Summary
Let(B t ) t≧0 be a Brownian Motion in ℝ3. We denote its Wiener sausage byw 1/n t :
Spitzer has proved in 64 that lim [n · Vol(w 1/n t )]=2πt.
IfK is a closed set in\(\mathbb{R}^{3^{n \to \infty } } \), we have extended this result to a localized Wiener sausagew 1/n t (K) and to a measure μ whose support lies inK. We define:
If there exists a function of “local capacity”,C K , with respect toK and if μ satisfies some integrability properties, then
where (A μ t ) t is the additive functional which is related to μ and to (B t ) t≧0.
Finally we have applied this result to solve an homogeneization problem concerning a Brownian motion when it is absorbed on a collection of small disks in a plane.
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Weinryb, S. Etude Asymptotique par des mesures de 135-1135-1135-1de saucisses de Wiener localisées. Probab. Th. Rel. Fields 73, 135–148 (1986). https://doi.org/10.1007/BF01845997
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DOI: https://doi.org/10.1007/BF01845997