Champagne subdomains with unavoidable bubbles

Abstract

A champagne subdomain of a connected open set U = ∅ in Ê d , d ≥ 2, is obtained omitting pairwise disjoint closed balls B(x, r x ), x ∈ X, the bubbles, where X is an infinite, locally finite set in U . The union A of these balls may be unavoidable, that is, Brownian motion, starting in U \ A and killed when leaving U , may hit A almost surely or, equivalently, A may have harmonic measure one for U \ A.Recent publications by Gardiner/Ghergu (d ≥ 3) and by Pres (d = 2) give rather sharp answers to the question how small such a set A may be, when U is the unit ball.In this paper, using a totally different approach, optimal results are obtained, results which hold as well for arbitrary connected open sets U .

Harmonic measure of the outer boundary of colander sets

“…One aspect is whether these sets are avoidable or not, i.e is the harmonic measure supported on this set, and assigns zero to the boundary of the disk. The question of when a set is avoidable has been investigated by many mathematicians: Akeroyd [2], Carrol and Ortega-Cerdà [5], O'Donovan [13], Gardiner and Ghergu [7], Pres [16], Hansen and Netuka [8] and more... An overturn of this question would be to ask when are such sets so small that the remaining harmonic measure, the one restricted to the boundary of the disk, is comparable with Lebesgue's measure. Questions such as these have been answered by Volberg [19], Essén [6] and by Aikawa and Lundh [1].…”

Unavoidable sets and harmonic measures living on small sets

“…The proof of Theorem 1.1 given in [21] is based on a criterion for unavoidable sets which, in probabilistic terms, relies on the continuity of the paths for Brownian motion (see [21,Proposition 2.1]). We shall use a criterion which, using entry times T E (ω) := inf{t ≥ 0 : X t (ω) ∈ E} for Borel measurable sets E, states the following.…”

“…Finally, in the very general setting of a balayage space (X, W) on which the function 1 is harmonic (which covers not only large classes of second-order partial differential equations, but also non-local situations as, for example, given by Riesz potentials, isotropic unimodal Lévy processes, or censored stable processes) a construction of champagne subsets X \ A of X with small unavoidable sets A is given which generalizes (and partially improves) recent constructions in the classical case. Mathematics Subject Classification 31A12, 31B05, 31B15, 31D05, 60J45, 60J65, 60J25 (primary), 28A78 (secondary).Both authors gratefully acknowledge support by CRC-701, Bielefeld.Recently, the following has been shown (see [20, Theorem 1.1]; cf. [14,37] for the case U = B(0, 1) and h(t) = (cap(t)) ε ).Theorem 1.1.…”

“…This follows by a slight modification of arguments in [14,37] (for the possibility of omitting finitely many bubbles, see (1b) in Lemma 2.2). Therefore, the condition lim inf t→0 h(t) = 0 is also necessary for the conclusion in Theorem 1.1.The proof of Theorem 1.1 given in [20] is based on a criterion for unavoidable sets which, in probabilistic terms, relies on the continuity of the paths for Brownian motion (see [20, Proposition 2.1]). We shall use a criterion which, using entry times T E (ω) := inf{t 0: X t (ω) ∈ E} for Borel measurable sets E, states the following.…”

“…Recently, the following has been shown (see [20, Theorem 1.1]; cf. [14,37] for the case U = B(0, 1) and h(t) = (cap(t)) ε ).…”