Stochastic Ordinary and Stochastic Partial Differential Equations

Transition from Microscopic to Macroscopic Equations

  • Peter Kotelenez

Table of contents

About this book


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.


Kotelenez Macroscopic Microscopic Ordinary Partial Differential Equations Stochastic Variance partial differential equation

Authors and affiliations

  • Peter Kotelenez
    • 1
  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York 2008
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-387-74316-5
  • Online ISBN 978-0-387-74317-2
  • Series Print ISSN 0172-4568
  • About this book