## 4 Numerical Algorithm and Results
### Algorithm  
For simplicity, we assume the dimension \(n=1\). In order to track the temporal evolution of the law of the stochastic variable \(x_{t}^{x}\) numerically, the discretization of both the state space and time is necessary, which is formulated as follows:  
First, we assume that a domain \(E=[a,b]\) containing points \(x_{0}\) and \(x_{f}\) could be chosen such that the vector field \(\mathbf{b}(\mathbf{x})\) at the boundary \(\partial E\) points towards its interior; meanwhile:  
\[\min_{x\in\partial E,0\leq t\leq T}V(x_{0};t,x)>V(x_{f};T,x_{0}) \tag{92}\]  
It follows that, for vanishing noise, almost all trajectories will stay in this domain over the interval \([0,T]\), and the event of escaping from \(E\) becomes more unlikely than that of transition between \(x_{0}\) and \(x_{f}\). In other words, the NOP connecting \(x_{0}\) and \(x_{f}\) is wholly contained in the domain \(E\), and neglecting all of the trajectories escaping before the moment \(T\) has almost no effect on the calculation of the NOP. Thus, focusing on the trajectories not having left from the domain \(E\) up to time \(T\) will be precise enough.  
Next, we divide the domain \(E\) into \(N\) subintervals with uniform sizes \(\Delta x=(b-a)/N\). The \(i\)-th cell is defined as the interval \([a+(i-1)\Delta x,a+i\Delta x]\). We choose a representative element in each cell as \(x(i)=a+(i-1/2)\Delta x\). The whole state space is then partitioned into \(N+1\) cells, with the \((N+1)\)-th cell being defined as the complement of \(E\).  
Similarly, we discretize the time interval \([0,T]\) into \(M\) steps of uniform sizes \(\Delta t=T/M\). A large enough \(M\) could be chosen such that the local Gaussian approximation could be applied, i.e.,  
\[x_{t+\Delta t}\simeq x_{t}+b(x_{t})\Delta t+\sqrt{\varepsilon}\sigma(x_{t})w_ {\Delta t} \tag{93}\]  
Let \(\{p_{m}(i),m=0,1,\cdots,M,i=1,2,\cdots,N+1\}\) denote the probability of observing the process in the \(i\)-th cell at the moment \(m\Delta t\). A Markov chain can then be defined to describe the discrete-time evolution of the absorbed diffusion, i.e.,  
\[p_{m+1}(j)=\sum_{i}p_{m}(i)P_{i,j},\;j=1,2,\cdots,N+1,\;m=0,1,\cdots,M-1 \tag{94}\]  
with its transition probability \(\{P_{i,j},i,j=1,2,\cdots,N+1\}\) given as follows:  
\[P_{i,j}=\frac{1}{\sqrt{2\pi t\Delta t}}\exp\biggl{(}-\frac{(x(i)-x(j)-b(x(i) )\Delta t)^{2}}{2\varepsilon M}\biggr{)}\Delta x,\;i,j=1,2,\cdots,N\] \[P_{i,N+1}=1-\sum_{j=1}^{N}P_{i,j},\;P_{N+1,i}=0,\;i=1,2,\cdots,N \tag{95}\] \[P_{N+1,N+1}=1\]  
To calculate the NOP, we need to explore the reversed evolution of the original process. The evolution \(p_{m+1}\to p_{m}\) is generally time-inhomogeneous, with its transition probability having the following form:  
\[\overline{P}_{i,j}^{(m)}=\left\{\begin{array}{ll}\frac{p_{m}(j)P_{i,j}}{p_ {m+1}(i)},&p_{m+1}(i)>0\\ 0,&p_{m+1}(i)=0\end{array}\right. \tag{96}\]  
Thus, if we choose the distribution \(p_{0}\) and \(\overline{p}_{M}\) to be the probability concentrated on the cells containing \(x_{0}\) and \(x_{f}\), respectively, the nonstationary PPD is then given as follows:  
\[\overline{p}_{m}(j)=\sum_{i}\overline{p}_{m+1}(i)\overline{P}_{i,j}^{(m)} \tag{97}\]  
Searching for the peak of \(\overline{p}_{m}\) and calculating its local Gaussian coefficient can yield the NOP and its dispersion. The numerical results are displayed next.