Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2（π）-space, denoted by gap（P）, is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap（P） and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap（P）. When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap（P）. These results are extended to L^∞ （π）-space.
In the paper, Harnack inequality and derivative formula are established for stochastic differential equation driven by fractional Brownian motion with Hurst parameter H < 1/2. As applications, strong Feller property, log-Harnack inequality and entropy-cost inequality are given.
