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Symmetry and functional inequalities for stable Lévy-type operators

Author

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  • Huang, Lu-Jing
  • Wang, Tao

Abstract

In this paper, we establish the sufficient and necessary conditions for the symmetry of the following stable Lévy-type operator L on R: L=a(x)Δα/2+b(x)ddx,where a is a continuous and strictly positive function, and b is a differentiable function. We then study the criteria for functional inequalities, such as logarithmic Sobolev inequalities, Nash inequalities and super-Poincaré inequalities under the assumption of symmetry. Our approach involves the Orlicz space theory and the estimates of the Green functions.

Suggested Citation

  • Huang, Lu-Jing & Wang, Tao, 2025. "Symmetry and functional inequalities for stable Lévy-type operators," Stochastic Processes and their Applications, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:spapps:v:183:y:2025:i:c:s0304414925000419
    DOI: 10.1016/j.spa.2025.104600
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    References listed on IDEAS

    as
    1. Feng-Yu Wang & Jian Wang, 2015. "Functional Inequalities for Stable-Like Dirichlet Forms," Journal of Theoretical Probability, Springer, vol. 28(2), pages 423-448, June.
    2. 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.
    3. Döring, Leif & Kyprianou, Andreas E. & Weissmann, Philip, 2020. "Stable processes conditioned to avoid an interval," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 471-487.
    4. 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.
    Full references (including those not matched with items on IDEAS)

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