Good morning everybody. My question is, as maybe already hinted in the title, rather philosopic.

We know that geometric properties of a riemannian manifold can be interpreted in terms of certain evolution processes; I'm thinking about all the relations between the expansion of the heat kernel and, for example, the gaussian curvature of the manifold.

Now, since I'm not a probabilist the question is as follows...

are there characterizations, or properties of riemannian geodesics which can be deduced from stochastic operators (like the Brownian motion)? Even more, can we give a characterization of (properties of) riemannian geodesics in these terms?

All references are welcomed.. this is just a soft question for me to know where I have to look at in literature.

Thanks again for the patience.

Guido


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