Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

M. Hairer and N. S. Pillai

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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 traîter 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 controllabilité 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.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 47, Number 2 (2011), 601-628.

First available in Project Euclid: 23 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G10: Stationary processes 60H07: Stochastic calculus of variations and the Malliavin calculus 26A33: Fractional derivatives and integrals

Ergodicity Fractional Brownian motion Hörmander’s theorem


Hairer, M.; Pillai, N. S. Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 601--628. doi:10.1214/10-AIHP377.

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