Probability and Partial Differential Equations in Modern Applied Mathematics (The IMA Volumes in Mathematics and its Applications, 140)

✍ Scribed by Edward C. Waymire (editor)

• Publisher: Springer
• Year: 2005
• Tongue: English
• Leaves: 265
• Category: Library

✦ Synopsis

"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a 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 the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.

This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.

✦ Table of Contents

Title Page
Copyright Page
FOREWORD
PREFACE
Table of Contents
NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS
1. Introduction
2. Examples
3. Random dynamical systems.
4. Stationary distributions for Markov chains satisfying (1)
5. Harris irreducibility.
REFERENCES
PHASE CHANGES WITH TIME AND MULTI-SCALE HOMOGENIZATIONS OF A CLASS OF ANOMALOUS DIFFUSIONS
1. Introduction
2. A general model with two spatial scales: The first phase of asymptotics and the time scale for its breakdown.
3. The second Gaussian phase and its time scale, examples of non-Gaussian int ermediat e phases.
4. Examples of stratified media with non-Gaussian intermediate phases.
5. The non-divergence free case:
REFERENCES
SEMI-MARKOV CASCADE REPRESENTATIONS OF LOCAL SOLUTIONS TO 3-D INCOMPRESSIBLE NAVIER-STOKES
1. Introduction and preliminaries.
2. Semi-Markov cascades and local representations.
3. Time-asymptotic steady state solutions.
REFERENCES
AMPLITUDE EQUATIONS FOR SPDES: APPROXIMATE CENTRE MANIFOLDS AND INVARIANT MEASURES
1. Introduction.
2. General setting.
2.1. Assumptions.
2.2. Examples of equations.
3. Amplitude equations, main results.
3.1. Attractivity.
3.2. Approximation.
4. Applications.
4.1. Approximate centre manifold.
4.2. Dynamics of the random attractor.
5. Approximation of the invariant measure.
6. What is so special about cubic nonlinearities?
REFERENCES
ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS
1. Geophysical background.
2. Mathematical model.
3. Cocycle property.
4. Dissipativity.
5. Random dynamics: Enstrophy and ergodicity.
REFERENCES
STOCHASTIC HEAT AND BURGERS EQUATIONS AND THEIR SINGULARITIES
1. Introduction.
2. Stochastic heat and Burgers equations.
3. Stochastic general case.
4. Closeness to classical.
5. The Burgers fluid.
6. Singularities and intermittence of turbulence.
REFERENCES
A GENTLE INTRODUCTION TO CLUSTER EXPANSIONS
1. Introduction.
2. The exponential and the combinatorial exponential.
3. The equilibrium lattice gas.
3.1. The Mayer equations.
3.2. Cluster estimates.
3.3. Abstract polymer systems.
4. Polymer systems.
5. Cluster expansions.
REFERENCES
CONTINUITY OF THE ITO-MAP FOR HOLDER ROUGH PATHS WITH APPLICATIONS TO THE SUPPORT THEOREM IN HOLDER NORM
1. Introduction.
1.1. Background in Rough Path theory.
1.2 . Rough Path theory and stochastic analysis.
1.3. Rough Path theory for p E [2,3).
1.4. Definitions and outline.
2. Holder-regularity of Enhanced Brownian motion.
3. Approximations to Brownian Rough Paths.
3.1. Piecewise linear nested approximations.
3.2. Adapted dyadic approximations.
4. A primer on the Universal Limit Theorem.
5. Lipschitz regularity of Ito-rnap for HOlder Rough Paths.
6. Application to the Support Theorem.
APPENDIX
REFERENCES
DATA-DRIVEN STOCHASTIC PROCESSES IN FULLY DEVELOPED TURBULENCE
1. Introduction.
2. A careful look at data analysis.
3. Binary random multiplicative cascade process.
4. RMCP-driven data analysis.
5. Outlook: more stochastic processes.
REFERENCES
STOCHASTIC FLOWS ON THE CIRCLE
1. Introduction.
2. Flows of diffeomorphisms.
3. The Krylov Veretennikov expansion.
3.1. Lipschitz case.
3.2. Non-Lipschitz case.
3.3. A flow of infinite matrices.
4. Flows of kernels and flow of maps.
4.1. n-point motions.
4.2. Flow of maps.
4.3. Diffusive flow of kernels.
4.4. Diffusive or coalescing?
5. Classification of the solutions of the SDE.
5.1. Solutions of the SDE.
5.2. Extension of the noise and weak solutions.
REFERENCES
PATH INTEGRATION: CONNECTING PURE JUMP AND WIENER PROCESSES
1. Introduction.
2. Infinitely divisible complex distributions and complex Markov processes.
3. Regularization and the Fock space lifting.
4. Two remarks on parabolic equations in momentum representation.
REFERENCES
RANDOM DYNAMICAL SYSTEMS IN ECONOMICS
1. Introduction.
1.1. The Solow model: A dynamical system with an increasing law of motion.
1.2. The quadratic family in dynamic optimization problems.
2. Random dynamical systems.
3. Evolution.
3.1. A general theorem under splitting.
3.2. Applications of splitting.
3.2.1. Stochastic turnpike theorems.
3.2.2. Uncountable I': an example.
3.2.3. An estimation problem.
4. Iterates of quadratic maps.
REFERENCES
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION OF THE NAVIER-STOKES EQUATIONS
1. Introduction.
2. The Navier-Stokes equations in the Galerkin approximation.
2.1. The noisy forcing term.
2.2. The Galerkin approximation.
3. The ergodic properties of the process.
4. Regularity of the transition probabilities.
5. Irreducibility and the control problem.
6. A toy model.
7. Some numerical results.
7.1. The numerical simulation.
7.2. Conclusions.
REFERENCES
INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS
1. Introduction.
2. Random dynamical systems.
3. random evolution equations.
4. The random graph transform and inertial manifolds.
6. Examples.
REFERENCES
EXISTENCE AND UNIQUENESS OF CLASSICAL, NONNEGATIVE, SMOOTH SOLUTIONS OF A CLASS OF SEMI-LINEAR SPDES
1. Introduction.
2. Linear SPDE.
3. Semi-linear SPDE.
REFERENCES
NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS AND RELATED STATISTICAL ISSUES
Introduction.
1. Selfsimilar solutions of evolutions equations as limits of general solutions.
2. Parabolic scaling limits for Burgers turbulence.
3. Parametric estimation in Burgers turbulence via parabolicrescaling.
4. Selfsimilar solutions and scaling limits for fractional conservationlaws.
REFERENCES
LIST OF WORKSHOP PARTICIPANTS