Ergodic theory for SDEs with extrinsic memory

Details Ergodic theory for SDEs with extrinsic memory Ergodic theory for SDEs with extrinsic memory

2006-03-28

arxiv.org/abs/math/0603658v2

Annals of Probability 2007, Vol. 35, No. 5, 1950-1977

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117906000001141 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

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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