An Eyring--Kramers Law for the Hypoelliptic Third-Order Langevin Diffusion

We prove an Eyring--Kramers law for metastable transitions of the hypoelliptic third-order Langevin diffusion in the low-temperature limit.

The model is motivated by the high-order Langevin diffusion of Mou et al.

(2021, J.

Mach.

Learn.

Res., 22(42), 1--41).

The model has a Gibbs invariant measure, but the noise reaches the position variable only through a step-three chain.

For a double-well potential with a unique index-one transition saddle, we determine both the Arrhenius exponential scale and the sharp prefactor of the mean transition time.

The prefactor is governed by the unique positive unstable rate of the third-order deterministic linearization at the saddle.

Our proof extends the underdamped Eyring--Kramers strategy of Lee--Ramil--Seo (2026, arXiv:2503.12610v2) to the step-three setting and combines the weak-capacity framework with a saddle-adapted boundary layer, an explicit Gaussian current calculation, committor localization, and intrawell flatness.

Under matched kinetic normalizations, the resulting metastable prefactor is strictly smaller than its underdamped counterpart.

A numerical experiment for a one-dimensional double well illustrates both the Arrhenius scaling and the predicted prefactor comparison.