The Annals of Probability

Ergodic theory for SDEs with extrinsic memory

M. Hairer and A. Ohashi

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Abstract

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.

Article information

Source
Ann. Probab., Volume 35, Number 5 (2007), 1950-1977.

Dates
First available in Project Euclid: 5 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1189000933

Digital Object Identifier
doi:10.1214/009117906000001141

Mathematical Reviews number (MathSciNet)
MR2349580

Zentralblatt MATH identifier
1129.60052

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G10: Stationary processes 26A33: Fractional derivatives and integrals

Keywords
Non-Markovian processes ergodicity fractional Brownian motion

Citation

Hairer, M.; Ohashi, A. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007), no. 5, 1950--1977. doi:10.1214/009117906000001141. https://projecteuclid.org/euclid.aop/1189000933


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