ABSTRACT
To provide a unified theory with a wider range of applications, this chapter considers stochastic evolution equations in a Hilbert space setting. It introduces the notion of Hilbert space-valued martingales and defines stochastic integrals in Hilbert spaces. The chapter provides a brief introduction to stochastic evolution equations and considers the mild solutions for the stochastic evolution equations by the semigroup approach. The strong solutions of linear and nonlinear stochastic evolution equations are studied mainly under the so-called coercivity and monotonicity conditions. These conditions are commonly assumed in the study of deterministic partial differential equations. The chapter adopts the function analytic approach to study the existence, uniqueness and regularity of strong solutions. Finally, it treats the strong solutions of second-order stochastic evolution equations.
