Simple Nonlinear Models with Rigorous Extreme Events and Heavy Tails

TL;DR

This paper develops a rigorous theoretical framework for understanding extreme events and heavy tails in stochastic models, revealing how stochastic damping influences tail behavior and providing bounds for long-term averages.

Contribution

It introduces a novel method using Lyapunov functions and Feynman-Kac formula to rigorously analyze tail densities in conditional Gaussian models, advancing understanding beyond numerical experiments.

Findings

Negative stochastic damping leads to polynomial tails.

Nonnegative damping with zeros results in tails between exponential and Gaussian.

Framework provides non-asymptotic large deviation bounds.

Abstract

Extreme events and the heavy tail distributions driven by them are ubiquitous in various scientific, engineering and financial research. They are typically associated with stochastic instability caused by hidden unresolved processes. Previous studies have shown that such instability can be modeled by a stochastic damping in conditional Gaussian models. However, these results are mostly obtained through numerical experiments, while a rigorous understanding of the underlying mechanism is sorely lacking. This paper contributes to this issue by establishing a theoretical framework, in which the tail density of conditional Gaussian models can be rigorously determined. In rough words, we show that if the stochastic damping takes negative values, the tail is polynomial; if the stochastic damping is nonnegative but takes value zero, the tail is between exponential and Gaussian. The proof is established by constructing a novel, product-type Lyapunov function, where a Feynman-Kac formula is applied. The same framework also leads to a non-asymptotic large deviation bound for long-time averaging processes.