On Lundh’s percolation diffusion

Abstract

A collection of spherical obstacles in the ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function depending only on the distance from the obstacle's centre to the centre of the ball. Lundh has given the name percolation diffusion to this process if avoidable configurations are generated with positive probability. An integral condition for percolation diffusion is derived in terms of the intensity of the Poisson point process and the function that determines the radii of the obstacles. 1

Unavoidable collections of balls for isotropic Lévy processes

“…The right-hand side inequality is proved exactly in the same way as in the proof of [9, Lemma 3]. To prove the left-hand side inequality we follow [9] and use the super-additivity property of capacity due to Aikawa and Borichev, [2,Theorem 3], which in our case reads as follows: For r ą 0 and a ą 0 the radius η a prq is chosen so that Cap a pBp0, rqq " |Bp0, η a prqq| " σ d η a prq d (σ d is the volume of the unit ball). By Lemma 2.7,…”

“…Theorem 1.3 can be used to study avoidability of a collection of balls with random centers given by the Poisson point process. In case of Brownian motion this question was recently studied in [9]. By adopting their method we are able to extend the results to subordinate Brownian motions satisfying (1.4).…”

“…In this section we consider collections of randomly located closed balls with centers coming from a Poisson point process. Since we closely follow the arguments from [9], most of the proofs are omitted. We precisely spell out the assumptions and give proofs when some (non-trivial) changes are needed.…”

“…Following [9] we say that we have percolation Lévy process if there is a positive probability that the realization of point from the Poisson point process results in an avoidable collection of balls. Moreover, in case percolation Lévy process occurs, the random collection of balls A P is avoidable with probability one.…”

Champagne subdomains with unavoidable bubbles

“…This generalizes the notions used in [3,8,12,13,14] for U = B(0, 1); see also [6] for the case, where U is Ê d , d ≥ 3. Avoidable unions of randomly distributed balls have been discussed in [11] and, recently, in [5]. It will be convenient to introduce the set X A for a champagne subdomain U \ A: X A is the set of centers of all the bubbles forming A (and r x , x ∈ X A , is the radius of the bubble centered at x).…”