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]

For a small-noise system with Freidlin--Wentzell action $I_T$, its quasipotential from $x$ to $y$ is the free-time minimum actionOnly admissible paths enter the inner infimum. The optimization over duration is essential: a finite-time action and a quasipotential answer different questions. A minimum need not be attained at a finite duration.^[ref]

For $dX=-aXdt+\sqrt{\epsilon }dW$, $a>0$, the action from zero is $\frac{1}{2}{\int }_0^T(\dot{\varphi }+a\varphi {)}^{2}dt$. Its minimum for endpoint $y$ is $a{y}^2/(1-e^{-2aT})$, obtained by minimizing the squared control driving $\dot{\varphi }=-a\varphi +u$. Taking $T\to \infty$ gives $V(0,y)=a{y}^2$. This shows how a free-time cost records the difficulty of leaving a stable equilibrium and why the noise normalization affects the numerical coefficient.

Geometric minimum action removes the time parametrization when minimizing the Freidlin--Wentzell action over both a path and its travel time. With a positive-definite diffusion covariance $D(\varphi )$, write $⟨u,v{⟩}_D=u^T{D}^{-1}v$ and $\Vert u{\Vert }_D=\sqrt{⟨u,u{⟩}_D}$. For an absolutely continuous curve $\varphi (s)$, the free-time geometric cost isIt is invariant under increasing changes of the curve parameter. This is a free-time cost used in the quasipotential, not a replacement for the action at a prescribed finite arrival time.^[ref]

To derive the integrand, set $r=dt/ds>0$ in the time action. Its density becomesMinimizing over $r$ uses $A/r+rB\ge 2\sqrt{AB}$ and gives the geometric expression. Where both norms are positive, the optimal speed satisfies $r=\Vert {\varphi }^{\prime }{\Vert }_D/\Vert a{\Vert }_D$. At equilibria the infimum may require an infinite travel time, explaining the usefulness of an arc-length-like parameter.

For gradient drift $a=-\nabla U$ and $D=I$, an uphill gradient curve has cost $2[U(\mathrm{end})-U(\mathrm{start})]$, while a downhill deterministic trajectory has zero cost. The factor two corresponds to noise $\sqrt{\epsilon }dW$; changing to $\sqrt{2\epsilon }dW$ changes the action normalization.