no. 2
On the volume of intersection of three independent Wiener sausages
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 313-337

Let K be a compact, non-polar set in ℝm, m≥3 and let SKi(t)={Bi(s)+y: 0≤s≤t, y∈K} be Wiener sausages associated to independent brownian motions Bi, i=1, 2, 3 starting at 0. The expectation of volume of ⋂i=13SKi(t) with respect to product measure is obtained in terms of the equilibrium measure of K in the limit of large t.

Soit K un ensemble compact, non-polaire dans ℝm (m≥3) et soit SKi(t)={Bi(s)+y: 0≤s≤t, y∈K} des saucisses de Wiener associées à des processus Browniens indépendants Bi, i=1, 2, 3 initialisés à 0. L'espérance des volumes de ⋂i=13SKi(t) par rapport à la mesure produit est obtenue en termes de la mesure d'équilibre de K lorsque t tend vers l'infini.

DOI : 10.1214/09-AIHP316
Classification : 35K20, 60J65, 60J45
Keywords: Wiener sausage, equilibrium measure 
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     year = {2010},
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van den Berg, M. On the volume of intersection of three independent Wiener sausages. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 313-337. doi: 10.1214/09-AIHP316

[1] S. Albeverio and X. Y. Zhou. Intersections of random walks and Wiener sausages in four dimensions. Acta Appl. Math. 45 (1996) 195-237. | Zbl | MR

[2] R. Fernández, J. Fröhlich and A. D. Sokal. Random Walks, Critical Phenomena and Triviality in Quantum Field Theory. Texts and Monographs in Physics. Springer, New York, 1992. | Zbl | MR

[3] P. B. Gilkey. Asymptotic Formulae in Spectral Geometry. Studies in Advanced Mathematics. Chapman & Hall, Boca Raton, 2004. | Zbl | MR

[4] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products. Academic Press, San Diego, 1994. | Zbl | MR

[5] G. Hardy, J. Littlewood and G. Polya. Inequalities. Cambridge Univ. Press, London, 1952. | Zbl | MR

[6] K. M. Khanin, A. E. Mazel, S. B. Shlosman and Ya. G. Sinai. Loop condensation effects in the behaviour of random walks. In The Dynkin Festschrift, Markov Processes and Their Applications 167-184. M. Freidlin (ed.). Progr. Probab. 34. Birkhäuser, Boston, 1994. | Zbl | MR

[7] G. Lawler. Intersections of Random Walks. Probability and Its Applications. Birkhäuser, Boston, 1991. | Zbl | MR

[8] J.-F. Le Gall. Sur une conjecture de M. Kac. Probab. Theory Related Fields 78 (1988) 389-402. | Zbl | MR

[9] J.-F. Le Gall. Wiener sausage and self-intersection local times. J. Funct. Anal. 88 (1990) 299-341. | Zbl | MR

[10] J.-F. Le Gall. Some properties of planar Brownian motion. In École d'Été de Probabilités de Saint-Flour XX, 1990 111-235. Lecture Notes in Mathematics 1527. Springer, Berlin, 1992. | Zbl | MR

[11] N. Madras and G. Slade. The Self-Avoiding Walk. Birkhäuser, Boston, 1993. | Zbl | MR

[12] S. C. Port. Asymptotic expansions for the expected volume of a stable sausage. Ann. Probab. 18 (1990) 492-523. | Zbl | MR

[13] S. C. Port and C. J. Stone. Brownian Motion and Classical Potential Theory. Academic Press, New York, 1978. | Zbl | MR

[14] F. Spitzer. Electrostatic capacity and Brownian motion. Z. Wahrsch. Verw. Gebiete 3 (1964) 110-121. | Zbl | MR

[15] A. S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer, Berlin, 1998. | Zbl | MR

[16] M. Van Den Berg. On the expected volume of intersection of independent Wiener sausages and the asymptotic behaviour of some related integrals. J. Funct. Anal. 222 (2005) 114-128. | Zbl | MR

[17] M. Van Den Berg and J.-F. Le Gall. Mean curvature and the heat equation. Math. Z. 215 (1994) 437-464. | Zbl | MR

[18] M. Van Den Berg, E. Bolthausen and F. Den Hollander. On the volume of intersection of two Wiener sausages. Ann. Math. 159 (2004) 741-782. | Zbl | MR

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