Simple nonlinear models with rigorous extreme events and heavy tails

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 at a point, 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.