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Hairer, M.; Pillai, N. S.

Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. (English. French summary) Zbl 1221.60083

Ann. Inst. Henri Poincaré, Probab. Stat. 47, No. 2, 601-628 (2011).
From the authors’ abstract: We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter \(H>\frac12 \) 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.
Reviewer: Nikolaos Halidias (Athens)

Cited in 35 Documents

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G10 Stationary stochastic processes
60H07 Stochastic calculus of variations and the Malliavin calculus
26A33 Fractional derivatives and integrals

Keywords:

ergodicity; fractional Brownian motion; Hörmander’s theorem
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\textit{M. Hairer} and \textit{N. S. Pillai}, Ann. Inst. Henri Poincaré, Probab. Stat. 47, No. 2, 601--628 (2011; Zbl 1221.60083)
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