Invariant Sublinear Expectations

Abstract

We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}={\sup }_{P\in \hat{I}˜}{\mathrm{E}}_P$, and show that each component ${\mathbb{E}}^{(d)}={\sup }_{P\in \hat{I}{˜}^{(d)}}{\mathrm{E}}_P$ 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 $\hat{I}{˜}^{(d)}$ 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.

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