Partial Differential Equations Ii: Qualitative Studies of Linear Equations

Partial differential equations is a many-faceted subject. Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, it has developed into a body of material that interacts with many branches of math­ ematics, such as differential geometry, complex analysis, and harmonic analysis, as weil as a ubiquitous factor in the description and elucidation of problems in mathematical physics. This work is intended to provide a course of study of some of the major aspects of PDE.

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.