= Freidlin--Wentzell action
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For a concrete nondegenerate small-noise model, let
$$
dX_t^\varepsilon=a(X_t^\varepsilon)dt+\sqrt\varepsilon\,\Sigma\,dW_t,\qquad X_0^\varepsilon=x_0,
$$
where $a$ is globally <Lipschitz continuity>[Lipschitz], $\Sigma$ is a constant invertible <matrix> and the time horizon $T$ is finite. The path laws on $C([0,T],\mathbb R^d)$ satisfy a <large deviation principle> of speed $1/\varepsilon$. Their <Freidlin--Wentzell action> is
$$
I_T(\phi)=\frac12\int_0^T\|\Sigma^{-1}(\dot\phi(t)-a(\phi(t)))\|^2dt
$$
for absolutely continuous paths with $\phi(0)=x_0$ and finite displayed <integral>, and $+\infty$ otherwise. Thus the initial condition and admissible path class are part of the rate, not optional qualifications.
The action measures the squared control needed to move the deterministic trajectory away from $\dot\phi=a(\phi)$. For zero drift and constant scalar noise coefficient $\sigma>0$, the least-action path from $x_0$ to $y$ in time $T$ is the straight line, with cost $(y-x_0)^2/(2\sigma^2T)$, by the <Cauchy-Schwarz inequality>. The noiseless solution has action zero. Neighborhood <probabilities> have leading exponential cost $I_T/\varepsilon$, in the precise open/closed-set sense of the <large deviation principle>.
For constant $\Sigma$, continuity of the additive-noise solution map under the uniform <norm> follows from Gronwall's inequality. The <Schilder theorem> for scaled <Brownian motion> and the <contraction principle for large deviations> explain the path-space rate. With state-dependent diffusion, the integrand uses the inverse <covariance> <matrix> $[\sigma(\phi)\sigma(\phi)^T]^{-1}$; regularity and nondegeneracy assumptions must be checked rather than using an inverse for a singular <covariance>.<The geometric minimum action method: A least action principle on the space of curves by Matthias Heymann and Eric Vanden-Eijnden>[Section 1.1, equations (1.1)–(1.4) and small-noise convention (1.3)]{ref}
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