Non-endogenous fixed points of the discounted tree sum equation: the regular case

Aldous and Bandyopadhyay showed that every nonzero solution $Y$ of the homogeneous equation $Y\stackrel{d}{=}\max_i \xi_i Y_i$ produces, for each $a\ge 0$, an invariant measure $\mu_a$ of the discounted tree sum equation $X\stackrel{d}{=}\eta+\max_i \xi_i X_i$, and conjectured that for $a>0$ the invariant recursive tree process with marginal $\mu_a$ is never endogenous. We show that the conditional essential supremum of $Y$ given all the innovations is an endogenous solution of the homogeneous equation with values in $[0,\infty]$, and that $\mu_a$ fails to be endogenous as soon as this supremum is infinite with positive probability. It follows that the conjecture, for all nonzero $Y$ at once, is equivalent to the absence of a nonnull finite endogenous solution of the homogeneous equation. When the weights have a characteristic exponent, the disintegration theorem of Alsmeyer, Biggins and Meiners shows that the conditional supremum is infinite on the survival event of an endogenous fixed point of the smoothing transform, and the conjecture holds.