Quasi-continuous random variables and processes under the G-expectation framework

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 quasi-continuous 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.