Michał Ryznar

Heat Kernel estimates for the fractional Laplacian with Dirichlet conditions.

We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains. AMS 2000 subject classifications: Primary 60J35, 60J50; secondary 60J75, 31B25. Keywords and phrases: fractional Laplacian, Dirichlet problem, heat kernel estimate, Lipschitz domain, boundary Harnack principle.

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.

First passage times for subordinate Brownian motions

Let Xt be a subordinate Brownian motion, and suppose that the Levy measure of the underlying subordinator has a completely monotone density. Under very mild conditions, we find integral formulae for the tail distribution P(τx>t) of first passage times τx through a barrier at x>0, and its derivatives in t. As a corollary, we examine the asymptotic behaviour of P(τx>t) and its t-derivatives, either as t→∞ or x→0+.

Hitting times of points and intervals for symmetric Levy processes

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function Px(TB>t), where TB is the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove sharp two-sided estimates of the transition density of the process killed after hitting B.

Kelvin transform for α-harmonic functions in regular domains

Abstract We investigate conditional stable processes in a Lipschitz domain D and conditional stable processes in the image of D under the Kelvin transform. We show that, with a suitable change of time, these processes are equal in distribution. As an application, we show the equivalence of the Hardy spaces and the relative Fatou theorem for D and its image.

CONDITIONING VS. STANDARDIZATION FOR CONTRASTS ON ERRORS ATTRACTED TO STABLE LAWS

In multiple regression and other settings one encounters the problem of estimating sampling distributions for contrast operations applied to i.i.d. errors. Permutation bootstrap applied to least squares residuals has been proven to consistently estimate conditionalsampling distributions of contrasts, conditional upon order statistics of errors, even for long-tailed error distributions. How does this compare with the unconditional sampling distribution of the contrast when standardizing by the sample s.d. of the errors (or the residuals)? For errors belonging to the domain of attraction of a normal we present a limit theorem proving that these distributions are far closer to one another than they are to the limiting standard normal distribution. For errors attracted to α-stable laws with α ≤ 2 we construct random variables possessing these conditional and unconditional sampling distributions and develop a Poisson representation for their a.s. limit correlation ρα. We prove that ρ2= 1, ρα→ 1 for α → 0 + or 2 −, and ρα< 1 a.s. for α < 2.

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

Let (E, ‖ · ‖) be a uniformly convex Banach space of power type. In this paper we investigate differentiability properties of the distribution function of the norm of a random series with one-dimensional independent components inE.