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.

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