Degenerate Dirichlet Problems Related to the Ergodic Property of an Elasto-Plastic Oscillator Excited by a Filtered White Noise

TL;DR

This paper models an elasto-plastic oscillator driven by filtered white noise using a stochastic variational inequality, proving its ergodic properties and characterizing the invariant measure through advanced degenerate Dirichlet problems.

Contribution

It extends previous methods to higher dimensions, addressing complex degenerate PDEs with nonlocal boundary conditions for the first time.

Findings

Proves ergodicity of the modeled process

Characterizes the invariant measure explicitly

Extends existing methods to two-dimensional PDEs

Abstract

A stochastic variational inequality is proposed to model an elasto-plastic oscillator excited by a filtered white noise. We prove the ergodic properties of the process and characterize the corresponding invariant measure. This extends Bensoussan-Turi's method (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008) 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 (Stochastic Stability of Differential Equations, Sijthoff and Noordhof,1980) leads to the study of degenerate Dirichlet problems with partial differential equations and nonlocal boundary conditions.