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.
The aim of these notes is to relate covariant stochastic integration in a vector bundle $E$ (as in Norris cite{Norris}) with the usual Stratonovich calculus via the connector $K:TE rightarrow E$ (cf. e.g. Paterson cite{Paterson} or Poor cite{Poor}) which carries the connection dependence.
In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic meas
ures. Other results include a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.
