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I am trying to understand how the two paradigms of differential geometry and probability theory can fruitfully be applied to each other.

The more suggestive direction is to use methods of differential geometry to understand the geometric structure of objects in probability theory, e.g. distributions, as done for example in Information Geometry (http://en.wikipedia.org/wiki/Information_geometry).

But what about the other direction? Does it make sense (meaning does one gain more insights) by regarding objects of differential geometry in a probabilistic setup? One could for example consider a distribution of manifolds and instead of deterministic time evolution of a manifold itself, one could investigate the time evolution of random variables that take values in a space of manifolds.

Unfortunately, I didn't find any literature on this second direction. If someone has encountered such problems, I would be thankful for literature suggestions.

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closed as unclear what you're asking by Benoît Kloeckner, Stefan Kohl, Andrey Rekalo, Ben Webster Feb 26 at 18:36

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

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The question seems too vague to me, in fact I can hardly tell what the question is. Right now I cannot tell whether random walks on manifolds answer the question. –  Benoît Kloeckner Feb 26 at 16:16
    
Sorry for being so vague. I don't mean random walks that take values on a manifolds, but to stay within your direction of thought e.g. random walks on the space of manifolds that take a whole manifold as a value. –  madison54 Feb 26 at 16:22
    
I'm not sure if this answers your question, but there is a nice book on applications of stochastic analysis to differential geometry: amazon.com/Stochastic-Analysis-Manifolds-Graduate-Mathematics/…. It seems to culminate in a stochastic proof of the index theorem. –  Paul Siegel Feb 26 at 17:47
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2 Answers 2

Let me just quote (in the CW mode) an abstract of a colloquium talk that I attended yesterday at the University of Michigan, Ann Arbor:

date: Tuesday, February 25, 2014 Location: 1360 East Hall (4:10 PM to 5:00 PM)

Speaker: Elton Hsu Institution: Northwestern University

Title: Stochastic Analysis and Its Applications to Geometric Problems

Abstract: Recent popularity of probability theory and stochastic analysis beyond their traditional confine is to a large extent due to successful applications of their methods and results in other areas of pure and applied mathematics. In this talk I will explain how probability theory and stochastic analysis can be applied to certain problems from analysis and differential geometry. The central object of these applications is Brownian motion on a Riemannian manifold, a diffusion process generated by the Laplace-Beltrami operator. Its transition density function is the fundamental solution of the attendant heat equation. This connection between stochastic analysis and classical analysis and differential geometry makes it possible to study certain geometric properties of Riemannian manifolds by techniques from stochastic analysis. In this context, the concept of stopping times play an important role. I will explain the basic framework of the theory and showcase some of the most interesting results from this fruitful union of stochastic analysis and differential geometry, including eigenvalue estimates, heat kernel asymptotics, harmonic functions on manifolds, escape rate of Riemannian Brownian motion, and diffusion and geometric models from financial mathematics.

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There are some examples in integral geometry. For example, a classical result is Crofton's formula, which gives a way to interpret arc length as the expected intersections with random lines.

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