  Diffusion limits of MCMC methods in high dimensions provide a useful
theoretical tool for studying computational complexity. In particular, they
lead directly to precise estimates of the number of steps required to explore
the target measure, in stationarity, as a function of the dimension of the
state space. However, to date such results have mainly been proved for target
measures with a product structure, severely limiting their applicability. The
purpose of this paper is to study diffusion limits for a class of naturally
occurring high-dimensional measures found from the approximation of measures on
a Hilbert space which are absolutely continuous with respect to a Gaussian
reference measure. The diffusion limit of a random walk Metropolis algorithm to
an infinite-dimensional Hilbert space valued SDE (or SPDE) is proved,
facilitating understanding of the computational complexity of the algorithm.
