MARTINGALE CHARACTERIZATIONS OF STOCHASTIC PROCESSES
ON COMPACT GROUPS
Abstract: By a classical result of P. Lévy, the Brownian motion on may be
characterized as a continuous process on such that and 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 locally compact groups with independent stationary increments that is based on
irreducible unitary representations. In particular, for Gaussian processes on compact Lie
groups, analogues of the Lévy-characterization above are obtained. It turns out that for
certain compact Lie groups even the continuity assumption in this characterization can be
dropped.
1991 AMS Mathematics Subject Classification: Primary: 60J30. Secondary: 60G44,
60H05, 60J65, 22D10.
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