The It\^o exponential on Lie Groups

Let $G$ be a Lie Group with a left invariant connection $\nabla^{G}$. Denote by $\g$ the Lie algebra of $G$, which is equipped with a connection $\nabla^{\g}$. Our main is to introduce the concept of the It\^o exponential and the It\^o logarithm, which take in account the geometry of the Lie group $G$ and the Lie algebra $\g$. This definition characterize directly the martingales in $G$ with respect to the left invariant connection $\nabla^{G}$. Further, if any $\nabla^{\g}$ geodesic in $\g$ is send in a $\nabla^{G}$ geodesic we can show that the It\^o exponential and the It\^o logarithm are the same that the stochastic exponential and the stochastic logarithm due to M. Hakim-Dowek and D. L\'epingle in [10]. Consequently, we have a Campbell-Hausdorf formula. From this formula we show that the set of affine maps from $(M,\nabla^{G})$ into $(G,\nabla^{G})$ is a subgroup of the Loop group. As in general, the Lie algebra is considered as smooth manifold with a flat connection, we show a Campbell-Hausdorf formula for a flat connection on $\g$ and a bi-invariant connection on $G$. To this main we introduce the definition of the null quadratic variation property. To end, we use the Campbell-Hausdorff formula to show that a product of harmonic maps with value in $G$ is a harmonic map.