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

TL;DR

This paper introduces the concept of uniformly gamma-radonifying families of operators, unifying existing notions, and applies it to analyze the stochastic Weiss conjecture related to invariant measures in stochastic Cauchy problems.

Contribution

It defines and studies uniformly gamma-radonifying families of operators and applies this to establish a criterion for invariant measures in stochastic systems, partially solving the stochastic Weiss conjecture.

Findings

Uniform gamma-radonification unifies R-boundedness and gamma-radonification.

Invariant measure existence is characterized by uniform gamma-radonification of a specific operator family.

Results apply when A and B are simultaneously diagonalisable, linking operator properties to stochastic system behavior.

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 $dU(t) = AU(t) dt + B dW(t); U(0) = 0$ 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.