A Pluzhnikov's Theorem, Brownian motions and Martingales in Lie Group with skew-symmetric connections

Let $G$ be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov’s Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian motions and Martingales in $G$, and these characterzations are used to prove Pluzhnikov’s Theorem.