On linear stochastic flows

Beniamin Goldys and Szymon Peszat

Abstract

We study the existence of the stochastic flow associated to a linear stochastic evolution equation $$\mathrm{d}X=AX\mathrm{d}t+\sum _k{B}_{k}X\mathrm{d}{W}_k,$$ 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 $\sum _k\Vert B_k\Vert <+\mathrm{\infty }$. 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.

Keywords: stochastic flow, stochastic equation with multiplicative noise, Schatten class.

AMS Subject Classification: Primary 60H15; secondary 60G15, 60G60.