Jump processes, \(\mathcal L\)-harmonic functions, continuity estimates and the Feller property
type: Item
instance of: scholarly article
title: Jump processes, \(\mathcal L\)-harmonic functions, continuity estimates and the Feller property
zbMATH Open document ID: 1203.60125
DOI: 10.1214/08-AIHP208
author: Ryad Husseini
author: Moritz Kassmann
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2010-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/78055
review text: In the paper [\textit{R.~F.~Bass} and \textit{M.~Kassmann}, Commun. Partial Differ. Equations 30, No.~8, 1249--1259 (2005; Zbl 1087.45004)], the authors proved the Hölder continuity of harmonic functions of an integro-differential operator of variable order satisfying certain conditions. Their approach was probabilistic, the main object being the corresponding Markov procees. The key to obtaining regularity results is the Krylov-Safonov-type estimate of hitting probabilities: There exists a constant \(c>0\) such that for all \(r\in (0,1/2)\), \(A\subset B(x_0,r)\) satisfying \(|A|\geq \frac12 |B(x_0,r)|\) and all \(y\in B(x_0,r/2)\) it holds that \(\mathbb P^y(T_A<\tau_{B(x_0,r)})\geq c\). Here \(T\) denotes the hitting time, and \(\tau\) the exit time from a set. In the paper under review, the authors discuss continuity a priori estimates of harmonic functions for similar integro-differential operators but with one of the conditions relaxed. The first consequence of this relaxation is that the Krylov-Safonov-type estimate need not hold -- the lower bound \(c\) may depend on the radius \(r\) and thus converges to zero as \(r\to 0\). The main continuity result for bounded functions \(u\) harmonic in the ball \(B(x_0,R)\) gives the following modulus of continuity: \[ \sup_{\substack{ x,y\in B(x_0,R/2)\\ |x-y|<t}} |u(x)-u(y)|\leq c ||u||_{\infty}|\log t|^{-\rho}\, ,\quad \forall t\in (0,1/2)\, , \] where the constant \(\rho >0\) depends on the constants appearing in the conditions.
reviewed by: Zoran Vondraček
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 5758872
zbMATH Keywords: jump processes
zbMATH Keywords: Lévy measure
zbMATH Keywords: Feller property
zbMATH Keywords: martingale problem
zbMATH Keywords: integro-differential operators
zbMATH Keywords: harmonic functions
zbMATH Keywords: a priori estimates
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5758872