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Langevin diffusions and the Metropolis-adjusted Langevin algorithm

We provide a clarification of the description of Langevin diffusions on Riemannian manifolds and of the measure underlying the invariant density. As a result we propose a new position-dependent Metropolis-adjusted Langevin algorithm (MALA) based upon a Langevin diffusion in $\mathbb{R}^d$ which has the required invariant density with respect to Lebesgue measure. We show that our diffusion and the diffusion upon which a previously-proposed position-dependent MALA is based are equivalent in some cases but are distinct in general. A simulation study illustrates the gain in efficiency provided by the new position-dependent MALA.