[Une approche analytique de la théorie ergodique dʼune inéquation variationnelle stochastique]
Dans un travail précédent du premier auteur en collaboration avec Janos Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), la solution dʼune inéquation variationnelle stochastique modélisant un oscillateur élastique-parfaitement-plastique a été étudiée. Lʼexistence et lʼunicité dʼune mesure invariante ont été prouvées. Des problèmes nonlocaux ont été introduits dans ce contexte. Le point clé est le lien entre des EDPs nonlocales et des EDPs locales qui peuvent être interprétées comme les cycles courts du processus de Markov solution de lʼinéquation variationnelle stochastique.
In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality.
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@article{CRMATH_2012__350_7-8_365_0,
author = {Bensoussan, Alain and Mertz, Laurent},
title = {An analytic approach to the ergodic theory of a stochastic variational inequality},
journal = {Comptes Rendus. Math\'ematique},
pages = {365--370},
publisher = {Elsevier},
volume = {350},
number = {7-8},
year = {2012},
doi = {10.1016/j.crma.2012.03.011},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.03.011/}
}
TY - JOUR AU - Bensoussan, Alain AU - Mertz, Laurent TI - An analytic approach to the ergodic theory of a stochastic variational inequality JO - Comptes Rendus. Mathématique PY - 2012 SP - 365 EP - 370 VL - 350 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.03.011/ DO - 10.1016/j.crma.2012.03.011 LA - en ID - CRMATH_2012__350_7-8_365_0 ER -
%0 Journal Article %A Bensoussan, Alain %A Mertz, Laurent %T An analytic approach to the ergodic theory of a stochastic variational inequality %J Comptes Rendus. Mathématique %D 2012 %P 365-370 %V 350 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.03.011/ %R 10.1016/j.crma.2012.03.011 %G en %F CRMATH_2012__350_7-8_365_0
Bensoussan, Alain; Mertz, Laurent. An analytic approach to the ergodic theory of a stochastic variational inequality. Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 365-370. doi : 10.1016/j.crma.2012.03.011. http://www.numdam.org/articles/10.1016/j.crma.2012.03.011/
[1] Contrôle impulsionnel et inéquations quasi variationnelles, Dunod, Paris, 1982
[2] Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators, Applied Mathematics and Optimization, Volume 58 (2008) no. 1, pp. 1-27
[3] On a class of partial differential equations with nonlocal Dirichlet boundary conditions, Applied and Numerical Partial Differential Equations, Volume 15 (2010), pp. 9-23
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☆ This research was partially supported by the ANRT, a grant from CEA, Commissariat à lʼénergie atomique and by the National Science Foundation under grant DMS-0705247. A large part of this work was completed while one of the authors was visiting the University of Texas at Dallas and the Hong Kong Polytechnic University. We wish to thank warmly these institutions for the hospitality and support.
