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Long-time behavior of stable-like processes

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  • Sandrić, Nikola

Abstract

In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol p(x,ξ)=−iβ(x)ξ+γ(x)|ξ|α(x), where α(x)∈(0,2), β(x)∈R and γ(x)∈(0,∞). More precisely, we give sufficient conditions for recurrence, transience and ergodicity of stable-like processes in terms of the stability function α(x), the drift function β(x) and the scaling function γ(x). Further, as a special case of these results we give a new proof for the recurrence and transience property of one-dimensional symmetric stable Lévy processes with the index of stability α≠1.

Suggested Citation

  • Sandrić, Nikola, 2013. "Long-time behavior of stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1276-1300.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:4:p:1276-1300
    DOI: 10.1016/j.spa.2012.12.004
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    References listed on IDEAS

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    1. Wang, Jian, 2008. "Criteria for ergodicity of Lévy type operators in dimension one," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1909-1928, October.
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    Cited by:

    1. Sandrić, Nikola, 2016. "On recurrence and transience of two-dimensional Lévy and Lévy-type processes," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 414-438.
    2. Song, Yan-Hong, 2016. "Algebraic ergodicity for SDEs driven by Lévy processes," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 108-115.
    3. Chen, Xin & Chen, Zhen-Qing & Wang, Jian, 2020. "Heat kernel for non-local operators with variable order," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3574-3647.
    4. Mikhail V. Menshikov & Dimitri Petritis & Andrew R. Wade, 2018. "Heavy-Tailed Random Walks on Complexes of Half-Lines," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1819-1859, September.
    5. Nikola Sandrić, 2016. "Ergodic Property of Stable-Like Markov Chains," Journal of Theoretical Probability, Springer, vol. 29(2), pages 459-490, June.
    6. 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.
    7. 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.

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