Growth-collapse and decay-surge evolutions, and geometric Langevin equations

A statistical exploration of these growth-collapse and decay-surge systems is conducted, with a focus on two special classes of systems: scale-free systems and generalized power-law systems. For stationary scale-free systems we explicitly compute the distribution of the pre-discontinuity, post-discontinuity, and equilibrium levels. Generalized power-law systems are proved to display three possible qualitative types of behavior: (i) super-critical-in which the system eventually explodes/freezes; (ii) critical-in which the system's underlying dynamical structure is that of a geometric random walk; and, (iii) sub-critical-in which the system reaches statistical equilibrium.