We will discuss a class of stochastic processes indexed by loops on a surface, which take their values in a compact matrix group and satisfy a property analogous to the classical Markov property. These processes are essentially in one-to-one correspondence with the classical Lévy processes on the matrix group under consideration. They generalize the Yang-Mills measure, which corresponds to the Brownian motion on the group. In the case where the matrix group is a finite group, there is a natural realization of these stochastic processes as the monodromies of random ramified coverings over the compact surface.