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Strong regularization by noise for a class of kinetic SDEs driven by symmetric α-stable processes

Author

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  • Lucertini, Giacomo
  • Menozzi, Stéphane
  • Pagliarani, Stefano

Abstract

We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric α-stable process under Hölder regularity conditions for the drift term. We partially recover the thresholds for the Hölder regularity that are optimal for weak uniqueness. In general dimension, we only consider α=2 and need an additional integrability assumption for the gradient of the drift: this condition is satisfied by Peano-type functions. In the one-dimensional case we do not need any additional assumption. In the multi-dimensional case, the proof is based on a first-order Zvonkin transform/PDE, while for the one-dimensional case we use a second-order Zvonkin/PDE transform together with a Watanabe–Yamada technique.

Suggested Citation

  • Lucertini, Giacomo & Menozzi, Stéphane & Pagliarani, Stefano, 2025. "Strong regularization by noise for a class of kinetic SDEs driven by symmetric α-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:spapps:v:189:y:2025:i:c:s0304414925001322
    DOI: 10.1016/j.spa.2025.104691
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    References listed on IDEAS

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    1. Zhang, Yong & Sun, HongGuang & Stowell, Harold H. & Zayernouri, Mohsen & Hansen, Samantha E., 2017. "A review of applications of fractional calculus in Earth system dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 29-46.
    2. Catellier, R. & Gubinelli, M., 2016. "Averaging along irregular curves and regularisation of ODEs," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2323-2366.
    3. Marino, L. & Menozzi, S., 2023. "Weak well-posedness for a class of degenerate Lévy-driven SDEs with Hölder continuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 106-170.
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