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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: Birkhäuser. 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].

MSC:
 60J65 Brownian motion 35P15 Estimates of eigenvalues in context of PDEs