  Consider a probability measure on a Hilbert space defined via its density
with respect to a Gaussian. The purpose of this paper is to demonstrate that an
appropriately defined Markov chain, which is reversible with respect to the
measure in question, exhibits a diffusion limit to a noisy gradient flow, also
reversible with respect to the same measure. The Markov chain is defined by
applying a Metropolis-Hastings accept-reject mechanism to an Ornstein-Uhlenbeck
proposal which is itself reversible with respect to the underlying Gaussian
measure. The resulting noisy gradient flow is a stochastic partial differential
equation driven by a Wiener process with spatial correlation given by the
underlying Gaussian structure.
