Stochastic Ordinary and Stochastic Partial Differential Equations Transition from Microscopic to Macroscopic Equations /

• Main Author: Kotelenez, Peter (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: New York, NY : Springer New York, 2008.
• Series: Stochastic Modelling and Applied Probability formerly: Applications of Mathematics, 58
• Subjects: Mathematics.; Mathematical analysis.; Analysis (Mathematics).; Probabilities.; Physics.; Analysis.; Probability Theory and Stochastic Processes.; Mathematical Methods in Physics.

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 ma...

Table of 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.