Let G be a Lie Group with a complete, left invariant connection ∇ G . Denote by g the Lie algebra of G which is equipped with a complete connection ∇ g . Our main goal is to introduce the concept of the Itô exponential and the Itô logarithm. As a result, we characterize the martingales in G with respect to the left invariant connection ∇ G . Also, assuming that connection function α : g × g → g associated to ∇ G satisfies α(M, M ) = 0 for all M ∈ g we obtain a stochastic Campbell-Hausdorff formula. Further, from this stochastic Campbell-Hausdorf formula we present a way to construct martingales in Lie group. In consequence, we show that a product of harmonic maps with value in G is a harmonic map. To end, we apply this study in some matrix Lie groups.

In this paper we study a classification of linear systems on Lie groups with respect to the conjugacy of the corresponding flows. We also describe stability according to Lyapunov exponents.

Let M be a smooth manifold endowed with a symmetric connection ∇. There are two important ways of lift the connection ∇ of M to the frame bundle BM, the canonical lift ∇ c and the horizontal lift ∇ h . The aim of this work is determine the ∇ c -martingales and the ∇ h -martingales on BM. Our results allow to establish new characterizations of harmonic maps from Riemannian manifolds to frame bundles.

Our first purpose is to study the stability of linear flows on real, connected, compact, semisimple Lie groups. Our second purpose is to study periodic orbits of linear and invariant flows. As an application, we present periodic orbits of linear or invariant flows on SO(3) and SU(2) and we study periodic orbits of linear or invariant flows on SO(4).

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