Synchronizing noise-frame realization for a prescribed diffusion matrix

Determine whether, for a diffusion matrix H selected for favorable one-point sampling properties, there exists a noise frame σ satisfying σσ^{\mathsf T}=2H such that the associated synchronous stochastic flow is synchronizing.

Background

For a fixed diffusion matrix H, different factorizations σσ^{\mathsf T}=2H generate the same one-point Markov semigroup but generally produce different synchronous two-point motions. The paper investigates whether this remaining freedom can be used to obtain synchronization, which is important for variance reduction in Richardson–Romberg extrapolation.

The paper provides contrasting realizations for a one-dimensional double-well sampler: the principal square-root realization is not weakly synchronizing, whereas an overcomplete realization is synchronizing. These examples establish the possibility in that particular model but do not resolve the question for arbitrary diffusion matrices chosen for one-point sampling performance.