Positive Pluriharmonic Functions on Symmetric Siegel Domains

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