RATE OF CONVERGENCE TO EQUILIBRIUM OF FRACTIONAL DRIVEN STOCHASTIC DIFFERENTIAL EQUATIONS WITH ROUGH MULTIPLICATIVE NOISE

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter H is an element of (1/3, 1) and multiplicative noise component sigma. When sigma is constant and for every H is an element of (0, 1), it was proved in [Ann. Probab. 33 (2005) 703-758] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order t(-alpha) where alpha is an element of (0, 1) (depending on H). In [Ann. Inst. Henri Poincare Probab. Stat. 53 (2017) 503-538], this result has been extended to the multiplicative case when H > 1/2. In this paper, we obtain these types of results in the rough setting H is an element of (1/3, 1/2). Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [Ann. Inst. Henri Poincare Probab. Stat. 53 (2017) 503-538] by deleting the gradient assumption on the noise coefficient sigma. The main theorems include some existence and uniqueness results for the invariant distribution.