 Geometric minimum action

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.