Towards mesoscopic ergodic theory
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. ARTICLES .
September 2020 Vol. 63 No. 9: 1853–1876 https://doi.org/10.1007/s11425-019-1642-5
Towards mesoscopic ergodic theory Dedicated with Admiration to the Memory of Professor Shantao Liao
Weiwei Qi1,2,3 , Zhongwei Shen3 , Shirou Wang3 & Yingfei Yi3,4,∗ 1Academy
of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China; 3Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T 6G 2G1, Canada; 4School of Mathematics, Jilin University, Changchun 130012, China 2School
Email: [email protected], [email protected], [email protected], [email protected] Received September 17, 2019; accepted January 9, 2020; published online August 6, 2020
Abstract
The present paper is devoted to a preliminary study towards the establishment of an ergodic theory
for stochastic differential equations (SDEs) with less regular coefficients and degenerate noises. These equations are often derived as mesoscopic limits of complex or huge microscopic systems. By studying the associated Fokker-Planck equation (FPE), we prove the convergence of the time average of globally defined weak solutions of such an SDE to the set of stationary measures of the FPE under Lyapunov conditions. In the case where the set of stationary measures consists of a single element, the unique stationary measure is shown to be physical. Similar convergence results for the solutions of the FPE are established as well. Some of our convergence results, while being special cases of those contained in Ji et al. (2019) for SDEs with periodic coefficients, have weaken the required Lyapunov conditions and are of much simplified proofs. Applications to stochastic damping Hamiltonian systems and stochastic slow-fast systems are given. Keywords
ergodic theory, stochastic differential equation, Fokker-Planck equation, stationary measure, phys-
ical measure, mesoscopic limit MSC(2010)
37A10, 35Q84, 35J25, 37B25, 60J60
Citation: Qi W W, Shen Z W, Wang S R, et al. Towards mesoscopic ergodic theory. Sci China Math, 2020, 63: 1853–1876, https://doi.org/10.1007/s11425-019-1642-5
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Introduction
The present paper aims at investigating ergodic properties of mesoscopic stochastic systems described by stochastic differential equations. Such a system arises in many scientific areas as the mesoscopic limit of a large or complex (deterministic) microscopic system involving both structured variables and noises due to uncertainties, lack of mechanisms, high degrees of freedom, dynamical complexities and so on [42]. The traditional statistical theory of large or complex microscopic systems was established within the framework of ergodic theory of measure-preserving dynamical systems. A fundamental result in this theory is the celebrated Birkhoff ergodic theorem. Let {P t } be a flow or semi-flow on a probability * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.sc
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