Ergodicity of Sublinear Markovian Semigroups

In this paper, we study the ergodicity of invariant sublinear expectation of sublinear Markovian semigroup. For this, we first develop an ergodic theory of an expectation-preserving map on a sublinear expectation space. Ergodicity is defined as any invariant set either has $0$ capacity itself or its complement has $0$ capacity. We prove, under a general sublinear expectation space setting, the equivalent relation between ergodicity and the corresponding transformation operator having simple eigenvalue $1$, and also with Birkhoff type strong law of large numbers if the sublinear expectation is regular. For sublinear Markov process, we prove that its ergodicity is equivalent to the Markovian semigroup having eigenvalue $1$ and it is simple in the space of bounded measurable functions. As an example we show that $G$-Brownian motion $\{B_t\}_{t\geq 0}$ on the unit circle has an invariant expectation and is ergodic if and only if ${\mathbb E}(-(B_1)^2)<0$. Moreover, it is also proved in this case that the invariant expectation is regular and the canonical stationary process has no mean-uncertainty under the invariant expectation.