Harmonic measures in embedded foliated manifolds

TL;DR

This paper investigates harmonic and invariant measures on embedded foliated manifolds, introducing stochastic calculus techniques to characterize measures and derive ergodic formulas related to leaf geometry.

Contribution

It develops geometrical stochastic calculus methods to explicitly construct foliated Brownian motion and characterizes invariant measures via flow of diffeomorphisms.

Findings

Explicit Stratonovich equation for foliated Brownian motion

Characterization of totally invariant measures

Ergodic formula relating Lyapunov exponents to leaf geometry

Abstract

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold isometrically embedded in an Euclidean space. We introduce geometrical techniques for stochastic calculus in this space. In particular, using these techniques we can construct explicitely an Stratonovich equation for the foliated Brownian motion (cf. L. Garnett \cite{LG} and others). We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms of associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.