Mixing rates for linear operators under infinitely divisible measures on Banach spaces

TL;DR

The paper develops a codifference-based framework to analyze mixing of linear operators under infinitely divisible measures on Banach spaces, extending beyond Gaussian dynamics. It provides a necessary-and-sufficient mixing characterization that avoids previous vanishing-support constraints and derives quantitative codifference decay bounds. For three non-Gaussian measure families—compound Poisson, $\alpha$-stable, and tempered stable—it gives explicit decay rates for weighted shifts and forward/backward operators, along with results for functionals of the processes. The work yields both abstract criteria and concrete rate estimates, enhancing the toolbox for ergodic and dynamical analysis in infinite-dimensional non-Gaussian settings. These results have potential implications for understanding non-Gaussian linear dynamics in stochastic systems and ergodic theory.

Abstract

We derive rates of convergence for the mixing of operators under infinitely divisible measures in the framework of linear dynamics on Banach spaces. Our approach is based on the characterization of mixing in terms of codifference functionals and control measures, and extends previous results obtained in the Gaussian setting via the use of covariance operators. Explicit mixing rates are obtained for weighted shifts under compound Poisson, α-stable, and tempered α-stable measures.

Theorems & Definitions (21)

• Definition 2.2
• Lemma 2.3
• Theorem 2.4
• Example 2.5
• Proposition 2.6
• Lemma 3.1: Lévy measure bounds.
• Lemma 3.2: Control measure bounds.
• Lemma 4.1
• Lemma 4.2
• Proposition 4.3