ABSTRACT
So far we have studied two principal types of stochastic partial differential equations, namely, the parabolic and hyperbolic equations. To provide a unified theory with a wider range of applications, we shall consider stochastic evolution equations in a Hilbert space driven by the Wiener and/or Le´vy noises. We first introduce the notion of Hilbert space-valued martingales in Section 6.2. Then stochastic integrals in Hilbert spaces with respect to the Wiener process and the Poisson measure are defined in Section 6.3. In Section 6.4, an Itoˆ’s formula is presented. After a brief introduction to stochastic evolution equations in Section 6.5, we consider, in Section 6.6, the mild solutions by the semigroup approach. This subject was treated extensively by Da Prato and Zabczyk [21] and the article by Walsh [94]. For further development, one is referred to this classic book. Next, in Section 6.7, 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 [61] and were introduced to the stochastic counterpart by Bensoussan and Teman [5], [6], and Pardoux [75]. Their function analytic approach will be adopted to study the existence, uniqueness, and regularity of strong solutions. We should also mention that a real analytical approach to strong solutions was developed by Krylov [51]. Finally, the strong solutions of second-order stochastic evolution equations will be treated separately, in Section 6.8, due to the fact that they are neither coercive nor monotone. Some simple examples are given and more interesting ones will be discussed in Chapter 8.
