Simple nonlinear models with rigorous extreme events and heavy tails

Andrew Majda, Xin T. Tong

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)1641-1674
Number of pages34
JournalNonlinearity
Volume32
Issue number5
DOIs
StatePublished - Apr 12 2019

Fingerprint

Extreme Events
Heavy Tails
Nonlinear Model
Damping
damping
Tail
Conditional Model
Gaussian Model
Liapunov functions
Lyapunov functions
Feynman-Kac Formula
polynomials
Large Deviations
Polynomials
engineering
Lyapunov Function
Rough
deviation
Averaging
Non-negative

Keywords

  • conditional Gaussian model
  • extreme event
  • heavy tail
  • intermittency

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

Simple nonlinear models with rigorous extreme events and heavy tails. / Majda, Andrew; Tong, Xin T.

In: Nonlinearity, Vol. 32, No. 5, 12.04.2019, p. 1641-1674.

Research output: Contribution to journalArticle

Majda, Andrew ; Tong, Xin T. / Simple nonlinear models with rigorous extreme events and heavy tails. In: Nonlinearity. 2019 ; Vol. 32, No. 5. pp. 1641-1674.
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