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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...

(Homogeneous) markovian bridges

Vincent Vigon (2011)

Annales de l'I.H.P. Probabilités et statistiques

(Homogeneous) Markov bridges are (time homogeneous) Markov chains which begin at a given point and end at a given point. The price to pay for preserving the homogeneity is to work with processes with a random life-span. Bridges are studied both for themselves and for their use in describing the transformations of Markov chains: restriction on a random interval, time reversal, time change, various conditionings comprising the confinement in some part of the state space. These bridges lead us to look...

Infinitely divisible processes and their potential theory. II

Sidney C. Port, Charles J. Stone (1971)

Annales de l'institut Fourier

This second part of our two part work on i.d. process has four main goals:(1) To develop a potential operator for recurrent i.d. (infinitely divisible) processes and to use this operator to find the asymptotic behavior of the hitting distribution and Green’s function for relatively compact sets in the recurrent case.(2) To develop the appropriate notion of an equilibrium measure and Robin’s constant for Borel sets.(3) To establish the asymptotic behavior questions of a potential theoretic nature...

Infinitely divisible processes and their potential theory. I

Sidney C. Port, Charles J. Stone (1971)

Annales de l'institut Fourier

We show that associated with every i.d. (infinitely divisible) process on a locally compact, non-compact 2nd countable Abelian group is a corresponding potential theory that yields definitive results on the behavior of the process in both space and time. Our results are general, no density or other smoothness assumptions are made on the process. In this first part of two part work we have four main goals.(1) To lay the probabilistic foundation of such processes. This mainly consists in giving the...

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