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Hitting hyperbolic half-space

Małecki J., Serafin G.

Demonstratio Mathematica

2012

Vol. 45, nr 2

337-360

Let (…) be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space (…) having the Laplace-Beltrami operator with drift as its generator. We prove the reflection principle for (…) which enables us to study the process (…) killed when exiting the hyperbolic half space, that is the set (…). We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process (…). Finally, we derive formula for the (…) Poisson kernel of the set D.

Bessel potentials, Green functions and exponential functionals on half-spaces

Byczkowski T., Ryznar M., Byczkowska H.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

155--173

The purpose of the paper is to provide precise estimates for the Green function corresponding to the operator (I—Δ)α/2, 0 <α< 2. The potential theory of this operator is based on Bessel potentials Jα=(I—Δ) -α/2. In probabilistic terms it corresponds to a subprobabilistic process obtained from the so-called relativistic a-stable process. We are interested in the theory of the killed process when exiting a fixed half-space. The crucial role in our research is played by (recently found) an explicit form of the Green function of a half-space. We also examine properties of some exponential functionals corresponding to the operator (I—Δ) α/2.

The estimates for the Green Function in Lipschitz domains for the symmetric stable processes

Jakubowski T.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

419--441

We give sharp global estimates for the Green function, Martin kernel and Poisson kernel in Lipschitz domains for symmetric α-stable processes. We give some applications of the estimates.

Estimates for the Poisson kernels on homogeneous manifolds of negative curvature and the boundary Harnack inequality in the noncoercive case

Urban R.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

213--229

Using a probabilistic technique we obtain upper and lower estimates for the Poisson kernels of the second order differential operators on a homogeneous manifold of negative curvature. Our results improve estimates obtained in the paper [5]. Moreover, for the noncoercive operator we proved the boundary Harnack inequality which turned out to be the same as in the coercive case.

Horocyclic Radon transform on Damek-Ricci space

Abouelaz A., El Fourchi O.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 2

107-140

We study the horocyclic Radon transform, defined in [5], of Damek-Ricci space. This transform R is obtained by integration over an orbitals family of NA space. We establish a Plancherel's formula for this transform. In particular, we characterize the range of horocyclic Radon transform of certain subspace of [L^2] ( NA, dx). The operators R and R*, where R* is the dual Radon transform of R, are inverted by a differential operator with constant coefficients (if dim NA is odd), or an integro-differential operator (if dim NA is even). The inversion formulas obtained are similar to the inversion formulas in symmetric space of noncompact type (see [7], [18], [19]). Also, we prove a radial Paley-Wiener-Schwartz's theorem for Damek-Ricci space.