# Stochastic Ordinary and Stochastic Partial Differential Equations

## Transition from Microscopic to Macroscopic Equations

Part of the Stochastic Modelling and Applied Probability formerly: Applications of Mathematics book series (SMAP, volume 58)

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Part of the Stochastic Modelling and Applied Probability formerly: Applications of Mathematics book series (SMAP, volume 58)

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

- DOI https://doi.org/10.1007/978-0-387-74317-2
- 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
- Buy this book on publisher's site