## 3 Prehistorical Description of the Optimal Fluctuations
### In the Nonstationary Cases
#### 3.2.2 For the Weak Noise Limit  
Note that the conditional probability \(p(x,t|x_{0})\) has the WKB approximation of the following form:  
\[p(\mathbf{x},t|\mathbf{x}_{0})=\frac{C(\mathbf{x},t,\mathbf{x}_{0},\epsilon) }{\epsilon^{n/2}}\exp\left(-\frac{V(\mathbf{x}_{0};t,\mathbf{x})}{\epsilon}\right) \tag{50}\]  
Substituting it into Equation (43), and expanding for \(\epsilon\ll 1\), we can determine the average dynamics of \(y_{t}^{\epsilon}\):  
\[\mathbf{y}_{t}^{0}=\bar{\mathbf{b}}^{0}(\mathbf{y}_{t}^{0},t)=-\mathbf{b}( \mathbf{y}_{t}^{0})-\mathbf{A}(\mathbf{y}_{t}^{0})\
abla_{\mathbf{y}}V( \mathbf{x}_{0},T-t,\mathbf{y}_{t}^{0}) \tag{51}\]  
After a time-reversed transformation \(t\to T-t\), this will have the same form as Equation (12). In this case, \(\overline{p}(\mathbf{y},t|\mathbf{x}_{f},0)\) will focus on \(\mathbf{y}_{t}^{0}\) if \(\mathbf{y}^{0}(0)=\mathbf{x}_{f}\). Consequently, the NOP will be achieved at the peak trajectory of the nonstationary PPD.  
If \(V(\mathbf{x}_{0};t,\mathbf{x})\) is a smooth function, there exists a unique path connecting \(\mathbf{x}_{0}\) with the prescribed point \(\mathbf{x}_{f}\); thus, the PPD has a single peak trajectory. However, what is similar to stationary cases is the troublesome non-differentiability. In nonstationary cases, this takes place more frequently--even in equilibrium systems. Here, we can also divide the space \(\mathbb{R}^{n}\times\mathbb{R}\) into three parts (\(F_{1}\cup F_{2}\cup F_{3}\)) according to the number of solutions to the constrained Hamiltonian problem (Equation (10)), as well as their value of the action (Equation (3)).  
For points \((\mathbf{x}_{f},T)\) located in these sets, the nonstationary PPD possesses a unique peak trajectory, a unique highest peak trajectory, and multiple highest peak trajectories with the same height. This is similar to the stationary situation, which will not be repeated here.  
In addition, it is also worth pointing out that the local deviation \(\zeta=\mathbf{x}-\mathbf{x}_{np}\) along the NOP is still Gaussian, i.e.,  
\[p_{h}(\mathbf{x}_{nop}+\zeta,t|\mathbf{x}_{f},T;\mathbf{x}_{0},0)=C\exp\left(- \frac{1}{2\epsilon}\epsilon^{T}\kappa(t)^{-1}\zeta\right),\ 0<t<T \tag{52}\]  
Here, the noise-independent dispersion parameter \(\kappa(t)\) is the finite-time solution of the time-varying Lyapunov equation:  
\[\dot{\mathbf{x}}=-\
abla_{\mathbf{x}}\bar{\mathbf{b}}^{0}\left(\mathbf{x}_{ nop}(t),T-t\right)\kappa-\kappa\left(\
abla_{\mathbf{x}}\bar{\mathbf{b}} \left(\mathbf{x}_{nop}(t),T-t\right)\right)^{T}+\tfrac{1}{2}\mathbf{A}\left( \mathbf{x}_{nop}(t)\right),\ 0<t<T \tag{53}\]  
Note that  
\[\underset{t\to 0}{\lim}p_{h}(\mathbf{x},t|\mathbf{x}_{f},T;\mathbf{x}_{0},0) =\delta(\mathbf{x}-\mathbf{x}_{0}) \tag{54}\] \[\underset{t\to T}{\lim}p_{h}(\mathbf{x},t|\mathbf{x}_{f},T; \mathbf{x}_{0},0) =\delta(\mathbf{x}-\mathbf{x}_{f}) \tag{55}\]  
If one of the initial conditions in Equation (53) is valid, the other will hold automatically.  
The results here show that the nonstationary PPD has a good property of being able to pinpoint the location of the NOP, and can also be applied to identify points where there are multiple coexisting NOPs. In particular, this negates the assertions made in [39], saying that the PPD is non-Gaussian and highly asymmetrical. To illustrate the above theoretical analysis more intuitively, a numerical algorithm and its application to a simple example, in this case, are presented in Section 4.