Stochastic hybrid systems in equilibrium: moment closure, finite-time blowup, and exact solutions

TL;DR

This paper analyzes stochastic hybrid systems near equilibrium, establishing conditions for moment behavior, exploring finite-time blowups, and connecting recurrence relations to number theory, thereby advancing understanding of their long-term dynamics.

Contribution

It introduces general conditions for moment stability, investigates finite-time blowup scenarios, and links recurrence relations in SHSs to classical number theory expressions.

Findings

Conditions for well-behaved moments identified

Finite-time blowup scenarios characterized

Recurrence relations connected to number theory

Abstract

We present a variety of results analyzing the behavior of a class of stochastic processes --- referred to as Stochastic Hybrid Systems (SHSs) --- in or near equilibrium, and determine general conditions on when the moments of the process will, or will not, be well-behaved. We also study the potential for finite-time blowups for these processes, and exhibit a set of random recurrence relations that govern the behavior for long times. In addition, we present a connection between these recurrence relations and some classical expressions in number theory.