In the first part, I study asymptotic invariants of Brownian motion on Riemannian manifolds (symmetric spaces, manifolds with a compact quotient, and generic leaves of foliations) and their relationship with the Liouville property (i.e. whether or not there are non-constant bounded harmonic functions). I do this by introducing a new definition, that of a stationary random manifold, and following a general plan first layed out by Vadim Kaimanovich.
The second part of the thesis is purely geometrical (no Brownian motion involved). In it I study the possible Gromov-Hausdorff limits of leaves of foliations, obtaining a nice identification of the largest and smallest limits possible which implies in particular classical results about stability (such as the Reeb local stability theorem).