Abstract. In this article we present an intrinsic construction of foliated Brownian motion (FoBM) via stochastic calculus adapted to a foliated Riemaniann manifold (M, F ). The stochastic approach together with this proposed foliated vector calculus provide a natural method to work with (L. Garnett's) harmonic measures in M . New results include, beside an explicit stochastic equation for the FoBM, a decomposition of the Laplacian of M in terms of the foliated and the basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.
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 [11] 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.
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