Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations (Stochastic Modelling and Applied Probability)

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.