Brownian Motion and Obstacles

Abstract

Our starting point is a “random cloud” of points in the d-dimensional space ℝ^d, d ≥ 1, that is a locally finite possibly empty collection of points in ℝ^d. Each point of the cloud is the center of a closed ball of radius a > 0. These random, possibly overlapping configurations will play the role of random obstacles. A fairly natural choice of probability governing the randomness of the cloud is the Poisson law with constant intensity v > 0, denoted by ℙ. Independence and translation invariance are built into ℙ: indeed if A ₁,…,A _k are pairwise disjoint Borel subsets of ℝ^d the random numbers of points N(A ₁),…N(A _k ) which respectively fall in A ₁,…, A _k are independent variables with Poisson distribution

$$\mathbb{P}\,\left[ {N\left( {{A_i}} \right)\, = \,\ell } \right]\, = \,\exp \left\{ { - v\left| {{A_i}} \right|} \right\}\,\frac{{{{\left ( {v\left| {{A_i}} \right|} \right)}^\ell }}}{{\ell !}},\,\ell \, \geqslant \,0\,,$$

((1.1))

if |·| stands for the usual Lebesgue volume on ℝ^d. We are first going to describe two model problems connected with this random medium, and later we shall see that these two problems turn out to be related.