First Order Stochastic Partial Differential Equations

Abstract

This chapter discusses the first order stochastic partial differential equations. The chapter studies the Cauchy problem of the first order stochastic partial differential equations of the parabolic type. The method of stochastic characteristic curve has been proposed to construct a solution of a suitable linear stochastic partial differential equation of first order. The chapter defines the first order stochastic partial differential equation rigorously and then introduces the associated stochastic characteristic equation. The proof of the existence and uniqueness of local solutions associated with a given initial condition is given. The chapter discusses quasi-linear, semi-linear and linear equation as special cases. The existence of the maximal or global solution is shown. The chapter also focuses on the regularity of Itô integral and the Stratonovich integral.