Invariant Sublinear Expectations

We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}={\sup }_{P\in \mathrm{\Theta }}{\mathrm{E}}_P$, and show that each component ${\mathbb{E}}^{(d)}={\sup }_{P\in {\mathrm{\Theta }}^{(d)}}{\mathrm{E}}_P$ of the decomposition has a finite period $p_d\in \mathbb{N}$, i.e., $${\mathbb{E}}^{(d)}[f-f\circ T^{p_d}]=0,f\in \mathcal{H}\mathrm{.}$$ Then we prove that a continuous invariant sublinear expectation that is strongly ergodic has a finite period $p_{\mathbb{E}}$, and each component ${\mathrm{\Theta }}^{(d)}$ of its periodic decomposition is the convex hull of a finite set of $T^{p_d}$-ergodic probabilities. As an application of the characterization, we prove an ergodicity result which shows that the limit of the $p_{\mathbb{E}}$-step time means achieves the upper expectation.