The purpose of this paper is to present a concise survey of the main properties of biholomorphically invariant pluricomplex Green functions and to describe a number of new examples of such functions. A concept of pluricomplex geodesics is also discussed.
Given a family of Lévy measures ν={ν(x, ⋅)}x∈ℝd, the present work deals with the regularity of harmonic functions and the Feller property of corresponding jump processes. The main aim is to establish continuity estimates for harmonic functions under weak assumptions on the family ν. Different from previous contributions the method covers cases where lower bounds on the probability of hitting small sets degenerate.
This work is concerned with the existence and regularity of solutions to the Neumann problem associated with a Ornstein–Uhlenbeck operator on a bounded and smooth convex set K of a Hilbert space H. This problem is related to the reflection problem associated with a stochastic differential equation in K.
On introduit les espaces fonctionnels dans lesquels l’opérateur potentiel satisfait au principe semi-complet du maximum si et seulement si la contraction module opère. Un tel espace fonctionnel sur la frontière de Martin d’un espace harmonique symétrique de Brelot est envisagé à l’aide du noyau de Naïm. Il est isomorphe à l’espace de Dirichlet des fonctions harmoniques. L’opérateur potentiel de cet espace donne la solution du problème de Neumann. On introduit l’espace de Dirichlet des fonctions...
In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be found e.gi̇n [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.
A compact set satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that if dist(z,K) ≤ 1. (LS) Here denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).