6 - Elliptic and Parabolic Partial Differential Equations and Their Relations to Stochastic Differential Equations

Abstract

This chapter presents elliptic and parabolic partial differential equations and discusses their relations to stochastic differential equations. It describes the square root of a non-negative definite matrix. The maximum principles for elliptic equations and parabolic equations are presented. The density function of the transition probability function is called the transition density function. The chapter presents the Cauchy problem and fundamental solutions for parabolic equations. It also presents a stochastic representation of solutions of partial differential equations.