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

We introduce and study an analytic model for physical systems exhibiting growth-collapse and decay-surge evolutionary patterns. We consider a generic system undergoing a smooth deterministic growth/decay evolution, which is occasionally interrupted by abrupt stochastic collapse/surge discontinuities. The deterministic evolution is governed by an arbitrary potential field. The discontinuities are multiplicative perturbations of random magnitudes, and their occurrences are state-dependent—governed by an arbitrary rate function. The combined deterministic-stochastic evolution of the system turns out to be governed by a geometric Langevin equation driven by a state-dependent noise. 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 systemʹs underlying dynamical structure is that of a geometric random walk; and, (iii) sub-critical—in which the system reaches statistical equilibrium.