Diffusive Scaling Limits of Forward Event-Chain Monte Carlo: Provably Efficient Exploration with Partial Refreshment
1 Collaboration
I’ve learned a lot from Luke. Especially, Proposition 2.1 is now what I’m proud of the most. At first, I was trying my best to come up with a general strategy to ensure invariance of PDMPs defined on manifolds, using the Riemannian gradient and geodesics. However, this is not how we are going to implement manifold PDMPs! We will most certainly prepare a suitable coordinate system to run ODEs in a desired manifold.
Moreover, old I would have complained that, Proposition 2.1 uses two difference quantities to describe one value.
2 A Geometric Account for PDMP
As soon as I started drafting an invariance proof for manifold PDMP & sticky manifold PDMP, I realised that the natural state space for PDMP was the tangent bundle and that there are loads of geometric ideas flowing out from this appreciation.
Especially, the reason of lifting the state space to the tangent or cotangent bundles is the availability of flows, or foliations induced from them. On \(T(M)\), we have geodesic flow induced from the metric, while on \(T^*(M)\) we have Hamiltonian flow.
Citation
@article{shiba2026,
author = {Shiba, Hirofumi and Kamatani, Kengo},
title = {Diffusive {Scaling} {Limits} of {Forward} {Event-Chain}
{Monte} {Carlo:} {Provably} {Efficient} {Exploration} with {Partial}
{Refreshment}},
journal = {Submitted to the Annals of Applied Probability},
date = {2026},
url = {https://arxiv.org/abs/2602.17087},
langid = {en}
}