Penalization of non-smooth dynamical systems with noise : ergodicity and asymptotic formulae for threshold crossings probabilities

TL;DR

This paper proves ergodicity and derives asymptotic formulas for threshold crossing probabilities in noisy nonlinear mechanical models, enabling better understanding of their long-term behavior and rare event probabilities.

Contribution

It introduces new methods to establish ergodicity and asymptotic formulas for complex nonlinear systems with noise, extending prior theoretical frameworks.

Findings

Established ergodicity for three nonlinear mechanical models with noise.

Derived asymptotic formulas for threshold crossing probabilities.

Connected theoretical results to practical applications in engineering and science.

Abstract

The purpose of this paper is to prove ergodicity and provide asymptotic formulae for probabilities of threshold crossing related to smooth approximations of three fundamental nonlinear mechanical models: (a) an elasto-plastic oscillator, (b) an oscillator with dry friction, (c) an oscillator constrained by an obstacle (one sided or two sided) and subject to impacts, all three in presence of white or colored noise. Relying on a groundbreaking result on density estimates for degenerate diffusions by Delarue and Menozzi (2010), we identify Lyapunov functions that satisfy appropriate conditions leading to ergodicity (invariant measure and Poisson equation) and a functional central limit theorem. These conditions appear in the very fundamental works of Down, Meyn and Tweedie (1995) and Glynn and Meyn (1996). From an applied mathematics perspective, an important consequence is the access to asymptotic formulae for quantities of interest in engineering and science.