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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Brownian motions of ellipsoids
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by J. R. Norris, L. C. G. Rogers and David Williams PDF
Trans. Amer. Math. Soc. 294 (1986), 757-765 Request permission

Abstract:

The object of this paper is to provide an elementary treatment (involving no differential geometry) of Brownian motions of ellipsoids, and, in particular, of some remarkable results first obtained by Dynkin. The canonical right-invariant Brownian motion $G = \{ G(t)\}$ on ${\text {GL}}(n)$ induces processes $X = G{G^T}$ and $Y = {G^T}G$ on the space of positive-definite symmetric matrices. The motion of the common eigenvalues of $X$ and $Y$ is analysed. It is further shown that the orthonormal frame of eigenvectors of $X$ ultimately behaves like Brownian motion on ${\text {O}}(n)$, while that of $Y$ converges to a limiting value. The $Y$ process is that studied by Dynkin and Orihara. From a naive standpoint, the $X$ process would seem to provide a more natural model.
References
  • E. B. Dynkin, Non-negative eigenfunctions of the Laplace-Beltrami operator and Brownian motion in certain symmetric spaces, Dokl. Akad. Nauk SSSR 141 (1961), 288–291 (Russian). MR 0132607
  • Freeman J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys. 3 (1962), 1191–1198. MR 148397, DOI 10.1063/1.1703862
  • Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
  • H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
  • Akio Orihara, On random ellipsoid, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 73–85. MR 275489
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 294 (1986), 757-765
  • MSC: Primary 60J60; Secondary 58G32, 60J65
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0825735-5
  • MathSciNet review: 825735