 Freidlin--Wentzell action

For a concrete nondegenerate small-noise model, letwhere $a$ is globally 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 $1/\epsilon$. Their Freidlin--Wentzell action isfor absolutely continuous paths with $\varphi (0)=x_0$ 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 $\dot{\varphi }=a(\varphi )$. 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 $(y-x_0{)}^2/(2{\sigma }^{2}T)$, by the Cauchy-Schwarz inequality. The noiseless solution has action zero. Neighborhood probabilities have leading exponential cost $I_T/\epsilon$, 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 $[\sigma (\varphi )\sigma (\varphi {)}^T{]}^{-1}$; regularity and nondegeneracy assumptions must be checked rather than using an inverse for a singular covariance.^[ref]