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On Small Noise Perturbations of DeterministicDynamical Systems


Academic Report

Title: On Small Noise Perturbations of DeterministicDynamical Systems

Reporter: Prof. JIANG Jifa (Shanghai Normal University)

Time: Nov 23, 2019 (Saturday) AM 9:00-9:40

Location: B1410# room, Innovation Park Building

Contact: Prof. LIU Zhenxin (tel:84708351-8039)

Abstract: This talk presents a general framework to deal with the limit behavior of stationary measures for small noise perturbations of deterministic dynamical systems. Underthe probability convergence assumption, we prove that any weak convergent limit of a sequence of stationary measures is an invariant measure of deterministic dynamicalsystem, whose support is contained in the Birkhoffcenter of the deterministic dynamical system. This result can be applied to SODEs, SPDEs and SFDEs driven by either Brown motion orLevy process with small noise intensity as well as stochastic approximation with constant step. We also discuss stochastic stability and provide a criterion for a repeller to be a null set of limit measures for SODEs with small nondegenerate noise and prove that any limit cycle of a planar system with positive characteristic exponent is always a null set of limit measures. We show that there is a distinct difference on stochastic stability between noise perturbed systems of structurally stable and structurally unstable deterministic systems.
This is a joint work with Dr. Chen Lifeng and Profs. Dong Zhao and Zhai Jianliang.