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.
We extend the concept of Lyapunov 1-forms for the case of diffu- sion processes to study its asymptotic behavior. We give some examples and a condition for the existence of these objects.
Given a control system $dot{p} = X_0(p) + sum_i u_i (t)X_i(p)$ on a compact manifold M we study conditions for the foliation defined by the accessible sets be dense in M . To do this we relate the control system to a stochastic differential equation
and, using the support theorem, we give a characterization of the density in terms of the infinitesimal generator of the diffusion and its invariant measures. Also we give a different proof of Kreners theorem.
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.
