Ergodicity of Sublinear Markovian Semigroups

Abstract

In this paper, we study the ergodicity of an invariant sublinear expectation of sublinear Markovian semigroups. 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 in which either 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 a sublinear Markov process, we prove that its ergodicity is equivalent to the Markovian semigroup having eigenvalue 1, and the eigenvalue is simple in the space of bounded measurable functions. As an example we show that G-Brownian motion \{ Bt\} t\geq 0 on the unit circle has an invariant expectation and is ergodic if and only if \BbbE ( - (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.

Optimal Control of Probability on a Target Set for Continuous-Time Markov Chains

“…This is in contrast to the traditional stochastic optimal control problems, where the value function is a sublinear expectation operator [23]. Given the recent progress of the ergodic theory of sublinear semigroup and capacity [10], [11], and that the superlinear semigroup and the sublinear semigroup are conjugate to each other, it would be very interesting to ask whether or not there exists an invariant superlinear distribution µ such that V t,µ = µ. Here V t,µ (B) = (µV t,• )(B).…”