Moment instabilities in multidimensional systems with noise

TL;DR

This paper systematically analyzes how moments evolve in multidimensional stochastic difference systems with noise, identifying conditions under which moments diverge and providing analytical expressions validated by simulations.

Contribution

It introduces a comprehensive framework for characterizing moment divergence in noisy multidimensional systems, including explicit formulas and critical noise thresholds.

Findings

Derived asymptotic distributions for small noise

Calculated second moments for larger noise cases

Identified critical noise levels leading to divergence

Abstract

We present a systematic study of moment evolution in multidimensional stochastic difference systems, focusing on characterizing systems whose low-order moments diverge in the neighborhood of a stable fixed point. We consider systems with a simple, dominant eigenvalue and stationary, white noise. When the noise is small, we obtain general expressions for the approximate asymptotic distribution and moment Lyapunov exponents. In the case of larger noise, the second moment is calculated using a different approach, which gives an exact result for some types of noise. We analyze the dependence of the moments on the system's dimension, relevant system properties, the form of the noise, and the magnitude of the noise. We determine a critical value for noise strength, as a function of the unperturbed system's convergence rate, above which the second moment diverges and large fluctuations are likely. Analytical results are validated by numerical simulations. We show that our results cannot be extended to the continuous time limit except in certain special cases.