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Dive into the research topics where Krzysztof Bogdan is active.

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Featured researches published by Krzysztof Bogdan.


Annals of Probability | 2010

Heat Kernel estimates for the fractional Laplacian with Dirichlet conditions.

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

Probabilistic Proof of Boundary Harnack Principle for α-Harmonic Functions

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

A data driven smooth test for circular uniformity

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

Majorization, 4G Theorem and Schrödinger perturbations

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

On Existence of Maximum Likelihood Estimators in Exponential Families

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

Hardy Inequalities and Non-explosion Results for Semigroups

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

Trace estimates for unimodal Lévy processes

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

Symmetric stable processes in parabola–shaped regions

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

On Nonlocal Perturbations of Integral Kernels

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

Harnack inequality for symmetric stable processes on fractals

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.

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Tomasz Grzywny

Wrocław University of Technology

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Tomasz Jakubowski

Wrocław University of Technology

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Michał Ryznar

Wrocław University of Technology

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Paweł Sztonyk

Wrocław University of Technology

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Bartłomiej Dyda

Wrocław University of Technology

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Karol Szczypkowski

Wrocław University of Technology

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Andrzej Stós

Wrocław University of Technology

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Tomasz Luks

University of Paderborn

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