The stochastic Weiss conjecture for bounded analytic semigroups

TL;DR

This paper proves the equivalence of conditions related to invariant measures, gamma-radonifying operators, and Gaussian sums for stochastic Cauchy problems involving bounded analytic semigroups, solving a recent conjecture.

Contribution

It establishes the stochastic Weiss conjecture by demonstrating the equivalence of three key conditions for stochastic Cauchy problems on Banach spaces.

Findings

Equivalence of invariant measure existence and gamma-radonifying operators.

Convergence of Gaussian sums in gamma(H,E).

Resolution of the stochastic Weiss conjecture.

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}} \gamma_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.