On Linear Stochastic Flows

TL;DR

This paper investigates the existence of stochastic flows for linear stochastic evolution equations on Hilbert spaces, providing new conditions and examples, especially in Schatten classes beyond Hilbert-Schmidt operators.

Contribution

It establishes new sufficient conditions for the existence of stochastic flows in various operator classes, extending previous results to broader settings.

Findings

Existence of stochastic flows when A generates a C_0-semigroup and B_k are summable bounded operators

Sufficient conditions for stochastic flows in Schatten classes beyond Hilbert-Schmidt

New results and examples in the commutative case

Abstract

We study the existence of the stochastic flow associated to a linear stochastic evolution equation $$d X= AX\,d t +\sum_{k} B_k X\,d W_k, $$ 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.