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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On weak solution of SDE driven by inhomogeneous singular Lévy noise
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by Tadeusz Kulczycki, Alexei Kulik and Michał Ryznar PDF
Trans. Amer. Math. Soc. 375 (2022), 4567-4618 Request permission

Abstract:

We study a time-inhomogeneous SDE in $\mathbb {R}^d$ driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit ‘principal part’ and a ‘residual part’, subject to certain $L^\infty (dx)\otimes L^1(dy)$ and $L^\infty (dx)\otimes L^\infty (dy)$-estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.
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Additional Information
  • Tadeusz Kulczycki
  • Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
  • MR Author ID: 628862
  • ORCID: 0000-0001-9855-1481
  • Email: tadeusz.kulczycki@pwr.edu.pl
  • Alexei Kulik
  • Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
  • MR Author ID: 339722
  • ORCID: 0000-0002-9371-6862
  • Email: oleksii.kulyk@pwr.edu.pl
  • Michał Ryznar
  • Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
  • ORCID: 0000-0001-7168-4049
  • Email: michal.ryznar@pwr.edu.pl
  • Received by editor(s): April 9, 2021
  • Received by editor(s) in revised form: September 2, 2021
  • Published electronically: April 21, 2022
  • Additional Notes: The first and third authors were supported in part by the National Science Centre, Poland, grant no. 2019/35/B/ST1/01633
    The second author has been supported through the DFG-NCN Beethoven Classic 3 programme, contract no. 2018/31/G/ST1/02252 (National Science Center, Poland) and SCHI-419/11–1 (DFG, Germany)
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4567-4618
  • MSC (2020): Primary 60H10, 60G51, 47G20
  • DOI: https://doi.org/10.1090/tran/8612
  • MathSciNet review: 4439486