In this paper, we extend the energy-Casimir stability method for deterministic Lie–Poisson Hamiltonian systems to provide sufficient conditions for stability in probability of stochastic dynamical systems with symmetries. We illustrate this theory with classical examples of coadjoint motion, including the rigid body, the heavy top, and the compressible Euler equation in two dimensions. The main result is that stable deterministic equilibria remain stable in probability up to a certain stopping time that depends on the amplitude of the noise for finite-dimensional systems and on the amplitude of the spatial derivative of the noise for infinite-dimensional systems.
Dans cet article, nous étendons la méthode d'énergie-Casimir de stabilité des systèmes déterministes hamiltoniens de Lie–Poisson afin de fournir des conditions suffisantes de stabilité en probabilité des systèmes dynamiques stochastiques par des symétries. Nous illustrons cette théorie par des exemples classiques de mouvements coadjoints, comme le corps solide, la toupie pesante et l'équation d'Euler compressible en deux dimensions. Le principal résultat de cette extension est que les équilibres relatifs déterministes stables restent stables en probabilité jusqu'à un certain temps d'arrêt. Ce dernier dépend, d'une part, de l'amplitude du bruit pour les systèmes de dimensions finies et, d'autre part, de l'amplitude de la dérivée spatiale du bruit pour les systèmes de dimensions infinies.
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Mot clés : Mécanique géométrique stochastique, Méthode d'énergie-Casimir, Stabilité stochastique
Alexis Arnaudon 1; Nader Ganaba 1; Darryl D. Holm 1
@article{CRMECA_2018__346_4_279_0,
author = {Alexis Arnaudon and Nader Ganaba and Darryl D. Holm},
title = {The stochastic {energy-Casimir} method},
journal = {Comptes Rendus. M\'ecanique},
pages = {279--290},
publisher = {Elsevier},
volume = {346},
number = {4},
year = {2018},
doi = {10.1016/j.crme.2018.01.003},
language = {en},
}
Alexis Arnaudon; Nader Ganaba; Darryl D. Holm. The stochastic energy-Casimir method. Comptes Rendus. Mécanique, Volume 346 (2018) no. 4, pp. 279-290. doi : 10.1016/j.crme.2018.01.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.01.003/
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