Exponential convergence of Langevin distributions and their discrete approximations

In this paper we consider a continuous-time method of approximating a given distribution using the Langevin di€usion dLtˆdWt‡ 1 2 r log (Lt)dt. We ®nd conditions under this di€usion converges exponentially quickly to or does not: in one dimension, these are essentially that for distributions with exponential tails of the form (x)/ exp (y |x| , 0< <1, exponential convergence occurs if and only if 1. We then consider conditions under which the discrete approximations to the di€usion converge. We ®rst show that even when the di€usion itself converges, naive discretizations need not do so. We then consider a `Metropolis-adjusted' version of the algorithm, and ®nd conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the di€usion does. We brie y discuss a truncated form of the algorithm which, in practice, should avoid the diculties of the other forms.