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arXiv · 1206.3656

The stochastic Weiss conjecture for bounded analytic semigroups

Jamil Abreu·Bernhard Haak·Jan van Neerven
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Abstract

Suppose -A admits a bounded H-infinity calculus of angle less than pi/2 on a Banach space E with Pisier's property (alpha), let B be a bounded linear operator from a Hilbert space H into the extrapolation space E_{-1} of E with respect to A, and let W_H denote an H-cylindrical Brownian motion. Let gamma(H,E) denote the space of all gamma-radonifying operators from H to E. We prove that the following assertions are equivalent: (i) the stochastic Cauchy problem dU(t) = AU(t)dt + BdW_H(t) admits an invariant measure on E; (ii) (-A)^{-1/2} B belongs to gamma(H,E); (iii) the Gaussian sum \sum_{n\in\mathbb{Z}} γ_n 2^{n/2} R(2^n,A)B converges in gamma(H,E) in probability. This solves the stochastic Weiss conjecture proposed recently by the second and third named authors.

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Jamil Abreu, Bernhard Haak, Jan van Neerven. 2012-06-16. The stochastic Weiss conjecture for bounded analytic semigroups. https://doi.org/10.1112/jlms%2Fjdt003

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