Invariant and ergodic nonlinear expectations for $G$-diffusion processes

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 (G-SDEs) 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.