Positive Pluriharmonic Functions on Symmetric Siegel Domains

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