Martingale characterizations of stochastic processes on locallycompact

By a classical result of P. L evy, the Brownian motion (B t) t0 on R may be characterized as a continuous process on R such that (B t) t0 and (B 2 t ?t) t0 are martingales. Generalizations of this result are usually obtained in the setting of the so-called martingale problem. This paper contains a variant of the martingale problem for stochastic processes on compact groups with independent stationary increments that is based on irreducible representations. For Gaussian processes on Lie groups, analogues of the L evy-characterization above are obtained, where in some cases even the continuity-assumption can be dropped.