Degenerate Dirichlet problems related to the ergodic property of an elasto-plastic oscillator excited by a filtered white noise

A stochastic variational inequality is proposed to model an elasto-plastic oscillator excited by a filtered white noise. We prove the ergodic property of the process and characterize the corresponding invariant measure. This extends Bensoussan-Turi's method (2008, Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Appl. Math. Optim., 58, 1-27) with a significant additional difficulty of increasing the dimension. Two points boundary value problem in dimension 1 is replaced by elliptic equations in dimension 2. In the present context, Khasminskii's method (1980, Stochastic Stability of Differential Equations. The Netherlands: Sijthoff and Noordhof) leads to the study of degenerate Dirichlet problems with partial differential equations and non-local boundary conditions.