Differential and Integral Equations

Noise-vanishing concentration and limit behaviors of periodic probability solutions

Min Ji, Weiwei Qi, Zhongwei Shen, and Yingfei Yi

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The present paper is devoted to the investigation of noisy impacts on the dynamics of periodic ordinary differential equations (ODEs). To do so, we consider a family of stochastic differential equations resulting from a periodic ODE perturbed by small white noises, and study noise-vanishing behaviors of their “steady states” that are characterized by periodic probability solutions of the associated Fokker-Plank equations. By establishing noise-vanishing concentration estimates of periodic probability solutions, we prove that any limit measure of periodic probability solutions must be a periodically invariant measure of the ODE and that the global periodic attractor of a dissipative ODE is stable under general small noise perturbations. For local periodic attractors (resp. local periodic repellers), small noises are constructed to stabilize (resp. de-stabilize) them. Our study provides an elementary step towards the understanding of stochastic stability of periodic ODEs.

Article information

Differential Integral Equations, Volume 33, Number 5/6 (2020), 273-322.

First available in Project Euclid: 16 May 2020

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Mathematical Reviews number (MathSciNet)

Primary: 35Q84: Fokker-Planck equations 37B25: Lyapunov functions and stability; attractors, repellers 60J60: Diffusion processes [See also 58J65] 93E15: Stochastic stability


Ji, Min; Qi, Weiwei; Shen, Zhongwei; Yi, Yingfei. Noise-vanishing concentration and limit behaviors of periodic probability solutions. Differential Integral Equations 33 (2020), no. 5/6, 273--322. https://projecteuclid.org/euclid.die/1589594454

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