Stochastic ordinary and stochastic partial differential equations : transition from microscopic to macroscopic equations / Peter Kotelenez.

Format:

Book

Author/Creator:

Kotelenez, P. (Peter), 1943-

Series:

Stochastic modelling and applied probability ; 58.

Stochastic modelling and applied probability ; 58

Language:

English

Subjects (All):

Stochastic differential equations.

Stochastic partial differential equations.

Physical Description:

1 online resource (462 pages)

Edition:

1st ed. 2008.

Place of Publication:

New York : Springer Science+Business Media, c2008.

Language Note:

English

Summary:

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.

Contents:

From Microscopic Dynamics to Mesoscopic Kinematics

Heuristics: Microscopic Model and Space—Time Scales

Deterministic Dynamics in a Lattice Model and a Mesoscopic (Stochastic) Limit

Proof of the Mesoscopic Limit Theorem

Mesoscopic A: Stochastic Ordinary Differential Equations

Stochastic Ordinary Differential Equations: Existence, Uniqueness, and Flows Properties

Qualitative Behavior of Correlated Brownian Motions

Proof of the Flow Property

Comments on SODEs: A Comparison with Other Approaches

Mesoscopic B: Stochastic Partial Differential Equations

Stochastic Partial Differential Equations: Finite Mass and Extensions

Stochastic Partial Differential Equations: Infinite Mass

Stochastic Partial Differential Equations:Homogeneous and Isotropic Solutions

Proof of Smoothness, Integrability, and Itô’s Formula

Proof of Uniqueness

Comments on Other Approaches to SPDEs

Macroscopic: Deterministic Partial Differential Equations

Partial Differential Equations as a Macroscopic Limit

General Appendix.

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 445-458) and index.

ISBN:

1-281-14053-8

9786611140533

0-387-74317-0

OCLC:

233971424