Ergodicity of stochastic differential equations driven by fractional Brownian motion

Abstract

We study the ergodic properties of finite-dimensional syste ms of SDEs driven by non-degenerate additive fractional Brownian motion with arbitrary Hurst parameter H 2 (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 solut ion towards the stationary one is (at least) algebraic. A lower bound on the exponent is also given.