The Kac formula and Poincar\'{e} recurrence theorem in Riesz spaces

TL;DR

This paper extends classical recurrence and ergodic theorems to Riesz spaces, providing new conditional versions of the Poincaré Recurrence Theorem and Kac formula, applicable to a broad class of non-pointwise processes.

Contribution

It introduces Riesz space generalizations for recurrence, ergodicity, and classical theorems, broadening their applicability beyond traditional measure-theoretic frameworks.

Findings

Conditional expectation preserving processes are conditionally ergodic.

New conditional versions of Poincaré Recurrence and Kac formula are established.

Results apply to processes in L^1 spaces with probability measures.

Abstract

Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincar\'{e} Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Ces\`aro mean associated with the iterates of the process. Applied to processes in $L^1(\Omega,{\mathcal A},\mu)$, where $\mu$ is a probability measure, new conditional versions of the above theorems are obtained.