Invariant Sublinear Expectations

We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}=\sup_{P\in\Theta}\mathrm{E}_P$, and show that each component $\mathbb{E}^{(d)}=\sup_{P\in\Theta^{(d)}}\mathrm{E}_P$ of the decomposition has a finite period $p_d\in\mathbb{N}$, i.e., $$\mathbb{E}^{(d)}\left[f-f\circ T^{p_d}\right]=0, \quad f\in\mathcal{H}.$$ Then we prove that a continuous invariant sublinear expectation that is strongly ergodic has a finite period $p_{\mathbb{E}}$, and each component $\Theta^{(d)}$ 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.