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Ergodic semi-implicit approximations to periodic measures of stochastic differential equations with locally Lipschitz drifts—Error analysis in Wasserstein distance (2025)
Journal Article
Feng, C., Liu, Y., Liu, Y., & Zhao, H. (2025). Ergodic semi-implicit approximations to periodic measures of stochastic differential equations with locally Lipschitz drifts—Error analysis in Wasserstein distance. Journal of Differential Equations, 441, Article 113472. https://doi.org/10.1016/j.jde.2025.113472

We study numerical approximations to the periodic measures of time-periodic stochastic differential equations. For those systems with locally Lipschitz coefficients, while the explicit Euler-Maruyama scheme does not work, we carry out semi-implicit E... Read More about Ergodic semi-implicit approximations to periodic measures of stochastic differential equations with locally Lipschitz drifts—Error analysis in Wasserstein distance.

Random periodic paths of stochastic periodic semi-flows through random attractors, synchronizations and Lyapunov exponents (2025)
Journal Article
Feng, C., & Luo, Y. (2025). Random periodic paths of stochastic periodic semi-flows through random attractors, synchronizations and Lyapunov exponents. Computational and Applied Mathematics, 44(2), Article 179. https://doi.org/10.1007/s40314-025-03135-9

This paper discusses the existence and uniqueness of random periodic paths of stochastic periodic semi-flows. Random periodic attractors are introduced and synchronization for stochastic periodic semi-flows is proved under some conditions to find the... Read More about Random periodic paths of stochastic periodic semi-flows through random attractors, synchronizations and Lyapunov exponents.

Existence of geometric ergodic periodic measures of stochastic differential equations (2023)
Journal Article
Feng, C., Zhao, H., & Zhong, J. (2023). Existence of geometric ergodic periodic measures of stochastic differential equations. Journal of Differential Equations, 359, 67-106. https://doi.org/10.1016/j.jde.2023.02.022

Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems and can be used to characterise the long-term periodic behaviour of stochastic systems. This paper gives sufficient conditions for the existence, uniquene... Read More about Existence of geometric ergodic periodic measures of stochastic differential equations.

Periodic measures and Wasserstein distance for analysing periodicity of time series datasets (2023)
Journal Article
Feng, C., Liu, Y., & Zhao, H. (2023). Periodic measures and Wasserstein distance for analysing periodicity of time series datasets. Communications in Nonlinear Science and Numerical Simulation, 120, Article 107166. https://doi.org/10.1016/j.cnsns.2023.107166

In this article, we establish the probability foundation of the periodic measure approach in analysing periodicity of a dataset. It is based on recent work of random periodic processes. While random periodic paths provide a pathwise model for time se... Read More about Periodic measures and Wasserstein distance for analysing periodicity of time series datasets.

Differentiable maps with isolated critical points are not necessarily open in infinite dimensional spaces (2021)
Journal Article
Feng, C., & Li, L. (2022). Differentiable maps with isolated critical points are not necessarily open in infinite dimensional spaces. Advances in operator theory, 7, Article 5. https://doi.org/10.1007/s43036-021-00170-1

Saint Raymond asked whether continuously differentiable maps with isolated critical points are necessarily open in infinite dimensional (Hilbert) spaces. We answer this question negatively by constructing counterexamples in various settings including... Read More about Differentiable maps with isolated critical points are not necessarily open in infinite dimensional spaces.

Ergodicity of Sublinear Markovian Semigroups (2021)
Journal Article
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

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 i... Read More about Ergodicity of Sublinear Markovian Semigroups.

Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations (2021)
Journal Article
Feng, C., Liu, Y., & Zhao, H. (2021). Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations. Journal of Computational and Applied Mathematics, 398, Article 113701. https://doi.org/10.1016/j.cam.2021.113701

In this paper, we consider numerical approximation to periodic measure of a time periodic stochastic differential equations (SDEs) under weakly dissipative condition. For this we first study the existence of the periodic measure ρt and the large time... Read More about Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations.

Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations (2021)
Journal Article
Feng, C., Qu, B., & Zhao, H. (2021). Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations. Journal of Differential Equations, 286, 119-163. https://doi.org/10.1016/j.jde.2021.03.022

In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic... Read More about Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations.

Expected exit time for time-periodic stochastic differential equations and applications to stochastic resonance (2020)
Journal Article
Feng, C., Zhao, H., & Zhong, J. (2021). Expected exit time for time-periodic stochastic differential equations and applications to stochastic resonance. Physica D: Nonlinear Phenomena, 417, https://doi.org/10.1016/j.physd.2020.132815

In this paper, we derive a parabolic partial differential equation for the expected exit time of non-autonomous time-periodic non-degenerate stochastic differential equations. This establishes a Feynman–Kac duality between expected exit time of time-... Read More about Expected exit time for time-periodic stochastic differential equations and applications to stochastic resonance.

A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations (2020)
Journal Article
Feng, C., Qu, B., & Zhao, H. (2020). A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations. Nonlinearity, 33(10), https://doi.org/10.1088/1361-6544/ab9584

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated... Read More about A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations.

Random periodic processes, periodic measures and ergodicity (2020)
Journal Article
Feng, C., & Zhao, H. (2020). Random periodic processes, periodic measures and ergodicity. Journal of Differential Equations, 269(9), 7382-7428. https://doi.org/10.1016/j.jde.2020.05.034

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincaré sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, i... Read More about Random periodic processes, periodic measures and ergodicity.