Non-Archimedean Green's functions and Zariski decompositions

We study the non-Archimedean Monge-Amp\`ere equation on a smooth projective variety over a discretely or trivially valued field. First, we give an example of a Green's function, associated to a divisorial valuation, which is not Q-PL (i.e. not a model function in the discretely valued case). Second, we produce an example of a function whose Monge-Amp\`ere measure is a finite atomic measure supported in a dual complex, but which is not invariant under the retraction associated to any snc model. This answers a question by Burgos Gil et al in the negative. Our examples are based on geometric constructions by Cutkosky and Lesieutre, and arise via base change from Green's functions over a trivially valued field; this theory allows us to efficiently encode the Zariski decomposition of a pseudoeffective numerical class.