## Appendix A Numerical Solutions of the Optimization Problems in Equations (9) and (13)  
According to the gradient descent method, in order to obtain the minimizer of a scalar function \(U(\mathbf{x})\), one needs only to let the variables evolve in the opposite direction of the gradient field, i.e.,  
\[\dot{\mathbf{x}}=-\
abla U(\mathbf{x}) \tag{10}\]  
The minimum action method [47] is essentially a gradient descent method to obtain the minimizer of the optimization problem:  
\[V(\mathbf{x}_{0};T,\mathbf{x}_{f})=\min_{\mathbf{x}(0)=\mathbf{x}_{0}, \mathbf{x}(T)=\mathbf{x}_{f}}S[\mathbf{x}(s)] \tag{11}\]  
That is, one needs only to solve the following partial differential equation:  
\[\frac{\partial\mathbf{x}(\tau,t)}{\partial\tau}=-\frac{\delta S[\mathbf{x}( \cdot)]}{\delta\mathbf{x}}(t)=-\left[\frac{\mathrm{d}L(\mathbf{x},\dot{ \mathbf{x}})}{\mathrm{d}\mathbf{x}}-\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\mathrm{d}L(\mathbf{x},\dot{\mathbf{x}})}{\mathrm{d}\dot{\mathbf{x}}} \right)\right]\text{, }0<t<T\text{, }\tau>0\text{,} \tag{12}\]  
With the constraints \(\mathbf{x}(0)=\mathbf{x}_{0},\mathbf{x}(T)=\mathbf{x}_{f}\).  
Specifically, we can discretize the time interval \([0,T]\) into \(M\) steps of uniform sizes \(\Delta t=T/M\). If \(\mathbf{x}_{m}^{k}=\mathbf{x}(k\Delta\tau,m\Delta t)\), the optimization problem can now be solved in an iterative manner.  
Starting from the \(k\)th guess \(\mathbf{x}_{m}^{k}\), compute a descent direction \(\mathbf{d}_{m}^{k}=-\left.\frac{\delta\mathcal{S}[\mathbf{x}(\cdot)]}{\partial \mathbf{x}}\right|_{\mathbf{x}=\mathbf{x}_{m}^{k}}\);  
Descend along the descent direction \(\mathbf{x}_{m}^{k+1}=\mathbf{x}_{m}^{k}+\Delta\tau\mathbf{d}_{m}^{k}\) to obtain the next guess \(\mathbf{x}_{m}^{k+1}\);  
Finally, iterate until convergence.  
As we can see, the algorithm here is fundamentally different from that described in Section 4. The algorithm here is based on the gradient descent method to solve the optimization problem, while the algorithm in Section 4 is designed to display the peak trajectory of the nonstationary PPD. The numerical results for distinct noise intensity and time gap are presented in Figures 3 and 4. It can be seen that the peak trajectory of the nonstationary PPD coincides with the NOP only if the noise intensity is small enough, justifying our theoretical analysis performed in Section 3.2.  
In addition, it is notable that the gradient descent method can also be applied to calculate the OP. Since additional minimization over the transition time is required, we first need to transform the optimization problem in Equation (13) to a form similar to Equation (9). Based on the Maupertuis principle,  
\[\begin{array}{ll}V(\mathbf{x}_{q;}\mathbf{x}_{f})&=\inf_{T>\mathbf{x}(0)= \mathbf{x}_{q;}\mathbf{x}(T)=\mathbf{x}_{f}}S[\mathbf{x}(\cdot)]\\ &=\min_{\mathbf{x}(0)=\mathbf{x}_{q;}\mathbf{x}(T)=\mathbf{x}_{f},\theta:H( \mathbf{x},\theta)=0}\int_{0}^{\mathbf{x}_{q}}\mathbf{x}_{q}^{T}\,\theta^{T} \mathbf{dx}\\ &=\max_{\mathbf{x}(0)=\mathbf{x}_{q;}\mathbf{x}(1)=\mathbf{x}_{f}}\sup_{ \mathbf{x}(H,\mathbf{x}_{f})=0}\int_{0}^{1}\theta^{T}\mathbf{x}^{\prime} \mathbf{ds}\\ &=\max_{\mathbf{x}(0)=\mathbf{x}_{q;}\mathbf{x}(1)=\mathbf{x}_{f}}\hat{S}[ \mathbf{x}(\cdot)]\end{array} \tag{16}\]  
where \(\hat{S}[\mathbf{x}(\cdot)]\) represents the geometric minimum action. As we can see, the action is invariant under reparametrization. Thus, it enables us to treat the infinite time--as displayed in the constraints in Equation (18)--with finite discretization points. The above algorithm can then be used to calculate the OP, which is known as the geometric minimum action method (gMAM). Since only the NOP is plotted in this paper, we do not repeat the gMAM here. One can refer to [48] for further details.