The convolution equation of Choquet and Deny on [IN]-groups

Let sigma be a probability measure on a locally compact group G. A real Bor el function f on G is called a-harmonic if it satisfies the convolution equ ation sigma * f = f. Given that sigma is nonsingular with its translates, w e show that the bounded sigma -harmonic functions are constant on a class o f groups including the almost connected [IN]-groups. If sigma is nondegener ate and absolutely continuous, we solve the more general equation sigma * m u = mu for positive measure mu on those groups which are metrizable and sep arable.