Gradient Flow from a Random Walk in Hilbert Space

TL;DR

This paper demonstrates that a specific Markov chain constructed via Metropolis-Hastings on a Hilbert space converges to a stochastic gradient flow described by a stochastic PDE, revealing deep connections between Markov processes and gradient flows in infinite-dimensional spaces.

Contribution

It introduces a novel diffusion limit for a Markov chain on a Hilbert space, linking it to a stochastic gradient flow driven by a Wiener process.

Findings

Markov chain exhibits a diffusion limit to a noisy gradient flow

The resulting stochastic PDE is driven by a Wiener process with Gaussian spatial correlation

Reversibility is preserved in the limiting gradient flow

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.