  We study point processes on the real line whose configurations $X$ are
locally finite, have a maximum and evolve through increments which are
functions of correlated Gaussian variables. The correlations are intrinsic to
the points and quantified by a matrix $Q=\{q_{ij}\}_{i,j\in\mathbb{N}}$. A
probability measure on the pair $(X,Q)$ is said to be quasi-stationary if the
joint law of the gaps of $X$ and of $Q$ is invariant under the evolution. A
known class of universally quasi-stationary processes is given by the Ruelle
Probability Cascades (RPC), which are based on hierarchically nested
Poisson--Dirichlet processes. It was conjectured that up to some natural
superpositions these processes exhausted the class of laws which are robustly
quasi-stationary. The main result of this work is a proof of this conjecture
for the case where $q_{ij}$ assume only a finite number of values. The result
is of relevance for mean-field spin glass models, where the evolution
corresponds to the cavity dynamics, and where the hierarchical organization of
the Gibbs measure was first proposed as an ansatz.
