Freidlin--Wentzell action

For a small-noise diffusion

d X = a (X) d t + ϵ σ (X) d W

, the Freidlin--Wentzell action of a smooth path is

I [ϕ] = \frac{1}{2} \int (\dot{ϕ} - a)^{T} (σ σ^{T})^{- 1} (\dot{ϕ} - a) d t .

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- Geometric minimum action Freidlin--Wentzell action

Geometric minimum action

On a zero-Hamiltonian infinite-time fluctuation path, the Freidlin--Wentzell action can be parametrized independently of time as

I = \int ∥ a ∥ d s - a \cdot d ϕ

in the diffusion metric.

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