# Percolation Diffusion Journal article, 2001

Let a Brownian motion in the unit ball be absorbed if it hits a set generated by a radially symmetric Poisson point process. The point set is fattened by putting a ball with a constant hyperbolic radius on each point. When is the probability non-zero that the Brownian motion hits the boundary of the unit ball? That is, manage to avoid all the Poisson balls and percolate diffusively all the way to the boundary. We will show that if the bounded Poisson intensity at a point z is ν(d(0,z)), where d(· ,·) is the hyperbolic metric, then the Brownian motion percolates diffusively if and only if $\nu \in L^1$.

Percolation

Poisson process

Brownian motion

minimal thinness

hyperbolicgeometry

## Author

#### Torbjörn Lundh

Department of Mathematics

University of Gothenburg

#### Stochastic Processes and their Applications

0304-4149 (ISSN)

Vol. 95 2 235-244

#### Subject Categories

Probability Theory and Statistics

#### DOI

10.1016/S0304-4149(01)00101-6