Random transformations and invariance of semimartingales on Lie groups

Invariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non trivial examples of symmetric semimartingales are provided and applications of this concept to stochastic analysis are discussed.