L\'evy Processes on $U_q(g)$ as Infinitely Divisible Representations

L'evy processes on bialgebras are families of infinitely divisible representations. We classify the generators of L'evy processes on the compact forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then we show how the processes themselves can be reconstructed from their generators and study several classical stochastic processes that can be associated to these processes.