ON CONVERGENT RATES OF ERGODIC HARRIS CHAINS INDUCED FROM DIFFUSIONS

Abstract

We construct an irreducible ergodic Harris chain $\{ X_n \}$ from a diffusion $\{ S_t \}$ and barriers $\rho^{\pm}(x)$. We show that $\{ X_n \}$ is exponentially uniformly ergodic in the sense of the operator norm under the Banach space $C_{\beta}$, where $\beta \in (0,1)$. Moreover, the sizes of the convergent rates $\alpha_{X}(\beta)$ and $\alpha_{S}(\beta)$ measured by the operator norm are studied. We give an upper bound of $\alpha_{X}(\beta)$ in terms of $\rho^{\pm}(x)$. The Ornstein-Uhlenbeck process and proper $\rho^{\pm}(x)$ are taken to show $\alpha_{X}(\beta) \lt \alpha_{S}(\beta)$ for $0 \lt \beta \lt 0.5$.