Mixing of linear operators under non-Gaussian measures

Abstract

The mixing and ergodicity of Gaussian measures have been characterized in terms of their covariances, first for random sequences, and then in the framework of linear dynamics on Banach spaces using covariance operators.

In this thesis, we extend the latter results to the setting of infinitely divisible measures on Banach spaces, by deriving necessary and sufficient conditions for the strong and weak mixing of linear operators. Our results are specialized in explicit form to stable and tempered stable measures, with examples of linear operators satisfying the required measure invariance conditions.

We also derive rates of convergence for the mixing of linear operators in the infinite-dimensional framework. Explicit mixing rates are obtained for weighted shifts under compound Poisson, α-stable, and tempered α-stable measures.

Our approach relies on characterizations of mixing for infinitely divisible stochastic processes, and replaces the use of using covariance operators with codifference functionals and control measures on Banach spaces.

We also investigate mixing in the setting of non-Frechet spaces and show that weak*-discontinuity is a necessary condition for the mixing of linear operators on duals of real countably Hilbert nuclear spaces equipped with a non-Gaussian Mittag-Leffler or Gamma-grey probability measure. For the Gaussian measure, we derive a partial characterization for weak*-continuous linear operators in terms of covariance operators.