We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the half-line (0, ∞) and 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 ψ λ of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in 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 ψ λ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λ n of A in the interval the asymptotic formula λ n = nπ 2 − π 8 + O( 1 n ) is derived, and all eigenvalues λ n are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λ n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point.

In this paper we study the supremum functional Mt = sup 0≤s≤t Xs, where Xt, t ≥ 0, is a one-dimensional Lévy 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 Lévy-Khintchin exponent of the process increases on (0, ∞). . This reprint differs from the original in pagination and typographic detail. 1 2 M. KWAŚNICKI, J. MA LECKI AND M. RYZNAR by Darling [11], for a compound Poisson process with Ψ(ξ) = 1 − cos ξ by Baxter and Donsker [3] and for the Poisson process with drift by Pyke [32].The development of the fluctuation theory for Lévy processes resulted in many new identities involving the supremum functional M t ; see, for example, [5,13,31,33]. There are numerous other representations for the distribution of M t , at least in the stable case; see [4,7,11,12,15,16,19,20,27,28,30,36]. The main goal of this article is to give a more explicit formula for P(M t < x) and simple sharp bounds for P(M t < x) in terms of the Lévy-Khintchin exponent Ψ(ξ) for a class of Lévy processes. Most estimates of the cumulative distribution function of M t are proved for very general Lévy processes, without symmetry assumptions.Let τ x denote the first passage time through a barrier at the level x for the process X t ,with the infimum understood to be infinity when the set is empty. We always assume that X 0 = 0. Since P(M t < x) = P(τ x > t), the problems of finding the cumulative distribution functions of M t and τ x are the same. The supremum functional and first passage time statistics are important in various areas of applied probability [1,2], as well as in mathematical physics [21,26]. The recent progress in the potential theory of Lévy processes is, in part, due to the application of fluctuation theory; see [9,10,18,[22][23][24][25]. The paper is organized as follows. Section 2 contains some preliminary material related to Bernstein functions, Stieltjes functions and estimates for the Laplace transform. In Section 3 (Theorem 3.1 and Corollary 3.2) we prove, under mild assumptions, the estimate P(M t < x) ≈ min(1, κ(1/t, 0)V (x)), t, x > 0,

be the first hitting time of the point 1 by the Bessel process with index μ ∈ R starting from x > 1. Using an integral formula for the density q 1 which exhibit the dependence both on time and space variables. Our result provides optimal uniform estimates for the density of the hitting time of the unit ball by the Brownian motion in R n , which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces.