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Transactions of the American Mathematical Society

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



Brownian motions of ellipsoids

Authors: J. R. Norris, L. C. G. Rogers and David Williams
Journal: Trans. Amer. Math. Soc. 294 (1986), 757-765
MSC: Primary 60J60; Secondary 58G32, 60J65
MathSciNet review: 825735
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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.

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Keywords: Brownian motions of ellipsoids, eigenvalues, eigenvectors, positivedefinite symmetric matrices, orthogonal group, Stratonovich differential, Itô's formula
Article copyright: © Copyright 1986 American Mathematical Society