In this paper, we first use PDE techniques and probabilistic methods to identify a kind of quasi-continuous random variables. Then we give a characterization of the G-integrable processes and get a kind of quasicontinuous processes by Krylov's estimates. This result is useful for the development of G-stochastic analysis theory. Moreover, it also provides a tool for the study of the non-Markovian Itô processes.

In this paper we study the problems of invariant and ergodic measures under G-expectation framework. In particular, the stochastic differential equations driven by G-Brownian motion (G-SDEs) have the unique invariant and ergodic measures. Moreover, the invariant and ergodic measures of G-SDEs are also sublinear expectations. However, the invariant measures may not coincide with ergodic measures, which is different from the classical case.

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