Linear Stochastic Evolution Systems in Hilbert Spaces

Abstract

Fix $T\in \mathbb {R}_+$ and consider a stochastic basis $\mathbb {F}=(\varOmega , \mathscr {F}, \{\mathscr {F}_t\}_{t\in [0,T]}, \mathbb {P})$ with the usual assumptions. Let $(\mathbb {X},\mathbb {H},\mathbb {X}')$ be a normal triple of separable Hilbert spaces with canonical bi-linear functional [⋅, ⋅], and let $\mathbb {Y}$ be another separable Hilbert space. As before, ∥⋅∥ is the norm in $\mathbb {H}$ and (⋅, ⋅) is the inner product in $\mathbb {H}$; the subscripts, such as $\|\cdot \|{ }_{\mathbb {X}}$, are used for all other Hilbert space.