No abstract

We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2.

We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.

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,