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September 2007 Ergodic theory for SDEs with extrinsic memory
M. Hairer, A. Ohashi
Ann. Probab. 35(5): 1950-1977 (September 2007). DOI: 10.1214/009117906000001141


We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob–Khas’minskii theorem.

The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition on the noise, such equations admit a unique adapted stationary solution.


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M. Hairer. A. Ohashi. "Ergodic theory for SDEs with extrinsic memory." Ann. Probab. 35 (5) 1950 - 1977, September 2007.


Published: September 2007
First available in Project Euclid: 5 September 2007

zbMATH: 1129.60052
MathSciNet: MR2349580
Digital Object Identifier: 10.1214/009117906000001141

Primary: 26A33 , 60G10 , 60H10

Keywords: ergodicity , fractional Brownian motion , non-Markovian processes

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 5 • September 2007
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