Applied Mathematics and Optimization

, Volume 58, Issue 1, pp 1–27 | Cite as

Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators



A stochastic variational inequality is proposed to model a white noise excited elasto-plastic oscillator. The solution of this inequality is essentially a continuous diffusion process for which a governing diffusion equation is obtained to study the evolution in time of its probability distribution. The diffusion equation is degenerate, but using the fact that the degeneracy occurs on a bounded region we are able to show the existence of a unique solution satisfying the desired properties. We prove the ergodic properties of the process and characterize the invariant measure. Our approach relies on extending Khasminskii’s method (Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980), which in the present context leads to the study of degenerate Dirichlet problems with nonlocal boundary conditions.


Random vibrations Elasto-plastic oscillators Ergodicity of degenerate diffusions 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.International Center for Decision and Risk Analysis, ICDRiA, School of ManagementUniversity of Texas at DallasRichardsonUSA
  2. 2.Programs in Mathematical SciencesUniversity of Texas at DallasRichardsonUSA

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