Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators |
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Authors: | Alain Bensoussan Janos Turi |
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Institution: | (1) International Center for Decision and Risk Analysis, ICDRiA, School of Management, University of Texas at Dallas, Richardson, TX 75083, USA;(2) Programs in Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75083, USA |
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Abstract: | 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.
This research was partially supported by a grant from CEA, Commissariat à l’énergie atomique and by the National Science Foundation
under grant DMS-0705247. |
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Keywords: | Random vibrations Elasto-plastic oscillators Ergodicity of degenerate diffusions |
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