Sampling conditioned hypoelliptic diffusions

TL;DR

This paper extends the construction of invariant SPDEs from elliptic to hypoelliptic diffusions, enabling modeling of more complex physical systems with non-gradient drifts and bridge conditioning.

Contribution

It introduces a novel methodology for constructing fourth-order invariant SPDEs for hypoelliptic diffusions, broadening applicability beyond elliptic cases.

Findings

Constructed invariant SPDEs for hypoelliptic diffusions with bridge conditioning.

Extended the framework to include non-gradient drifts.

Applied to model transitions in physical systems with friction and noise.

Abstract

A series of recent articles introduced a method to construct stochastic partial differential equations (SPDEs) which are invariant with respect to the distribution of a given conditioned diffusion. These works are restricted to the case of elliptic diffusions where the drift has a gradient structure and the resulting SPDE is of second-order parabolic type. The present article extends this methodology to allow the construction of SPDEs which are invariant with respect to the distribution of a class of hypoelliptic diffusion processes, subject to a bridge conditioning, leading to SPDEs which are of fourth-order parabolic type. This allows the treatment of more realistic physical models, for example, one can use the resulting SPDE to study transitions between meta-stable states in mechanical systems with friction and noise. In this situation the restriction of the drift being a gradient can also be lifted.