Krzysztof Bogdan
Wrocław University of Technology
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Featured researches published by Krzysztof Bogdan.
Annals of Probability | 2010
Krzysztof Bogdan; Tomasz Grzywny; Michał Ryznar
We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains. AMS 2000 subject classifications: Primary 60J35, 60J50; secondary 60J75, 31B25. Keywords and phrases: fractional Laplacian, Dirichlet problem, heat kernel estimate, Lipschitz domain, boundary Harnack principle.
Potential Analysis | 1999
Krzysztof Bogdan; Tomasz Byczkowski
The boundary Harnack principle for fractional Laplacian is now known to hold in Lipschitz domains [5]. It states that if two nonnegative functions, harmonic with respect to a symmetric stable Lévy process vanish continuously outside a Lipschitz domain, near a part of its boundary, then the ratio of the functions is bounded inside the domain, near this part of the boundary. We give a probabilistic proof of the assertion using elementary properties of the stable process.
Annals of the Institute of Statistical Mathematics | 2002
Małgorzata Bogdan; Krzysztof Bogdan; A. Futschik
We propose a new omnibus test for uniformity on the circle. The new test is based upon the idea of data driven smooth tests as presented in Ledwina (1994, J. Amer. Statist. Assoc., 89, 1000–1005). Our simulations indicate that the test performs very well for multifarious alternatives. In particular, it seems to outperform other known omnibus tests when testing against multimodal alternatives. We also investigate asymptotic properties of our test and we prove that it is consistent against every departure from uniformity.
Journal of Evolution Equations | 2016
Krzysztof Bogdan; Yana A. Butko; Karol Szczypkowski
Schrödinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance, this is so for the transition density of a subordinator perturbed by any time-independent unbounded potential. In order to estimate such perturbations, it is convenient to use an auxiliary transition density as a majorant and the 4G inequality for the original transition density and the majorant. We prove the 4G inequality for the 1/2-stable and inverse Gaussian subordinators, discuss the corresponding class of admissible potentials and indicate estimates for the resulting transition densities of Schrödinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution.
Statistics | 2000
Krzysztof Bogdan; Małgorzata Bogdan
We propose a simple necessary and sufficient condition for existence of maximum likelihood estimators in a large class of canonical exponential families. We give an application to log-spline families.
Potential Analysis | 2016
Krzysztof Bogdan; Bartłomiej Dyda; Panki Kim
We prove non-explosion results for Schrödinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.
Journal of Evolution Equations | 2016
Krzysztof Bogdan; Bartłomiej Siudeja
We give two-term approximation for the trace of the Dirichlet heat kernel of bounded smooth open set for unimodal Lévy processes satisfying the weak scaling conditions.
Proceedings of the American Mathematical Society | 2005
Rodrigo Bañuelos; Krzysztof Bogdan
We identify the critical exponent of integrability of the first exit time of the rotation invariant stable Levy process from a parabola-shaped region.
arXiv: Probability | 2015
Krzysztof Bogdan; Sebastian Sydor
We give sufficient conditions for nonlocal perturbations of integral kernels to be locally in time comparable with the original kernel.
Comptes Rendus Mathematique | 2002
Krzysztof Bogdan; Andrzej Stós; Paweł Sztonyk
Abstract We study nonnegative harmonic functions of symmetric α-stable processes on d-sets F. We prove the Harnack inequality for such functions when α∈(0,2/dw)∪(ds,2). Furthermore, we investigate the decay rate of harmonic functions and the Carleson estimate near the boundary of a region in F. In the particular case of natural cells in the Sierpinski gasket we also prove the boundary Harnack principle. To cite this article: K. Bogdan et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 59–63.