Penalization of nonsmooth dynamical systems with noise: Ergodicity and asymptotic formulae for threshold crossings probabilities

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, and (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 [J. Funct. Anal., 259 (2010), pp. 1577-1630], 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 [Ann. Probab., 23 (1995), pp. 1671-1691] and Glynn and Meyn [Ann. Probab., 24 (1996), pp. 916-931]. From an applied mathematics perspective, an important consequence is the access to asymptotic formulae for quantities of interest in engineering and science.