Jacek Malecki

Spectral properties of the Cauchy process on half-line and interval

We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0, ∞ )a nd the interval ( −1, 1). This process is related to the square root of one-dimensional Laplacian A = − � −(d 2 /dx 2 ) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions ψλ of A is derived, and then used to construct a spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process on the half-line (or the heat kernel of A in (0, ∞)), and for the distribution of the first exit time from the half-line follow. The formula for ψλ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λn of A in the interval the asymptotic formula λn = nπ/2 − π/ 8+ O(1/n) is derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point.

Suprema of Lévy processes

In this paper we study the supremum functional Mt=sup0≤s≤tXs, where Xt, t≥0, is a one-dimensional Levy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of Mt. In the symmetric case we find an integral representation of the Laplace transform of the distribution of Mt if the Levy–Khintchin exponent of the process increases on (0,∞).

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.

Multidimensional Yamada-Watanabe theorem and its applications to particle systems

We prove a multidimensional version of the Yamada-Watanabe theorem, i.e., a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions. It implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrix-valued stochastic processes, called a “spectral” matrix Yamada-Watanabe theorem. The multidimensional Yamada-Watanabe theorem is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes, Wishart and Jacobi matrix processes. The β-versions of these particle systems are also considered.

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

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

First passage times for subordinate Brownian motions

\mathbb{D}^{n}

Hitting hyperbolic half-space

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

Sharp estimates of the Green function of hyperbolic Brownian motion

\mathbb{D}^{4}

Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval ☆

and

Spectral properties of the massless relativistic harmonic oscillator

\mathbb{D}^{6}