TITLE: Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds
QUESTION [5 upvotes]: I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a probabilistic proof of the Atiyah-Singer index theorem.
I am also curious about the industrial applications (if any yet) given that Riemannian geometry is used in Machine learning and statistics.
Thanks anyhow
In case you are interested
[1]https://www.math.kyoto-u.ac.jp/probability/sympo/PSS03abstract.pdf
[2]http://www.math.northwestern.edu/~ehsu/Brownian%20Motion%20and%20Riemannian%20Geometry.pdf (proof of Atiyah Singer index thm)

REPLY [4 votes]: The earliest "industrial" application I know is in the context of microwave engineering: the eigenvalues of the transmission matrix through a waveguide with random scatterers perform a Brownian motion in hyperbolic space as a function of the length of the waveguide.
Waveguides with Random Inhomogeneities and Brownian Motion In the Lobachevsky Plane (1959).

REPLY [3 votes]: A somewhat recent engineering application is in study of dynamics and control on manifolds such as SO(3), which is the manifold on which the (position) state of a rotating rigid body (such as a satellite) evolves.
For inference/filtering problems (i.e. trying to estimate the position on SO(3) given some noisy measurements), it is required to perform uncertainty propagation on these manifolds.
Here's a reference:
http://arxiv.org/abs/0803.1515