Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's

summary:We consider a stochastic process $X_t^x$ which solves an equation \[ {\mathrm d}X_t^x = AX_t^x\mathrm{d}t + \Phi {\mathrm d}B^H_t,\quad X_0^x = x \] where $A$ and $\Phi $ are real matrices and $B^H$ is a fractional Brownian motion with Hurst parameter $H \in (1/2,1)$. The Kolmogorov backward equation for the function $u(t,x) = \mathbb{E} f(X^x_t)$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established