The Annals of Probability

Ergodicity of stochastic differential equations driven by fractional Brownian motion

Martin Hairer

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We study the ergodic properties of finite-dimensional systems of SDEs driven by nondegenerate additive fractional Brownian motion with arbitrary Hurst parameter H∈(0,1). A general framework is constructed to make precise the notions of “invariant measure” and “stationary state” for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solution toward the stationary one is (at least) algebraic. A lower bound on the exponent is also given.

Article information

Ann. Probab. Volume 33, Number 2 (2005), 703-758.

First available in Project Euclid: 3 March 2005

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Zentralblatt MATH identifier

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

Ergodicity fractional Brownian motion memory


Hairer, Martin. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 (2005), no. 2, 703--758. doi:10.1214/009117904000000892.

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