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Malliavin calculus for stable processes on homogeneous groups

100%
Graczyk P.
Studia Mathematica
|
1991
|
tom 100
|
nr 3
183-205
EN
Let ${μ_t}_{t>0}$ be a symmetric semigroup of stable measures on a homogeneous group, with smooth Lévy measure. Applying Malliavin calculus for jump processes we prove that the measures $μ_t$ have smooth densities.
2
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On the product formula on noncompact Grassmannians

63%
Graczyk P., Sawyer P.
Colloquium Mathematicum
|
2013
|
tom 133
|
nr 2
145-167
EN
We study the absolute continuity of the convolution $δ_{e^X}^{♮} * δ_{e^Y}^{♮}$ of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ 𝔞 for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to the study of absolute continuity of convolution powers of an orbital measure $δ_{e^X}^♮$.
3
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Hitting half-spaces or spheres by Ornstein-Uhlenbeck type diffusions

51%
Byczkowski T., Chorowski J., Graczyk P., Małecki J.
Colloquium Mathematicum
|
2012
|
tom 129
|
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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