On Lundh's percolation difussion

TL;DR

This paper investigates conditions under which Brownian motion can avoid randomly placed spherical obstacles in a Euclidean ball, defining a percolation diffusion process based on obstacle distribution and size.

Contribution

It derives an integral condition linking obstacle distribution and size function that determines the occurrence of percolation diffusion.

Findings

Provides a criterion for avoidability of obstacles in terms of Poisson process intensity and obstacle size function.

Establishes a connection between obstacle configuration and Brownian motion reachability.

Extends understanding of percolation phenomena in stochastic geometric settings.

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.