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

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