In this paper we study the problems of invariant and ergodic expectations under $G$-expectation framework. In particular, the stochastic differential equations driven by $G$-Brownian motion have the unique invariant and ergodic expectations. Moreover, the invariant and ergodic expectations of $G$-SDEs are also sublinear expectations. However, the invariant expectations may not coincide with the ergodic expectations, which is different from the classical case.
"Invariant and ergodic nonlinear expectations for $G$-diffusion processes." Electron. Commun. Probab. 20 1 - 15, 2015. https://doi.org/10.1214/ECP.v20-3886