An analytic approach to stochastic partial differential equations and its applications

The objective of this thesis is three fold. First, we consider linear stochastic partial differential equations (SPDEs) of divergence and nondivergence forms. In chapter 4, we study {dollar}Lsb{lcub}p{rcub}{dollar} -regularity for generalized solutions of equations with discontinuous coefficients. In chapter 5, we develop an {dollar}Lsb{lcub}p{rcub}{dollar}-theory of SPDEs of divergence form with continuous coefficients. Second, in chapter 6, the Cauchy problem for one-dimensional nonlinear SPDEs is studied. Third, we study numerical approximation of SPDEs. In chapter 7, we develop an {dollar}Lsb2{dollar}-theory for discrete stochastic evolution equations, in particular, stochastic partial difference equations obtained by discretizing SPDEs of divergence form. Chapter 8 concerns the finite difference approximation of SPDEs. We study the difference between the solutions of SPDEs and their discretizations using {dollar}Lsb2{dollar}-theory and {dollar}Lsb{lcub}p{rcub}{dollar}-theory. Error estimates and the rates of convergence are obtained.