= Quasipotential
{title2=$V(x,y)$}
For a small-noise system with <Freidlin--Wentzell action> $I_T$, its quasipotential from $x$ to $y$ is the free-time minimum action
$$
V(x,y)=\inf_{T>0}\ \inf_{\substack{\phi(0)=x\\\phi(T)=y}}I_T(\phi).
$$
Only 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.<The geometric minimum action method: A least action principle on the space of curves by Matthias Heymann and Eric Vanden-Eijnden>[Section1.2, equation(1.9)]{ref}
For $dX=-aX\,dt+\sqrt\varepsilon\,dW$, $a>0$, the action from zero is $\frac12\int_0^T(\dot\phi+a\phi)^2dt$. Its minimum for endpoint $y$ is $ay^2/(1-e^{-2aT})$, obtained by minimizing the squared control driving $\dot\phi=-a\phi+u$. Taking $T\to\infty$ gives $V(0,y)=ay^2$. This shows how a free-time cost records the difficulty of leaving a stable equilibrium and why the noise normalization affects the numerical coefficient.
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