  The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm
which makes local moves by incorporating information about the gradient of the
logarithm of the target density. In this paper we study the efficiency of MALA
on a natural class of target measures supported on an infinite dimensional
Hilbert space. These natural measures have density with respect to a Gaussian
random field measure and arise in many applications such as Bayesian
nonparametric statistics and the theory of conditioned diffusions. We prove
that, started in stationarity, a suitably interpolated and scaled version of
the Markov chain corresponding to MALA converges to an infinite dimensional
diffusion process. Our results imply that, in stationarity, the MALA algorithm
applied to an N-dimensional approximation of the target will take
$\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing
favorably with the Random Walk Metropolis which was recently shown to require
$\mathcal{O}(N)$ steps when applied to the same class of problems.
