Mixing of linear operators under infinitely divisible measures on Banach spaces

• Main Authors: Mau, Camille, Privault, Nicolas
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2024
• Subjects: Mathematical Sciences; Gaussian measures; Infinitely divisible measures
• Online Access: https://hdl.handle.net/10356/175812
• Institution: Nanyang Technological University

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. In this paper, 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 approach relies on characterizations of mixing for infinitely divisible random sequences, and replaces the use of using covariance operators with codifference functionals and control measures on Banach spaces. Our results are then specialized in explicit form to α-stable measures, with examples of linear operators satisfying the required measure invariance conditions.