Applications of such results to growth under
uncertainty and stochastic dynamic programming will be indicated.
Salah-Eldin
A. Mohammed (Department of Mathematics, Southern
Illinois University, Carbondale) salah@sfde.math.siu.edu
http://sfde.math.siu.edu
The
Stable Manifold Theorem for Semi-Linear Stochastic Partial Differential
Equations
We
give a characterization of the pathwise local structure of solutions
of semi-linear stochastic evolution equations (see's) and stochastic
partial differential equations (spde's) near stationary solutions.
The characterization is expressed in terms of the almost sure
large-time behavior of trajectories of the equation in the vicinity
of a stationary solution. More specifically, we establish local
stable manifold theorems for semi-linear see's and spde's. These
results give smooth stable and unstable manifolds in the neighborhood
of a hyperbolic stationary solution of the underlying stochastic
equation. The stable and unstable manifolds are stationary,
live in a stationary tubular neighborhood of the stationary
solution and are asymptotically invariant under the stochastic
semiflow of the see/spde. Examples covered by the theorems include
semilinear stochastic evolution equations, semilinear parabolic
spde's, stochastic reaction-diffusion equations and the stochastic
Burgers equation, all driven by infinite-dimensional noise.
Results
are joint work with Tusheng Zhang
and Huaizhong Zhao.
Charles
Newman
(Courant Institute of Mathematics, New York University) newman@courant.nyu.edu
The
Brownian Web and Scaling Limits
Arratia,
and later Toth and Werner, constructed random processes that formally
correspond to coalescing one-dimensional Brownian motions starting
from every space-time point. In joint work with L.R.G. Fontes,
M. Isopi and K. Ravishankar, we extend this earlier work by constructing
and characterizing what we call the Brownian Web as a random variable
taking values in an appropriate space whose points are sets of
paths. This leads to general convergence criteria and, in particular,
to convergence in distribution of coalescing random walks in the
scaling limit to the Brownian Web.
In further work, these results can be applied to scaling limits
of stochastic flows in one dimension. In this case the limit
is an extension of the Brownian Web, which includes both coalescence
and bifurcation, corresponding to regions of compression and
expansion in the original flow.
Refs.:
Fontes-Isopi-Newman-Ravishankar, PNAS 99 (2002) 15894-15897
(math.PR/0203184) math.PR/0304119
Keith
Nordstrom
(C4-CIRES, University of Colorado at Boulder) knordstrom@comcast.net
Critical
Scaling in a Physical Model of Convective Rainfall
Over
the last two decades, concepts of scale invariance have come
to the fore in both modeling and data analysis in hydrological
precipitation research. With the advent of the use of the multiplicative
random cascade model, these concepts have become increasingly
more important. However, unifying this statistical view of the
phenomenon with the physics of rainfall has proven to be a rather
nontrivial task. In this paper we present a simple model, developed
entirely from qualitative physical arguments, without invoking
any statistical assumptions, to represent tropical atmospheric
convection over the ocean. The model is analyzed numerically.
It shows that the data from the model rainfall look very spiky,
as if generated from a random field model. They look qualitatively
similar to real rainfall data sets from Global Atmospheric Research
Program (GARP) Atlantic Tropical Experiment [GATE].
A
critical point (in the sense of the physical theory of critical
phenomena) is found in a model parameter corresponding to the
Convective Inhibition (CIN), at which rainfall changes abruptly
from non-zero to a uniform zero value over the entire domain.
Near the critical value of this parameter, the model rainfall
field exhibits multifractal scaling determined from a fractional
wetted area analysis and a moment scaling analysis. It therefore
must exhibit long-range spatial correlations at this point,
a situation qualitatively similar to that shown by multiplicative
random cascade models and GATE rainfall data sets analyzed previously
(Gupta and Waymire, 1993; Over and Gupta, 1994; Over, 1995).
Scaling exponents associated with the model data are quantitatively
different from those estimated with real data. Such comparison
identifies a new theoretical framework for testing diverse physical
hypotheses governing rainfall based in empirically observed
scaling statistics. For example, considerations of the failure
of this model to reproduce observed geometries suggest model
generalizations based on the relaxation of certain assumptions.
Such generalizations may self-organize to a corresponding critical
point, a situation analogous to that proposed for the real system
by Peters, et al. (2002) and others.
Keith
Nordstrom
(C4-CIRES, University of Colorado at Boulder) knordstrom@comcast.net
Critical
Scaling in a Physical Model of Toprical Atmospheric Convection
over the Ocean (poster session)
Preprint: npg03001.pdf
Over
the last two decades, concepts of scale invariance have come
to the fore in both modeling and data analysis in hydrological
precipitation research. With the advent of the use of the multiplicative
random cascade model, these concepts have become increasingly
more important. However, unifying this statistical view of the
phenomenon with the physics of rainfall has proven to be a rather
nontrivial task. In this paper we present a simple model, developed
entirely from qualitative physical arguments, without invoking
any statistical assumptions, to represent tropical atmospheric
convection over the ocean. The model is analyzed numerically.
It shows that the data from the model rainfall look very spiky,
as if generated from a random field model. They look qualitatively
similar to real rainfall data sets from Global Atmospheric Research
Program (GARP) Atlantic Tropical Experiment [GATE].
A
critical point is found in a model parameter corresponding to
the Convective Inhibition (CIN), at which rainfall changes abruptly
from non-zero to a uniform zero value over the entire domain.
Near the critical value of this parameter, the model rainfall
field exhibits multifractal scaling determined from a fractional
wetted area analysis and a moment scaling analysis. It therefore
must exhibit long-range spatial correlations at this point,
a situation qualitatively similar to that shown by multiplicative
random cascade models and GATE rainfall data sets analyzed previously
(Gupta and Waymire, 1993; Over and Gupta, 1994; Over, 1995).
However, the scaling exponents associated with the model data
are different from those estimated with real data. This comparison
identifies a new theoretical framework for testing diverse physical
hypotheses governing rainfall based in empirically observed
scaling statistics.
Mina
Ossiander (Department of Mathematics, Oregon State
University) ossiand@MATH.ORST.EDU
Short
time existence of solutions to the incompressible Navier-Stokes
equations
Broadly
speaking, there are two scenarios in which existence and uniqueness
of solutions for the incompressible Navier-Stokes equations
can be demonstrated. In the first scenario, the initial data
and forcing are assumed to be `small enough' in some appropriate
space to guarantee existence and uniqueness of solutions for
all time. The methods used in deriving these results range from
energy estimates to LeJan-Sznitman type probabilistic representations.
The second scenario permits `large' initial data and forcing,
but existence of solutions is only guaranteed for a finite time
period. In the second scenario it is possible to show that,
for large initial data and forcing, mild solutions of the Navier-Stokes
equations can be represented as expectations of random multiplicative
functionals. This representation arises as an refinement of
the LeJan-Sznitman method of representing solutions probabilistically.
Cecile
Penland (NOAA-CIRES/Climate Diagnostics Center) Cecile.Penland@noaa.gov
Do
We Really Need to Describe Every Single Leaf in a Climate Model?:
Applications of the Central Limit Theorem
The
idea of of treating climate as a stochastic system perturbed
by rapidly-varying weather "noise" has been around for over
a quarter cen- tury. However, the idea is only now beginning
to be implemented by climate modelers. In this workshop, I will
summarize the mathematical theory allowing this to be done,
some of the current attempts at sto- chastic climate models,
and a few numerical considerations. I will also discuss some
common pitfalls and how severe they can be when quantitative
results (e.g., El Nino, global warming experiments) are important.
Marco
Romito (Dipartimento di Matematica U. Dini, Università
di Firenze) romito@math.unifi.it
A Probabilistic Representation for the Vorticity of a 3D
Viscous Fluid and for General Systems of Parabolic Equations
(poster session)
A
probabilistic representation formula for general systems of
linear parabolic equations, coupled only through the zero-order
term, is given. On this basis, an implicit probabilistic representation
for the vorticity in a $3$D viscous fluid (described by the
Navier-Stokes equations) is carefully analysed, and a theorem
of local existence and uniqueness is proved (joint work with
B. Busnello, Pisa, and F. Flandoli, Pisa).
Boris
L. Rozovskii (Center for Applied Mathematical Sciences,
University of Southern California, Denney Research Center) rozovski@math.usc.edu
http://www.usc.edu/dept/LAS/CAMS/usr/facmemb/boris
Stochastic
Navier-Stokes Equations for Turbulent Flows: Propagation of
Chaos and Moments Problem
A
perspective will be presented on stochastic equations of fluid
dynamics for turbulent velocities. A linear combination of Kraichnan
velocity and a regular component is a representative example
of the velocity fields in question. The motivation for this
setting is to understand the motion of fluid parcels in turbulent
and randomly forced fluid flows. Stochastic Euler and Navier-Stokes
equations for the smooth component of the velocity will be derived
from the first principles. Propagation of the Wiener chaos (generated
by Kraichnan velocity) by the Navier-Stokes equation and its
utility for the closure problem for the moments will be investigated.
Björn
Schmalfuss
(FB 1, Department 1, FH Merseburg, University of Applied Sciences
Merseburg) bjoern.schmalfuss@in.fh-merseburg.de
Stochastic
Partial Differential and Random Dynamical Systems Slides:
html
The
intention of the talk ist to give a description of the qualitative
analysis of the behavior of spde's. This analysis is based on
the theory of random dynamical systems.
At
the beginning of the talk we explain the concept of random dynamical
systems. In addition, we formulate examples for spde's which
generate a random dynamical system.
We
introduce the term random fixed point for random dynamical systems.
Such a random fixed point is generated by a random variable.
The stationary solutions starting in this random variable attract
other trajectories exponentially fast. Conditions for the existence
of random fixed points are formulated. In particular the stochastic
Navier Stokes equation has such a random fixed point if the
viscosity is large.
Under
weaker assumptions we can only prove the existence of a random
attractor which is a compact set in the phase space attracting
particular families of random sets. Particular spde's from climate
theory have a random attractor. For these systems we can prove
that the random attractor has a finite dimension.
At
the end of the talk we will discuss particular questions with
respect to inertial manifolds for spde's.
Michael
Scheutzow (Institute of Mathematics, MA 7-5, Technische
Universität Berlin, Str. des 17. Juni 136 D-10623 Berlin Germany)
ms@math.TU-Berlin.DE
On
the Dispersion of Sets Under the Action of an Isotropic Brownian
Flow
Isotropic
Brownian flows can be used as an approximate model to describe
the motion of passive tracers in a turbulent fluid. We start
by introducing this class of flows and then state some results
about the asymptotic growth of the diameter and the volume of
the image of a bounded subset of Euclidean space under the action
of such flows. This is joint work with Mike
Cranston, Georgi Dimitroff,
Hannelore Lisei and David
Steinsaltz.
Richard
Sowers
(Department of Mathematics, University of Illinois, Urbana Champaign)
r-sowers@uiuc.edu
http://www.math.uiuc.edu/~r-sowers
Stochastic
Averaging for Certain Systems with Conservation Laws
We
discuss several issues in stochastic averaging. These problems
all arise due to the presence of glueing. We recall that glueing
conditions arise where periodic oscillations bifurcate. The
glueing coefficients can be understood as solvability conditions
for the construction of a certain corrector function. We develop
some of the ideas behind these corrector functions and how they
are used in convergence issues arising in the martingale problem.
We then apply these ideas to some nontraditional problems in
stochastic averaging.
Michael
Tehranchi
(Department of Mathematics, University of Texas at Austin) tehranch@mail.ma.utexas.edu
A
Characterization of Hedging Portfolios for Interest Rate Contingent
Claims (poster session)
Reprint: HJMhedge.pdf
We
consider the problem of hedging a European interest rate contingent
claim with a portfolio of zero-coupon bonds and show that an
HJM type Markovian model driven by an infinite number of sources
of randomness does not have some of the shortcomings found in
the classical finite factor models. Indeed, under natural conditions
on the model, we find that there exists a unique hedging strategy,
and that this strategy has the desirable property that at all
times it consists of bonds with maturities that are less than
or equal to the longest maturity of the bonds underlying the
claim.
Enrique
Thomann
(Department of Mathematics, Oregon State University) thomann@MATH.ORST.EDU
Partial
Differential Equations and Multiplicative Processes Slides:
ima.pdf
ima.ps
ima3.pdf
ima3.ps
In
this talk, a survey of results and methods for representing
solutions of partial differential equations as an expected value
of a random multiplicative processes will be presented. This
method applies to linear, semilinear and quasilinear evolution
equations. Examples include the KPP equation, Burgers equation
and the incompressible Navier-Stokes equations. This work is
joint work with R. Bahttacharya, L. Chen,
S. Dobson, R. Guenther, C. Orum, M. Ossiander and E. Waymire.
Hao
Wang
(Department of Mathematics, University of Oregon) haowang@darkwing.uoregon.edu
A
Class of Conditional Independent Branching Particle Systems
and Their Interacting Limit Superprocesses
Papers:
DLW_rev.pdf
SinDeg.pdf
This talk will present recent progress in the research of a
model of a class of interacting branching particle systems and
their corresponding limit superprocesses (SDSM). In the non-degenerate
case, this model includes Super-Brownian motion as a special
case. For given coefficients, the limit superprocess has density
process which is a unique weak solution of a new type of stochastic
partial differential equation (SPDE). In the degenerate case,
for given coefficients the limit superprocess is of purely-atomic-measure-valued
process which can be characterized as unique strong solution
of a degenerate SPDE. In the singular, degenerate case, we also
can derive a SPDE. However, in this case, the uniqueness of
the strong solution fails. After modification of the singular,
degenerate SPDE, its strong solution has uniqueness and the
location processes can be identified by coalescing Brownian
motions. This class of superprocesses has conditional independence.
The conditional log-Laplace functional serves as an important
tool to discover deep properties of this class of superprocesses
same as log-Laplace functional does for Super-Brownian motion.
As examples, we will show that for both degenerate and the non-degenerate
cases, several difficult problems can be solved for immigration
SDSM by using conditional log-Laplace functional.
Edward
C. Waymire (Mathematics Department, Oregon State
University) waymire@MATH.ORST.EDU
Remarks
on Steady State Limits for NS in Majorizing Spaces: A Probabilistic
View
While
it is somewhat clear that results obtained from either LeJan-Sznitman
type probabilistic representations or contraction mapping arguments
are on nearly equal, if not in fact equal, footing, it may be
at least of interest to probabilists to see how simply certain
results may be obtained. The existtence of time-asymptotic steady
state solutions will be obtained for NS in majorizing spaces.
This is based on continued joint work with R.
Bhattacharya, University of Arizona, and L.
Chen, R. Guenther, C.
Orum, M. Ossiander, and
E. Thomann at Oregon State University.
Wojbor
A. Woyczynski (Department of Statistics, Case Western
Reserve University) waw@cwru.edu
http://laplace.stat.cwru.edu/~Wojbor
Nonlinear
partial differential equations driven by Levy diffusions and
related statistical issues
Using
nonlinear partial differential equations with random fields
initial data as models of real physical phenomena requires solving
statistical inference issues for parameters appearing in the
equations and the initial data. Such estimation problems can
be handled with help of the scaling limit results for the evolving
random fields governed by such equations. This approach requires
study and understanding of selfsimilar solutions of the equations
in question. We will illustrate it by discussion of conservation
laws driven by diffusive processes of Levy type.
Thaleia
Zariphopoulou (Department of Mathematics, University
of Texas at Austin) zariphop@mail.ma.utexas.edu
A
valuation algorithm in incomplete markets
A
new probabilistic valuation algorithm will be presented for
derivative prices in incomplete markets. The algorithm yields
prices in terms of nonlinear expectations under the correct
pricing measure. The valuation operator has desirable pricing
properties, among which, model universality, time consistency
and translation invariance with respect to hedgeable risks.
A byproduct of this work is the probabilistic representation
of solutions to quasilinear partial differential equations via
nonlinear iterative functionals.
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Probability
and Statistics in Complex Systems: Genomics, Networks, and Financial
Engineering, September 1, 2003 - June 30, 2004
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