Uniformly gamma-radonifying families of operators and and the stochastic Weiss conjecture

Abstract

We introduce the notion of uniform gamma-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and gamma-radonification of an individual operator. We study the the properties of uniformly gamma-radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem $dU(t)=AU(t)dt+BdW(t);U(0)=0$ Here, $A$ is the generator of a strongly continuous semigroup of operators on a Banach space $E$, $B$ is a bounded linear operator from a separable Hilbert space $H$ into $E$, and $W$ is an $H$-cylindrical Brownian motion. When $A$ and $B$ are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family $\{\sqrt{\lambda }R(\lambda ,A)B:\lambda >0\}$ is uniformly gamma-radonifying. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.

arXiv commentVersion accepted for publication in Operators and Matrices

Cited by in corpus (2)

• The stochastic Weiss conjecture for bounded analytic semigroups
• Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems