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Ergodicity for time-changed symmetric stable processes

Author

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  • Chen, Zhen-Qing
  • Wang, Jian

Abstract

In this paper we study ergodicity and related semigroup property for a class of symmetric Markov jump processes associated with time-changed symmetric α-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric α-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.

Suggested Citation

  • Chen, Zhen-Qing & Wang, Jian, 2014. "Ergodicity for time-changed symmetric stable processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2799-2823.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:9:p:2799-2823
    DOI: 10.1016/j.spa.2014.04.003
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    References listed on IDEAS

    as
    1. Sandrić, Nikola, 2013. "Long-time behavior of stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1276-1300.
    2. Chen, Xin & Wang, Jian, 2014. "Functional inequalities for nonlocal Dirichlet forms with finite range jumps or large jumps," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 123-153.
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    Cited by:

    1. Wang, Tao, 2022. "Ergodic convergence rates for time-changed symmetric Lévy processes in dimension one," Statistics & Probability Letters, Elsevier, vol. 183(C).
    2. Kumar, Rohini & Popovic, Lea, 2017. "Large deviations for multi-scale jump-diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1297-1320.
    3. Jian Wang, 2019. "Compactness and Density Estimates for Weighted Fractional Heat Semigroups," Journal of Theoretical Probability, Springer, vol. 32(4), pages 2066-2087, December.
    4. Yong-Hua Mao & Tao Wang, 2022. "Convergence Rates in Uniform Ergodicity by Hitting Times and $$L^2$$ L 2 -Exponential Convergence Rates," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2690-2711, December.
    5. Xinghu Jin & Tian Shen & Zhonggen Su, 2023. "Using Stein’s Method to Analyze Euler–Maruyama Approximations of Regime-Switching Jump Diffusion Processes," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1797-1828, September.
    6. Huang, Lu-Jing & Wang, Tao, 2023. "Dirichlet eigenvalues and exit time moments for symmetric Markov processes," Statistics & Probability Letters, Elsevier, vol. 193(C).

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