Abstract
We consider the stochastic differential equation dXt = A(Xt−)dZt, X0 = x, driven by cylindrical α-stable process Zt in , where α ∈ (0,1) and d ≥ 2. We assume that the determinant of A(x) = (aij(x)) is bounded away from zero, and aij(x) are bounded and Lipschitz continuous. We show that for any fixed γ ∈ (0,α) the semigroup Pt of the process Xt satisfies \(|P_{t} f(x) - P_{t} f(y)| \le c t^{-\gamma /\alpha } |x - y|^{\gamma } ||f||_{\infty }\) for arbitrary bounded Borel function f. Our approach is based on Levi’s method.
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Acknowledgments
We thank prof. J. Zabczyk for communicating to us the problem of the strong Feller property for solutions of SDEs driven by cylindrical α-stable processes. We also thank A. Kulik for discussions on the problem treated in the paper. We are very grateful to the referees for valuable comments and remarks which greatly improved the presentation of the paper. In particular we thank one of the referees who offered an alternative approach to the strong Markov property based on the theory of Besov spaces and the results of [10], see Remark 4.23.
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T. Kulczycki was supported in part by the National Science Centre, Poland, grant no. 2015/17/B/ST1/01233, M. Ryznar was supported in part by the National Science Centre, Poland, grant no. 2015/17/B/ST1/01043, P. Sztonyk was supported in part by the National Science Centre, Poland, grant no. 2017/27/B/ST1/01339. This work was supported in part by Wrocław University of Science and Technology grant 049U/0052/19.
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Kulczycki, T., Ryznar, M. & Sztonyk, P. Strong Feller Property for SDEs Driven by Multiplicative Cylindrical Stable Noise. Potential Anal 55, 75–126 (2021). https://doi.org/10.1007/s11118-020-09850-8
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DOI: https://doi.org/10.1007/s11118-020-09850-8
Keywords
- Strong Feller property
- Stochastic differential equations
- Cylindrical stable processes
- Semigroups of operators