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Brownian motions of ellipsoids. (English) Zbl 0613.60072
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 E. B. Dynkin [Dokl. Akad. Nauk SSSR 141, 288-291 (1961; Zbl 0116.361)].
The canonical right-invariant Brownian motion $$G=\{G(t)\}$$ on GL(n) induces processes $$X=GG^ T$$ and $$Y=G^ TG$$ 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 O(n), while that of Y converges to a limiting value.
The Y process is that studied by Dynkin and A. Orihara [J. Fac. Sci., Univ. Tokyo, Sect. I A 17, 73-85 (1970; Zbl 0231.60069)]. From a naive standpoint, the X process would seem to provide a more natural model.

##### MSC:
 60J65 Brownian motion 60J60 Diffusion processes 58J65 Diffusion processes and stochastic analysis on manifolds 15B52 Random matrices (algebraic aspects)
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##### References:
 [1] 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). · Zbl 0116.36106 [2] Freeman J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys. 3 (1962), 1191 – 1198. · Zbl 0111.32703 [3] 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. · Zbl 0684.60040 [4] H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. · Zbl 0191.46603 [5] Akio Orihara, On random ellipsoid, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 73 – 85. · Zbl 0231.60069
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