## 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 $$\sum_k\|B_k\|<+\infty$$. 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.

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

: Primary 60H15; secondary 60G15, 60G60.

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 Wednesday, May 5, 2021