Stochastic Ordinary and Stochastic Partial Differential Equations

This book analyzes mathematical models of time-dependent physical phenomena on microscopic, macroscopic and mesoscopic levels. It provides a rigorous derivation of each level from the preceding one and examines the resulting mesoscopic equations in detail.

Stochastic Partial Differential Equations analyzes mathematical models of time-dependent physical phenomena on microscopic, macroscopic and mesoscopic levels. It provides a rigorous derivation of each level from the preceding one and examines the resulting mesoscopic equations in detail. Coverage first describes the transition from the microscopic equations to the mesoscopic equations. It then covers a general system for the positions of the large particles.

Includes supplementary material: sn.pub/extras

Klappentext

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.

Inhalt
From Microscopic Dynamics to Mesoscopic Kinematics.- Heuristics: Microscopic Model and SpaceTime 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.