Quasipotential ($V(x,y)$)

For a small-noise system with Freidlin--Wentzell action $I_T$, its quasipotential from $x$ to $y$ is the free-time minimum 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.