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Construction and heat kernel estimates of general stable-like Markov processes

Volume 569 / 2021

Victoria Knopova, Alexei Kulik, René L. Schilling Dissertationes Mathematicae 569 (2021), 1-86 MSC: Primary 60J35; Secondary 60J25, 60G52, 35A08, 35A17, 35S05. DOI: 10.4064/dm824-8-2021 Published online: 3 November 2021


A stable-like process is a Feller process $(X_t)_{t\geq 0}$ taking values in ${\mathbb{R}^d}$ and whose generator behaves, locally, like an $\alpha$-stable Lévy generator, but the index $\alpha$ and all other characteristics may depend on the state space. More precisely, the jump measure need not be symmetric, and it strongly depends on the current state of the process; moreover, we do not require the gradient term to be dominated by the pure jump part. Our approach is to understand the above phenomena as suitable microstructural perturbations.

We show that the corresponding martingale problem is well-posed, and its solution is a strong Feller process which admits a transition density. For the transition density we obtain a representation as a sum of an explicitly given principal term—this is essentially the density of an $\alpha$-stable random variable whose parameters depend on the current state $x$—and a residual term; the $L^\infty\otimes L^1$-norm of the residual term is negligible and so is, under an additional structural assumption, the $L^\infty\otimes L^\infty$-norm. Concrete examples illustrate the relation between the assumptions and possible transition density estimates.


  • Victoria KnopovaDepartment of Probability, Statistics and Actuarial Mathematics
    Taras Shevchenko National University of Kyiv
    01601 Kyiv, Ukraine
  • Alexei KulikFaculty of Pure and Applied Mathematics
    Wrocław University of Science and Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
  • René L. SchillingInstitut für Mathematische Stochastik
    Fakultät Mathematik
    TU Dresden
    01062 Dresden, Germany

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