Positive Pluriharmonic Functions on Symmetric Siegel Domains

Given a symmetric Siegel domain D and a positive plurihamonic function f on D, we study the largest positive Radon measure μ on the Silov boundary b D of D whose Poisson integral P μ is ≤ f. If D has no tubular irreducible factors of rank ≥ 2, we show that P μ is plurihamonic, and that f- P μ is linear. As an application, we describe a possible analogue of the family of Clark measures associated with a holomorphic function from D into the unit disc in C.