Brownian motion and obstacles.

*(English)*Zbl 0815.60077
Joseph, A. (ed.) et al., First European congress of mathematics (ECM), Paris, France, July 6-10, 1992. Volume I: Invited lectures (Part 1). Basel: BirkhĂ¤user. Prog. Math. 119, 225-248 (1994).

This expository paper deals with the following situation. A random Poissonian cloud of points in \(\mathbb{R}^ d\) is given; each point of the cloud being the center of a closed ball of radius \(a\). The author considers two problems:

1. A spectral problem. A small number \(\lambda > 0\) is given; consider a large ball \(B_ N\) of \(\mathbb{R}^ d\) centered at the origin with radius \(N\). What is the order of magnitude of the number of Dirichlet eigenvalues of \(-{1\over 2} \Delta\) smaller or equal to \(\lambda\) in \(B_ n\backslash\)obstacles?

2. A diffusion problem. \(Z\) is a Brownian motion starting from the origin with law \(P_ 0\) independent of the law \(P\) of the cloud. \(Z\) is absorbed at the time \(T\) it reaches one of the obstacles. What is the large \(t\) behaviour of the probability \(S(t) = P \times P_ 0\) \((T > t)\) that \(Z\) is not absorbed by time \(t\)?

The author explains how these two problems are related and also, how they are linked to the Wiener sausage. He introduces a new technique for these questions: “the method of enlargement of obstacles”. The ideas contained in this method apply in a variety of contexts. The paper also includes a study of the same problems for a Brownian motion with a drift. Estimates for the survival probability are given.

For the entire collection see [Zbl 0807.00007].

1. A spectral problem. A small number \(\lambda > 0\) is given; consider a large ball \(B_ N\) of \(\mathbb{R}^ d\) centered at the origin with radius \(N\). What is the order of magnitude of the number of Dirichlet eigenvalues of \(-{1\over 2} \Delta\) smaller or equal to \(\lambda\) in \(B_ n\backslash\)obstacles?

2. A diffusion problem. \(Z\) is a Brownian motion starting from the origin with law \(P_ 0\) independent of the law \(P\) of the cloud. \(Z\) is absorbed at the time \(T\) it reaches one of the obstacles. What is the large \(t\) behaviour of the probability \(S(t) = P \times P_ 0\) \((T > t)\) that \(Z\) is not absorbed by time \(t\)?

The author explains how these two problems are related and also, how they are linked to the Wiener sausage. He introduces a new technique for these questions: “the method of enlargement of obstacles”. The ideas contained in this method apply in a variety of contexts. The paper also includes a study of the same problems for a Brownian motion with a drift. Estimates for the survival probability are given.

For the entire collection see [Zbl 0807.00007].

Reviewer: M.Chaleyat-Maurel (Paris)