Résumé
The spectral empirical function of a homogeneous, isotropic, Poisson mosaic process is a functional of the perimeter of the convex hull of planar Brownian bridge. Some geometrical identities follow.
Bibliographie
T. Bonnesen, W. Fenchel: Theorie der Konvexer Körper. 1971, Chelsea
S. Brassesco: A note on planar Brownian motion. Ann. Probab. 3,20, 1498–1503 (1992)
R. Cowan: The use of the ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10, 47–57 (1978)
R. Cowan: Properties of ergodic random mosaic processes. Math. Nachr.97, 89–102 (1980)
M. El Bachir: Thèse Univ. Paul Sabatier, Toulouse, 1983
A. Goldman: Le spectre de certaines mosaiques poisonniennes du plan et l'enveloppe convexe du pont brownien. C. R. Acad. Sci. Paris,320, 463–468 (1995)
S.A. Goudsmit: Random distributions of lines in a plane. Rev. Mod. Phys.17, 321–322 (1945)
P. Hall: Introduction to the theory of coverage processes. New York: Wiley 1988
J. Hersch: Sur la fréquence fondamentale d'une menbrane vibrante: évaluation par défaut et principe de maximum. Z. Angew. Math. Mech.11, 387–413 (1960)
M.N. Huxley: Exponential Sums and Lattice Points. Proc. Lond. Math. Soc. (à paraître)
M. Kac: Can one hear the shape of a drum? Amer. Math. Monthly73, 1–23 (1966)
J.R. Kuttler, V.G. Sigilitto: Eingenvalues of the Laplacian in two dimension. SIAM Rev,26, 163–193 (1984)
P. Li, S.T. Yau: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys.88, 309–318 (1983)
W. Magnus, F. Oberhettinger, F.G. Tricomi: Tables of integral trasforms. New York: McGraw-Hill 1954
H.P. McKean, I.M. Singer: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom.1, 43–69 (1967)
R.E. Miles: Random polygons determined by random lines in a plane I. Proc. Natl. Acad. Sci. USA. 52, 901–907 (1964)
R.E. Miles: Random polygons determined by random lines in a plane II. Proc. Natl. Acad. Sci. USA. 52, 1157–1160 (1964)
G. Polya: On the eigenvalues of vibrating membranes. Proc. London Math. Soc.3, 419–433 (1961)
H. Solomon: Geometric Probability, 1978, SIAM
L. Takacs: Expected perimeter length. Amer. Math. Monthly87, (1980)
M. Van den Berg, S. Srisatkunarajah: heat Equation for a region inR 2 with polygonal boundary. J. Lond. Math. Soc.3, 119–127 (1988)
D.G. Vassiliev: Two term asymptotics of the spectrum of a boundary value problem in the case of a piecewise smooth boundary: Soviet. Math. Dokl.33, 227–230 (1986)
N. Wiener: The ergodic theorem. Duke Math. J.5, 1–18 (1939)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goldman, A. Le spectre de certaines mosaïques poissoniennes du plan et l'enveloppe convexe du pont brownien. Probab. Th. Rel. Fields 105, 57–83 (1996). https://doi.org/10.1007/BF01192071
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192071
Mathematics Subject Classification (1991)
- 35P15
- 35P20
- 52A22
- 60D05
- 60J65