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

TL;DR

This paper extends Pluzhnikov's Theorem to Lie groups with skew-symmetric, left-invariant connections, using stochastic logarithms to characterize Brownian motions and martingales in this setting.

Contribution

It introduces a version of Pluzhnikov's Theorem for Lie groups with skew-symmetric connections, utilizing stochastic logarithms for characterization.

Findings

Characterization of Brownian motions via stochastic logarithm

Martingale characterizations in Lie groups with skew-symmetric connections

Extension of Pluzhnikov's Theorem to this geometric setting

Abstract

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.