On linear stochastic flows

Abstract

We study the existence of the stochastic flow associated to a linear stochastic evolution equation d ⁡ X = A X d ⁡ t + ∑ k B k X d ⁡ W k , \begin{equation*} \operatorname {d} X= AX\operatorname {d} t +\sum _{k} B_k X\operatorname {d} W_k, \end{equation*} on a Hilbert space. Our first result covers the case where A A is the generator of a C 0 C_0 -semigroup, and ( B k ) (B_k) is a sequence of bounded linear operators such that ∑ k ‖ B k ‖ > + ∞ \sum _k\|B_k\|>+\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.