The stochastic Weiss conjecture for bounded analytic semigroups

Jamil Abreu (1) , Bernhard Hermann Haak (2) , Jan van Neerven (3)

1 Instituto de Matem atica
2 IMB - Institut de Mathématiques de Bordeaux
3 Analysis group

Suppose that A admits a bounded H^infty-calculus of angle less than pi/2 on a Banach space E which has 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: (a) the stochastic Cauchy problem dU(t) = AU(t) dt + B dWH(t) admits an invariant measure on E; (b) (-A)^{-1/2} B \in gamma(H;E); (c) the Gaussian sum \sum \ga_n 2^{n/2} R(2^n;A)B converges in gamma(H;E) in probability. This solves the stochastic Weiss conjecture of [8].

Domaines: Analyse fonctionnelle [math.FA]

Jamil Abreu, Bernhard Hermann Haak, Jan van Neerven. The stochastic Weiss conjecture for bounded analytic semigroups. Journal London Mathematical Society, 2013, 88 (1), pp.181-201. ⟨10.1112/jlms/jdt003⟩. ⟨hal-00771883⟩