## 3 Prehistorical Description of the Optimal Fluctuations
### In the Stationary Cases
#### 3.1.2 For the Weak Noise Limit  
According to the large deviation theory, the stationary density \(\rho_{s}(\mathbf{x})\) is exponentially dominated by the quasi-potential. We have the following WKB-type approximation:  
\[\rho_{s}^{\varepsilon}(\mathbf{x})=\frac{C_{s}^{\varepsilon}(\mathbf{x})}{ \varepsilon^{n/2}}\exp\left(-\frac{V(\mathbf{x}_{eq},\mathbf{x})}{\varepsilon }\right),\;\mathbf{x}\in D \tag{31}\]  
where \(C_{s}^{\varepsilon}(\mathbf{x})=C_{s}(\mathbf{x})+o(\varepsilon)\). Substituting it into Equation (24), we get:  
\[\mathbf{b}\;(\mathbf{x})=-\mathbf{b}(\mathbf{x})-\mathbf{A}(\mathbf{x})\
abla V (\mathbf{x}_{eq},\mathbf{x})+\varepsilon\left(\frac{\mathbf{A}(\mathbf{x}) \
abla C_{s}(\mathbf{x})}{C_{s}(\mathbf{x})}+\
abla\cdot\mathbf{A}(\mathbf{x}) \right)+o(\varepsilon) \tag{32}\]  
For weak noise limit, the \(\varepsilon\)-independent part represents the average dynamics of \(y_{t}^{\varepsilon}\):  
\[\dot{\mathbf{y}}_{t}^{0}=-\mathbf{b}(\mathbf{y}_{t}^{0})-\mathbf{A}(\mathbf{ y}_{t}^{0})\
abla_{\mathbf{y}}V(\mathbf{x}_{eq},\mathbf{y}_{t}^{0}) \tag{33}\]  
Compared with Equation (19), we can see that this is exactly the time reversal of the optimal fluctuation field. According to the law of large numbers,  
\[\forall T>0,\forall\delta>0,\;\lim_{\varepsilon\to 0}\mathbf{x}_{\delta} \left(\max_{0\leq t\leq T}\left|\mathbf{y}_{t}^{\varepsilon}-\mathbf{y}_{t}^{0 }\right|>\delta\right)=0 \tag{34}\]  
As a result, \(\overline{p}(\mathbf{y},t|\mathbf{x}_{f})\) focuses on \(\mathbf{y}_{t}^{0}\) if \(\mathbf{y}^{0}(0)=\mathbf{x}_{f}\); that is, the OP \(\mathbf{x}_{op}(t)\) follows the peak trajectory of the PPD \(p_{h}(\mathbf{x},t|\mathbf{x}_{f})\). If the quasi-potential is smooth, for each point \(\mathbf{x}_{f}\), there exists a unique OP connecting it with the fixed point \(\mathbf{x}_{eq}\). This satisfies Equation (19); thus, the PPD has a single peak trajectory. This is a trivial situation. The main complexity comes from the case where the quasi-potential is non-differentiable. This is a frequent occurrence for systems far from equilibrium. Since the quasi-potential is a solution to the optimization problem (Equation (13)), the principle of variation is applied to transform it into the constrained Hamiltonian problem (Equation (18)). We can divide the domain \(D\) into three parts (\(D=D_{1}\cup D_{2}\cup D_{3}\)) as follows:  
1. We say that \(\mathbf{x}_{f}\in D_{1}\) if there exists a unique solution to the constrained Hamiltonian problem with \(\mathbf{x}(0)=\mathbf{x}_{f}\). This is the OP; thus, the PPD has a single peak trajectory. 2. We say that \(\mathbf{x}_{f}\in D_{2}\) if there exist multiple solutions to the constrained Hamiltonian problem with \(\mathbf{x}(0)=\mathbf{x}_{f}\) and, among them, there exists a unique one that has the minimum action. This is the OP. In this case, the PPD will have multiple peak trajectories. Each of them corresponds to one solution to the constrained Hamiltonian problem. As \(\varepsilon\to 0\), the exponential convergence rate (Equation (2)) highlights the OP, as it has the minimum action. As a result, the PPD possesses a unique highest peak trajectory located at the OP. 3. We say that \(\mathbf{x}_{f}\in D_{3}\) if there exist multiple solutions to the constrained Hamiltonian problem with \(\mathbf{x}(0)=\mathbf{x}_{f}\) and there exists more than one path with the same minimum action. This is the case where multiple OPs coexist, and each of them satisfies Equation (19). Let \(\mathbf{x}_{op1}\) and \(\mathbf{x}_{op2}\) denote two of the OPs. \((\mathbf{x}_{op1}(t),\
abla V(\mathbf{x}_{eq},\mathbf{x}_{op1}(t)))\) and \((\mathbf{x}_{op2}(t),\
abla V(\mathbf{x}_{eq},\mathbf{x}_{op2}(t)))\) are then the corresponding Hamiltonian trajectories. Since Equation (7) is a smooth vector field, these trajectories are also smooth. This suggests that non-differentiability does not occur along these paths except at the endpoint.