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Hardy spaces for the Laplacian with lower order perturbations

Tomasz Luks (2011)

Studia Mathematica

We consider Hardy spaces of functions harmonic on smooth domains in Euclidean spaces of dimension greater than two with respect to the Laplacian perturbed by lower order terms. We deal with the gradient and Schrödinger perturbations under appropriate Kato conditions. In this context we show the usual correspondence between the harmonic Hardy spaces and the L p spaces (or the space of finite measures if p = 1) on the boundary. To this end we prove the uniform comparability of the respective harmonic...

Harmonic measures for symmetric stable processes

Jang-Mei Wu (2002)

Studia Mathematica

Let D be an open set in ℝⁿ (n ≥ 2) and ω(·,D) be the harmonic measure on D c with respect to the symmetric α-stable process (0 < α < 2) killed upon leaving D. We study inequalities on volumes or capacities which imply that a set S on ∂D has zero harmonic measure and others which imply that S has positive harmonic measure. In general, it is the relative sizes of the sets S and D c ∖ S that determine whether ω(S,D) is zero or positive.

Harnack inequality for stable processes on d-sets

Krzysztof Bogdan, Andrzej Stós, Paweł Sztonyk (2003)

Studia Mathematica

We investigate properties of functions which are harmonic with respect to α-stable processes on d-sets such as the Sierpiński gasket or carpet. We prove the Harnack inequality for such functions. For every process we estimate its transition density and harmonic measure of the ball. We prove continuity of the density of the harmonic measure. We also give some results on the decay rate of harmonic functions on regular subsets of the d-set. In the case of the Sierpiński gasket we even obtain the Boundary...

Hitting distributions domination and subordinate resolvents; an analytic approach

Nicu Boboc, Gheorghe Bucur (2006)

Open Mathematics

We give an analytic version of the well known Shih's theorem concerning the Markov processes whose hitting distributions are dominated by those of a given process. The treatment is purely analytic, completely different from Shih's arguments and improves essentially his result (in the case when the given processes are transient

Hitting half-spaces or spheres by Ornstein-Uhlenbeck type diffusions

Tomasz Byczkowski, Jakub Chorowski, Piotr Graczyk, Jacek Małecki (2012)

Colloquium Mathematicae

The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2Δ + F·∇, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method developed in this paper is based on stochastic calculus and...

Hitting probabilities and potential theory for the brownian path-valued process

Jean-François Le Gall (1994)

Annales de l'institut Fourier

We consider the Brownian path-valued process studied in [LG1], [LG2], which is closely related to super Brownian motion. We obtain several potential-theoretic results related to this process. In particular, we give an explicit description of the capacitary distribution of certain subsets of the path space, such as the set of paths that hit a given closed set. These capacitary distributions are characterized as the laws of solutions of certain stochastic differential equations. They solve variational...

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