Stochastic Parabolic Equations

Abstract

This chapter is devoted to stochastic evolution equations in Hilbert spaces, in particular stochastic parabolic type equations of the form $$\displaystyle{du(t,x,\omega ) = \mathcal{L}(t,\omega )u(t,x,\omega )\,dt + \mathcal{M}(t,\omega )u(t,x,\omega )\,dW_{Q}(t),}$$ where $$\mathcal{L}$$ and $$\mathcal{M}$$ are second-order elliptic and first-order differential operators and W Q is a colored Wiener process (see Example 5.1). These equations are generalization of finite-dimensional SDEs and appear in the study of random phenomena in natural sciences and the unnormalized conditional probability of finite-dimensional diffusion processes (see Sect. 5.5), related to filtering equations derived in Fujisaki et al. (Osaka J Math 9:19–40, 1972) and Kushner (J Differ Equ 3:179–190, 1967). In Sect. 5.1 we collect basic definitions and results for Hilbert space-valued processes; in particular, for continuous martingales, quadratic variations and correlation operators are treated. Stochastic integrals are introduced in Sect. 5.2. Section 5.3 is devoted to the study of stochastic parabolic equations from the viewpoint of Hilbert space-valued SDEs, following Rozovskii (Stochastic evolution systems. Kluwer Academic, Dordrecht/Boston, 1990). By using the results presented, we also consider a semilinear stochastic parabolic equation with Lipschitz nonlinearity in Sect. 5.3.4. Section 5.4 deals with Itô’s formula and in Sect. 5.5 Zakai equations related to filtering problems are given.