Percolation Diffusion

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 $\nu \in L^1$