Methods of probability theory in differential geometry fruitful? 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.
 A: 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. 
A: 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.
