Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations*

We describe algorithms for estimating a given measure p known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which p is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to p, and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at veri®able rates. We then consider methods of discretizing the diffusion in time, and ®nd methods which inherit the convergence rates of the continuous time process. These contrast with the behavior of the naive or Euler discretization, which can behave badly even in simple cases. Our results are described in detail in one dimension only, although extensions to higher dimensions are also brie ̄y described.