Intermittency induced by long memory under stochastic regime switching

TL;DR

This paper investigates how long memory and stochastic regime switching can cause intermittent bursts and instability in nonlinear dynamical systems, revealing a fundamental annealed-quenched stability dichotomy.

Contribution

It formalizes the annealed-quenched stability dichotomy in non-Markovian switching systems and links microscopic burst mechanisms to macroscopic long-memory dynamics.

Findings

Identifies an averaged memory gain leading to mean-square control.

Shows rare excursions into supercritical regimes cause heavy-tailed bursts.

Establishes convergence of point processes to a Volterra limit, linking microscopic and macroscopic dynamics.

Abstract

We study a fundamental instability mechanism in nonlinear, nonlocal dynamical systems arising from the interaction of long-range memory and stochastic regime switching. The dynamics are governed by network-coupled, operator-valued Volterra evolutions with completely monotone memory kernels whose excitation operators and kernel parameters are modulated by an ergodic finite-state continuous-time Markov chain. We formalize a sharp separation between annealed stability (in expectation) and quenched behaviour (along typical sample paths). On the annealed side, we identify an averaged memory gain that yields uniform moment bounds and a memory-adapted Lyapunov functional implying mean-square control under an averaged subcriticality condition. On the quenched side, we show that rare but persistent excursions into supercritical regimes are amplified by memory, producing intermittent macroscopic bursts with heavy-tailed statistics and a deterministic almost sure growth exponent obtained via a subadditive ergodic argument. This establishes an annealed--quenched dichotomy specific to non-Markovian switching systems, where stability in expectation can coexist with pathwise growth and metastable burst phases. We further derive a micro--macro correspondence by proving that a population of regime-modulated self-exciting point processes converges, both annealed and quenched, to the random-coefficient Volterra limit, transferring the burst mechanism from microscopic branching dynamics to macroscopic long-memory flows. Numerical experiments illustrate how burst localization depends on graph geometry and on noncommuting excitation operators.