# Diffusion limits of the random walk Metropolis algorithm in high dimensions - Mathematics > Probability

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Abstract: Diffusion limits of MCMC methods in high dimensions provide a usefultheoretical tool for studying computational complexity. In particular, theylead directly to precise estimates of the number of steps required to explorethe target measure, in stationarity, as a function of the dimension of thestate space. However, to date such results have mainly been proved for targetmeasures with a product structure, severely limiting their applicability. Thepurpose of this paper is to study diffusion limits for a class of naturallyoccurring high-dimensional measures found from the approximation of measures ona Hilbert space which are absolutely continuous with respect to a Gaussianreference measure. The diffusion limit of a random walk Metropolis algorithm toan infinite-dimensional Hilbert space valued SDE or SPDE is proved,facilitating understanding of the computational complexity of the algorithm.

Author: ** Jonathan C. Mattingly, Natesh S. Pillai, Andrew M. Stuart**

Source: https://arxiv.org/