On Linear Stochastic Flows

We study the existence of the stochastic flow associated to a linear stochastic evolution equation dX = AXdt+ ∑ k BkXdWk, on a Hilbert space. Our first result covers the case where A is the generator of a C0semigroup, and (Bk) is a sequence of bounded linear operators such that ∑ k ‖Bk‖ < +∞. 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.

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