On Linear Stochastic Flows

Abstract

We study the existence of the stochastic flow associated to a linear stochastic evolution equation $$dX=AXdt+\sum _k{B}_{k}Xd{W}_k,$$ on a Hilbert space. Our first result covers the case where $A$ is the generator of a $C_0$-semigroup, and $(B_k)$ is a sequence of bounded linear operators such that ${\sum }_k\Vert B_k\Vert <+\infty$. We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert-Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well.