Ergodic theory under nonlinear expectations
时间  Datetime
2026-04-21 14:00 — 15:30
地点  Venue
会议室(703)
报告人  Speaker
Prof. Huaizhong Zhao
单位  Affiliation
Durham University, UK
邀请人  Host
肖冬梅
备注  remarks
报告摘要  Abstract

In this talk, I will discuss the ideas of ergodic theory of a dynamical system under the nonlinear expectation space/nonadditive probabilities. Ergodicity is defined as that any invariant set or its complement has upper probability 0 (Feng-Zhao (SIMA 2021)). It was proved that the ergodicity is equivalent to irreducibility, the eigenvalue 1 of the Koopman operator being simple, and Birkhoff’s law of large numbers with a single value. For a stochastic system under a sublinear Markov setup, the theory was also developed via lifting to canonical dynamical systems. Following this initial work, many progresses have been made recently. They include Zhao-Zhao (Preprint 2025) on the existence of invariant expectations of G-SDEs; Ma-Zhao (Preprint 2025) on ergodic optimal controls; Feng-Huang-Liu-Zhao (AAP 2026 and Preprint 2026) on the equivalence of the ergodicity and mixing of upper probability with the ergodicity of the invariant skeleton measure. If time permits, I will also talk about a weaker regime that any invariant set has upper probability 0 or 1 of Cerreia-Vioglio, Maccheroni and Marinacci (2016). We prove that this case does not give the irreducibility but is equivalent to a decomposition of finite ergodic components. I will also discuss Hurewicz’s problem on Birkhoff’s ergodic theorem for noninvariant measures and its necessary and sufficient conditions in terms of sublinear probabilities.
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