Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations

Peter Kotelenez (auth.)0387743162, 978-0-387-74316-5

This book provides the first rigorous derivation of mesoscopic and macroscopic equations from a deterministic system of microscopic equations. The microscopic equations are cast in the form of a deterministic (Newtonian) system of coupled nonlinear oscillators for N large particles and infinitely many small particles. The mesoscopic equations are stochastic ordinary differential equations (SODEs) and stochastic partial differential equatuions (SPDEs), and the macroscopic limit is described by a parabolic partial differential equation.

A detailed analysis of the SODEs and (quasi-linear) SPDEs is presented. Semi-linear (parabolic) SPDEs are represented as first order stochastic transport equations driven by Stratonovich differentials. The time evolution of correlated Brownian motions is shown to be consistent with the depletion phenomena experimentally observed in colloids. A covariance analysis of the random processes and random fields as well as a review section of various approaches to SPDEs are also provided.

An extensive appendix makes the book accessible to both scientists and graduate students who may not be specialized in stochastic analysis.

Probabilists, mathematical and theoretical physicists as well as mathematical biologists and their graduate students will find this book useful.

Peter Kotelenez is a professor of mathematics at Case Western Reserve University in Cleveland, Ohio.

Table of contents :
Front Matter….Pages i-xvi
Heuristics: Microscopic Model and Space—Time Scales….Pages 9-13
Deterministic Dynamics in a Lattice Model and a Mesoscopic (Stochastic) Limit….Pages 15-30
Proof of the Mesoscopic Limit Theorem….Pages 31-55
Stochastic Ordinary Differential Equations: Existence, Uniqueness, and Flows Properties….Pages 59-84
Qualitative Behavior of Correlated Brownian Motions….Pages 85-131
Proof of the Flow Property….Pages 133-149
Comments on SODEs: A Comparison with Other Approaches….Pages 151-159
Stochastic Partial Differential Equations: Finite Mass and Extensions….Pages 163-201
Stochastic Partial Differential Equations: Infinite Mass….Pages 203-219
Stochastic Partial Differential Equations:Homogeneous and Isotropic Solutions….Pages 221-227
Proof of Smoothness, Integrability, and Itô’s Formula….Pages 229-272
Proof of Uniqueness….Pages 273-290
Comments on Other Approaches to SPDEs….Pages 291-309
Partial Differential Equations as a Macroscopic Limit….Pages 313-331
Appendix….Pages 335-429
Back Matter….Pages 431-459