THEORY-1309: For a small-noise diffusion the Freidlin–Wentzell quasi-potential equals 2 inf ∫ |b| sin²(θ/2) ds over curves, θ the angle between curve and drift, and the minimizing curve is the path a noise-driven transition follows with probability tending to one; with gradient drift and identity noise that curve runs against the drift, the time-reversed relaxation path, while a far-from-gradient drift, as in the Maier–Stein model above β = 4, makes it depart from the heteroclinic orbit

Source

Heymann and Vanden-Eijnden (2008), LIT-1981; read in NOTE-1504.

What was actually shown

• The formula (eqs. 1.16–1.18): from ½|ψ̇ − b|²_a ≥ |ψ̇|_a|b|_a − ⟨ψ̇, b⟩_a, with equality at |ψ̇| = |b|, V(x₁, x₂) = 2 inf_γ ∫ |b|_a sin²(θ/2) ds. Re-derived.
• Concentration (Proposition 2.3): given a transition, the process stays in any tube around the unique minimizing curve with probability → 1 as ε → 0.
• Gradient case (p. 1058): with b = −∇U and σ = I, “transition paths are always parallel to the drift”. Re-derived: along an uphill curve ∫ |∇U|(1 − cos θ) ds ≥ ΔU, with equality only when the curve runs against the drift everywhere, so V to the saddle is 2ΔU and the path is the reversed relaxation path.
• Maier–Stein (§4.1): gradient only at β = 1, yet the minimum action path is the heteroclinic orbit for β ≤ 4; above 4 it is not.

What this does not say

• That the path is distinguishable at finite noise from one sample.
• That any nongradient drift changes the path: in Maier–Stein it takes β > 4.
• That it applies to degenerate noise (Assumption 3 requires ellipticity).