Noisy gradient flow from a random walk in hilbert space

Abstract

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 (Tierney, Ann Appl Probab 8:1–9, 1998) to an Ornstein– Uhlenbeck (OU) 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. There are two primary motivations for this work. The first concerns insight into Monte Carlo Markov Chain (MCMC) methods for sampling of measures on a Hilbert space defined via a density with respect to a Gaussian measure. These measures must be approximated on finite dimensional spaces of dimension N in order to be sampled. A conclusion of the work herein is that MCMC methods based on prior-reversible OU proposals will explore the target measure in O(1) steps with respect to dimension N. This is to be contrasted with standard MCMC methods based on the random walk or Langevin proposals which require O(N) and O(N1/3 ) steps respectively (Mattingly et al., Ann Appl Prob 2011; Pillai et al., Ann Appl Prob 22:2320–2356.