Ruelle–Pollicott resonances of stochastic systems and a new approach to stochastic bifurcations
主讲人: Mickaël D. Chekroun，Weizmann Institute of Science & UCLA
报告地点：青岛校区华岗苑东楼119报告厅/ZOOM: 159 801 0503
The work of Dr. Chekroun is at the confluence zone between the theory of stochastic/nonlinear dynamics and functional analysis (semigroup theory).His work at the interface of Mathematics and climate science has been crowned by the several grants awarded by major US governmental agencies including the Department of Energy (DOE), the Mathematical Division of the National Science Foundation (NSF), the Machine Learning, Reasoning and Intelligence program of the Office of Naval Research (ONR), and the Applied and Computational Analysis program of the ONR.
In this talk, a theory of Ruelle–Pollicott (RP) resonances for systems of stochastic differential equations (SDEs) will be presented, valid for a broad class of stochastic systems. Roughly speaking, RP resonances are defined, in the stochastic context, as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups and the spectral theory of semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances will be then presented (Chekroun et al., 2020).
By extending the results of (Chekroun et al., 2014), it will be explained how a notion of reduced RP resonances can be rigorously framed, as soon as the dynamics is partially observed within a reduced state space V. Applications to the detection and characterization of such stochastic nonlinear oscillations in a high-dimensional stochastic system, namely the Cane-Zebiak model of El Niño-Southern Oscillation (ENSO) (Cao et al., 2019) subject to noise modeling fast atmospheric fluctuations, will be finally discussed.