The stochastic Weiss conjecture for bounded analytic semigroups

Suppose A admits a bounded H 1 -calculus of angle less than =2 on a Banach space E which has Pisier's property ( ), 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 WH denote an H-cylindrical Brownian motion. Let (H;E) denote the space of all -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 = 2B2 (H;E); (c) the Gaussian sum P n2Z n2 n =2R(2n;A)B converges in (H;E) in prob- ability. This solves the stochastic Weiss conjecture of (8).

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