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Etude Asymptotique par des mesures de 135-1135-1135-1de saucisses de Wiener localisées
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  • Published: March 1986

Etude Asymptotique par des mesures de 135-1135-1135-1de saucisses de Wiener localisées

  • Sophie Weinryb1 

Probability Theory and Related Fields volume 73, pages 135–148 (1986)Cite this article

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Summary

Let(B t ) t≧0 be a Brownian Motion in ℝ3. We denote its Wiener sausage byw 1/n t :

$$w_t^{1/n} = \left\{ {m \in \mathbb{R}^3 /\exists s \leqq t,^ \parallel B_s - m\parallel \leqq \frac{{1}}{n}} \right\}.$$

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:

$$w_t^{1/n} (K) = \left\{ {m \in \mathbb{R}^3 /\exists s \leqq t,\parallel B_s - m\parallel \leqq \frac{{1}}{n}andB_s \in K} \right\}.$$

If there exists a function of “local capacity”,C K , with respect toK and if μ satisfies some integrability properties, then

$$\mathop {\lim }\limits_{n \to \infty } n\mu (w_t^{1/n} (K)) = \int\limits_0^t {C_K (B_s )dA_s^\mu ,}$$

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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Bibliographie

  1. Spitzer, F.: Electrostatic capacity, heat flow and Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 110–121 (1964)

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  3. Del Grosso, G., Figari, R., Orlandi, E.: Diffusion processes in a region with perforated boundary. (à paraître)

  4. Papanicolaou, G.C., Varadhan, S.R.: Diffusion in regions with many small holes. Stochastic Differential Systems Proceedings Vilnius (1978)

  5. Dellacherie, C., Meyer, P.A.: Probabilities et potentiels (théorie des martingales). Hermann, Publication de l'Institut de Mathématiques de l'Université de Strasbourg XVII 1980

  6. Kac, M.: Probabilistic methods in some problems of scattering theory. Rocky Mountain Journal (1974)

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Authors and Affiliations

  1. Ecole Polytechnique, Centre de Mathématiques Appliquées, F-91128, Palaiseau Cédex, France

    Sophie Weinryb

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  1. Sophie Weinryb
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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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  • Issue Date: March 1986

  • DOI: https://doi.org/10.1007/BF01845997

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