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.”
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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.
@article{1300887284,
author = {Hairer, M. and Pillai, N. S.},
title = {Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {47},
number = {1},
year = {2011},
pages = { 601-628},
language = {en},
url = {http://dml.mathdoc.fr/item/1300887284}
}
Hairer, M.; Pillai, N. S. Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist., Tome 47 (2011) no. 1, pp. 601-628. http://gdmltest.u-ga.fr/item/1300887284/