HOME    »    PROGRAMS/ACTIVITIES    »    Annual Thematic Program
July 21-August 1, 2003

Probability and Statistics in Complex Systems: Genomics, Networks, and Financial Engineering, September 1, 2003 - June 30, 2004

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

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 Liapunovs 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.

Zhenqing Chen (Department of Mathematics, University of Washington)  zchen@math.washington.edu

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:

1. Z.-Q. Chen and P. Kim: Green function estimate for censored stable processes. Probab. Theory Relat. Fields, 124 (2002), 595-610.
2. Z.-Q. Chen and R. Song: General gauge and conditional gauge theorems. Ann. Probab. 30 (2002), 1313-1339.
3. Z.-Q. Chen and R. Song: Conditional gauge theorem for non-local Feynman-Kac transforms. Probab. Theory Relat. Fields, 125 (2003), 45-72.
4. Z.-Q. Chen: Gaugeability and conditional gaugeability. Trans. Ameri. Math. Soc. 354 (2002), 4639-4679.
5. Z.-Q. Chen: Analytic characterization of conditional gaugeability for non-local Feynman-Kac transforms. J. Funct. Anal. 202 (2003), 226-246.
6. 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)

1. K. Bogdan, K. Burdzy and Z.-Q. Chen: Censored stable processes. To appear in Probab. Theory Relat. Fields.
2. Z.-Q. Chen and T. Kumagai: Heat kernel estimates for stable-like processes on d-sets. To appear in Stoch. Proc. Appl.
3. 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

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.

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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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