Gradient Flow from a Random Walk in Hilbert Space

2011/08/06 by Natesh S. Pillai, Andrew M. Stuart, Pillai, Natesh S. +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1108.1494

openalex publication_date 2011/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

Citations

• Asymptotics of the spectral gap with applications to the theory of simulated annealing
• Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm

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