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.