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

Journal Article · Fri Aug 15 04:00:00 EDT 2008 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-007-9027-4· OSTI ID:21241989

Bensoussan, Alain ^[1]; Turi, Janos ^[2]

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1. University of Texas at Dallas, International Center for Decision and Risk Analysis, ICDRiA, School of Management (United States)
2. University of Texas at Dallas, Programs in Mathematical Sciences (United States)

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.

• OSTI ID: 21241989
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 1 Vol. 58; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
99 GENERAL AND MISCELLANEOUS
BOUNDARY CONDITIONS
DIFFUSION EQUATIONS
DIRICHLET PROBLEM
MATHEMATICAL EVOLUTION
MATHEMATICAL SOLUTIONS
MEASURE THEORY
NOISE
OSCILLATORS
PROBABILITY
STOCHASTIC PROCESSES
VARIATIONAL METHODS