Probability And Partial Differential Equations in Modern Applied Mathematics

by Waymire, Edward C.; Duan, Jinqiao

• ISBN13: 9780387258799
• ISBN10: 0387258795
• Format: Hardcover
• Copyright: 2005-08-01
• Publisher: Springer Verlag

Summary

The IMA Summer Program on Probability and Partial Differential Equations in Modern Applied Mathematics took place July 21-August 1, 2003. The program was devoted to the role of probabilistic methods in modern applied mathematics from perspectives of both a tool for analysis and as a tool in modeling. There is a growing recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in analysis, as well as offer new perspectives on the phenomena for modeling purposes. In addition, such approaches can be effective in sorting out multiple scale structure and in the development of both non-Monte Carlo as well as Monte Carlo type numerical methods. There is also a growing recognition of a role in the inclusion of stochastic terms in the modeling of complex flows, and the addition of such terms has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations. This volume consists of original contributions by researchers with a common interest in the problems, but with diverse mathematical expertise and perspective. The volume will be useful to researchers and graduate students who are interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in engineering and sciences.

Table of Contents

Foreword	v
Preface	vii
Nonnegative Markov chains with applications	1	(10)
K.B. Athreya
Phase changes with time and multi-scale homogenizations of a class of anomalous diffusions	11	(16)
Rabi Bhattacharya
Semi-Markov cascade representations of local solutions to 3-D incompressible Navier-Stokes	27	(14)
Rabi Bhattacharya, Larry Chen, Ronald B. Guenther, Chris Orum, Mina Ossiander, Enrique Thomann, and Edward C. Waymire
Amplitude equations for SPDEs: Approximate centre manifolds and invariant measures	41	(20)
Dirk Blömker and Martin Hairer
Enstrophy and ergodicity of gravity currents	61	(18)
Vena Pearl Bongolan-Walsh, Jingiao Duan, Hongjun Gao, Tamay Özgökmen, Paul Fischer, and Traian Iliescu
Stochastic heat and Burgers equations and their singularities	79	(18)
Ian M. Davies, Aubrey Truman, and Huaizhong Zhao
A gentle introduction to cluster expansions	97	(20)
William G. Faris
Continuity of the Ito-map for Holder rough paths with applications to the Support Theorem in Milder norm	117	(20)
Peter K. Friz
Data-driven stochastic processes in fully developed turbulence	137	(14)
Martin Greiner, Jochen Cleve, Jürgen Schmiegel, and Katepalli R. Sreenivasan
Stochastic flows on the circle	151	(12)
Yves Le Jan and Olivier Raimond
Path integration: connecting pure jump and Wiener processes	163	(18)
Vassili N. Kolokoltsov
Random dynamical systems in economics	181	(16)
Mukul Majumdar
A geometric cascade for the spectral approximation of the Navier-Stokes equations	197	(16)
M. Romito
Inertial manifolds for random differential equations	213	(24)
Björn Schmalfuss
Existence and uniqueness of classical, nonnegative, smooth solutions of a class of semi-linear SPDEs	237	(10)
Hao Wang
Nonlinear PDE's driven by Levy diffusions and related statistical issues	247	(12)
Wojbor A. Woyczynski
List of workshop participants	259