  We study point processes on the real line whose configurations X can be
ordered decreasingly and evolve by increments which are functions of correlated
gaussian variables. The correlations are intrinsic to the points and quantified
by a matrix Q={q_ij}. Quasi-stationary systems are those for which the law of
(X,Q) is invariant under the evolution up to translation of X. It was
conjectured by Aizenman and co-authors that the matrix Q of robustly
quasi-stationary systems must exhibit a hierarchal structure. This was
established recently, up to a natural decomposition of the system, whenever the
set S_Q of values assumed by q_ij is finite. In this paper, we study the
general case where S_Q may be infinite. Using the past increments of the
evolution, we show that the law of robustly quasi-stationary systems must obey
the Ghirlanda-Guerra identities, which first appear in the study of spin glass
models. This provides strong evidence that the above conjecture also holds in
the general case.
