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Volume 14
Issue 2
10.3390/axioms14020098
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Article

Ergodicity and Mixing Properties for SDEs with α-StableLévy Noises

by
Siyan Xu
1 and
Huiyan Zhao
2,*
1
College of Science, North China University of Technology, Beijing 100144, China
2
Faculty of Arts and Sciences, Beijing Normal University at Zhuhai, Zhuhai 519087, China
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(2), 98; https://doi.org/10.3390/axioms14020098
Submission received: 19 November 2024 / Revised: 14 January 2025 / Accepted: 26 January 2025 / Published: 28 January 2025
Versions Notes

Abstract

In this paper, we consider a class of stochastic differential equations driven by multiplicative α-stable (0<α<2) Lévy noises. Firstly, we show that there exists a unique strong solution under a local one-sided Lipschitz condition and a general non-explosion condition. Next, the weak Feller and stationary properties are derived. Furthermore, a concrete sufficient condition for the coefficients is presented, which is different from the conditions for SDEs driven by Brownian motion or general squared-integrable martingales. Finally, some ergodic and mixing properties are obtained by using the Foster–Lyapunov criteria.
Keywords: local one-sided Lipschitz condition; α-stable process; weak Feller property; ergodicity; β-mixing property local one-sided Lipschitz condition; α-stable process; weak Feller property; ergodicity; β-mixing property

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MDPI and ACS Style

Xu, S.; Zhao, H. Ergodicity and Mixing Properties for SDEs with α-StableLévy Noises. Axioms 2025, 14, 98. https://doi.org/10.3390/axioms14020098

AMA Style

Xu S, Zhao H. Ergodicity and Mixing Properties for SDEs with α-StableLévy Noises. Axioms. 2025; 14(2):98. https://doi.org/10.3390/axioms14020098

Chicago/Turabian Style

Xu, Siyan, and Huiyan Zhao. 2025. "Ergodicity and Mixing Properties for SDEs with α-StableLévy Noises" Axioms 14, no. 2: 98. https://doi.org/10.3390/axioms14020098

APA Style

Xu, S., & Zhao, H. (2025). Ergodicity and Mixing Properties for SDEs with α-StableLévy Noises. Axioms, 14(2), 98. https://doi.org/10.3390/axioms14020098

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