Ergodicity of hypoelliptic SDEs driven by fractional brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, p. 601-628

We demonstrate that stochastic differential equations (SDEs) driven by fractional brownian motion with Hurst parameter H>½ have similar ergodic properties as SDEs driven by standard brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander's condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition they admit a unique stationary solution that is physical in the sense that it does not “look into the future.” The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.

Nous démontrons que les équations différentielles stochastiques (EDS) conduites par des mouvements browniens fractionnaires à paramètre de Hurst H>½ ont des propriétés ergodiques similaires aux EDS usuelles conduites par des mouvements Browniens. L'intérêt principal du présent article est de pouvoir traiter également des systèmes hypoelliptiques satisfaisant la condition de Hörmander. Nous montrons qu'une version adéquate de la propriété de Feller forte est vérifiée par de tels systèmes et nous en déduisons que, sous une propriété de contrôlabilité usuelle, ils admettent une unique solution stationnaire qui ait un sens physique. L'ingrédient principal de notre analyse est une borne supérieure sur les moments inverses de la matrice de Malliavin associée, conditionnée au passé du bruit.

DOI : https://doi.org/10.1214/10-AIHP377
Classification:  60H10,  60G10,  60H07,  26A33
Keywords: ergodicity, fractional brownian motion, Hörmander's theorem
@article{AIHPB_2011__47_2_601_0,
     author = {Hairer, Martin and Pillai, N. S.},
     title = {Ergodicity of hypoelliptic SDEs driven by fractional brownian motion},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     pages = {601-628},
     doi = {10.1214/10-AIHP377},
     zbl = {1221.60083},
     mrnumber = {2814425},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_2_601_0}
}
Hairer, M.; Pillai, N. S. Ergodicity of hypoelliptic SDEs driven by fractional brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, pp. 601-628. doi : 10.1214/10-AIHP377. http://www.numdam.org/item/AIHPB_2011__47_2_601_0/

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