Ergodicity and Mixing of Sublinear Expectation System and Applications

TL;DR

This paper develops an ergodic theory framework for sublinear expectation systems, establishing pointwise ergodic theorems, relaxing laws of large numbers, and applying results to $G$-SDEs like $G$-Ornstein-Uhlenbeck processes.

Contribution

It introduces a novel ergodic theory approach to sublinear expectations, extending classical results and applying them to stochastic differential equations driven by $G$-Brownian motion.

Findings

Pointwise Birkhoff's ergodic theorem for invariant sublinear systems

Relaxed conditions for law of large numbers under $ ext{alpha}$-mixing

Applications to $G$-SDEs such as $G$-Ornstein-Uhlenbeck processes

Abstract

We utilize an ergodic theory framework to explore sublinear expectation theory. Specifically, we investigate the pointwise Birkhoff's ergodic theorem for invariant sublinear expectation systems. By further assuming that these sublinear expectation systems are ergodic, we derive stronger results. Furthermore, we relax the conditions for the law of large numbers and the strong law of large numbers under sublinear expectations from independent and identical distribution to $\alpha$-mixing. These results can be applied to a class of stochastic differential equations driven by $G$-Brownian motion (i.e., $G$-SDEs), such as $G$-Ornstein-Uhlenbeck processes. As byproducts, we also obtain a series of applications for classical ergodic theory and capacity theory.