A metric-affine geometry about the Uhlenbeck-Ornstein (UO) process and about a random walk (RW) is investigated. It is found that the time development of the UO process is a geodesic motion in the metric-affine space spanned with two parameters, the mean and standard deviation, and that the time development of the RW is an expansion of a spherical space spanned with jump probabilities.
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Obata et al. (1992) studied this question.
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