Tomasz Byczkowski

BESSEL POTENTIALS, HITTING DISTRIBUTIONS AND GREEN FUNCTIONS

The purpose of the paper is to find explicit formulas for basic objects pertaining to the potential theory of the operator (I- Δ) α/2 , which is based on Bessel potentials J α = (I- Δ) -α/2 , 0 < α < 2. We compute the harmonic measure of the half-space and obtain a concise form for the corresponding Green function of the operator (I- Δ) α/2 . As an application we provide sharp estimates for the Green function of the half-space for the relativistic process.

Probabilistic Proof of Boundary Harnack Principle for α-Harmonic Functions

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.

Poisson kernels of half-spaces in real hyperbolic spaces

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space

Poisson Kernel and Green Function of the Ball in Real Hyperbolic Spaces

\H^n

Zeros of the densities of infinitely divisible measures on R n

. This enables us to find asymptotic properties of the kernel. We also show convergence to the Poisson kernel of the whole space

Tail Behavior for Semigroups of Measures on Groups

\H^n

Sharp estimates of the Green function of hyperbolic Brownian motion

. For n=3, 4 or 6 we compute explicit formulas for the Poisson kernel itself.

Smoothness of the distribution of the norm in uniformly convex Banach spaces

Let (Xt)t⩾0 be the n-dimensional hyperbolic Brownian motion, that is the diffusion on the real hyperbolic space

Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains

\mathbb{D}^{n}

Hitting Times of Bessel Processes

having the Laplace–Beltrami operator as its generator. The aim of the paper is to derive the formulas for the Gegenbauer transform of the Poisson kernel and the Green function of the ball for the process (Xt)t⩾0. Under additional hypotheses we prove integral representations for the Poisson kernel. This yields explicit formulas in