The Annals of Probability

Ergodicity of stochastic differential equations driven by fractional Brownian motion

Martin Hairer

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Abstract

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

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

Dates
First available in Project Euclid: 3 March 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1109868598

Digital Object Identifier
doi:10.1214/009117904000000892

Mathematical Reviews number (MathSciNet)
MR2123208

Zentralblatt MATH identifier
1071.60045

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

Keywords
Ergodicity fractional Brownian motion memory

Citation

Hairer, Martin. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 (2005), no. 2, 703--758. doi:10.1214/009117904000000892. https://projecteuclid.org/euclid.aop/1109868598


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