Ergodicity of Sublinear Markovian Semigroups

Feng, Chunrong; Zhao, Huaizhong

doi:10.1137/20m1356518

Ergodicity of Sublinear Markovian Semigroups

Feng, Chunrong; Zhao, Huaizhong

Authors

Dr Chunrong Feng chunrong.feng@durham.ac.uk
Associate Professor

Professor Huaizhong Zhao huaizhong.zhao@durham.ac.uk
Professor

Abstract

In this paper, we study the ergodicity of invariant sublinear expectation of sublinear Markovian semigroup. 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 either has 0 capacity 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 sublinear Markov process, we prove that its ergodicity is equivalent to the Markovian semigroup having eigenvalue 1 and it is simple in the space of bounded measurable functions. As an example we show that G-Brownian motion {Bt}t≥0 on the unit circle has an invariant expectation and is ergodic if and only if E(−(B1) 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.

Citation

Feng, C., & Zhao, H. (2021). Ergodicity of Sublinear Markovian Semigroups. SIAM Journal on Mathematical Analysis, 53(5), 5646-5681. https://doi.org/10.1137/20m1356518

Journal Article Type	Article
Acceptance Date	Jul 6, 2021
Online Publication Date	Oct 5, 2021
Publication Date	2021
Deposit Date	Jul 29, 2021
Publicly Available Date	Oct 7, 2021
Journal	SIAM Journal on Mathematical Analysis
Print ISSN	0036-1410
Electronic ISSN	1095-7154
Publisher	Society for Industrial and Applied Mathematics
Peer Reviewed	Peer Reviewed
Volume	53
Issue	5
Pages	5646-5681
DOI	https://doi.org/10.1137/20m1356518