Ergodicity of stochastic differential equations driven by fractional Brownian motion

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.