Brownian motions of ellipsoids

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.