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Hitting half-spaces or spheres by Ornstein-Uhlenbeck type diffusions

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Byczkowski T., Chorowski J., Graczyk P., Małecki J.
Colloquium Mathematicum
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2012
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tom 129
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nr 2
145-171
EN
The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2Δ + F·∇, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method developed in this paper is based on stochastic calculus and on the skew product representation of multidimensional Brownian motion and yields more complete results than those based on the Feynman-Kac technique.
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Feynman-Kac formula, λ-Poisson kernels and λ-Green functions of half-spaces and balls in hyperbolic spaces

100%
Byczkowski T., Małecki J., Żak T.
Colloquium Mathematicum
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2010
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tom 118
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nr 1
201-222
EN
We apply the Feynman-Kac formula to compute the λ-Poisson kernels and λ-Green functions for half-spaces or balls in hyperbolic spaces. We present known results in a unified way and also provide new formulas for the λ-Poisson kernels and λ-Green functions of half-spaces in ℍⁿ and for balls in real and complex hyperbolic spaces.
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Sharp estimates of the Green function of hyperbolic Brownian motion

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Bogus K., Byczkowski T., Małecki J.
Studia Mathematica
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2015
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tom 228
|
nr 3
197-221
EN
The main objective of the work is to provide sharp two-sided estimates of the λ-Green function, λ ≥ 0, of the hyperbolic Brownian motion of a half-space. We rely on the recent results obtained by K. Bogus and J. Małecki (2015), regarding precise estimates of the Bessel heat kernel for half-lines. We also substantially use the results of H. Matsumoto and M. Yor (2005) on distributions of exponential functionals of Brownian motion.
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