Abstract
Let ν be a probability measure on a semi-simple Lie group G with finite center. Under the hypothesis that the semigroup S generated by ν has non-empty interior, we identify the Poisson space Π = G/MνAN, where bounded (l.u.c.) ν-harmonic functions in G have a one-to-one correspondence with measurable (continuous) functions in Π. This paper extends a classical result (see Furstenberg [7], Azencott [1] and others), where the semigroup generated by ν was assumed to be the whole (connected) group. We present two detailed examples.
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