## 3 Prehistorical Description of the Optimal Fluctuations
### Interpretation of the PPD  
#### 3.4.1 Time Reversal of the Diffusion Process  
In the following section, we fix the time interval as \([0,T]\). Let \(\{\mathbf{x}_{t}^{\varepsilon},0\leq t\leq T\}\) be a solution to the stochastic differential Equation (1), with its laws at each moment represented by \(\{p(\mathbf{x},t),0\leq t\leq T\}\). The law of the forward evolution process  
\[p(\mathbf{x},t_{0})\to p(\mathbf{x},t_{1})\to\dots\to p(\mathbf{x},t_{M}),\ 0=t_{0}<t_{1}<\dots<t_{M}=T \tag{67}\]  
is given by the following forward equation:  
\[\frac{\partial p(\mathbf{x},t)}{\partial t}=\mathcal{L}_{\mathbf{x}}^{*}p( \mathbf{x},t) \tag{68}\]  
It is natural to ask how the backward process, i.e.,  
\[p(\mathbf{x},t_{0})\gets p(\mathbf{x},t_{1})\leftarrow\dots\gets p( \mathbf{x},t_{M}),\ 0=t_{0}<t_{1}<\dots<t_{M}=T \tag{69}\]evolves. If we define the reversed process as \(\mathbf{y}_{t}^{t}=\mathbf{x}_{T-t}^{\varepsilon}\), it is equivalent to asking how the laws \(\overline{p}(\mathbf{y},t)=p(\mathbf{y},T-t)\) of the reversed process evolve.  
It was proven that the reversed Markov process is also Markovian, while the reversed diffusion is not necessarily a diffusion. A sufficient condition for the reversibility of the diffusion property requires smoothness of the coefficients as well as the uniform non-degeneracy of the diffusion term [44]. Under this condition, the reversed process \(\mathbf{y}_{t}^{\varepsilon}\) is also a diffusion, whose generator is given as follows [45; 46]:  
\[\overline{L}_{\mathbf{y},t}=\sum_{i,j}\frac{\varepsilon}{2}A_{ij}(\mathbf{y}) \frac{\partial^{2}}{\partial y_{i}\partial y_{j}}+\sum_{i}\overline{b}_{i}^{ \varepsilon}(\mathbf{y},t)\frac{\partial}{\partial y_{i}} \tag{70}\]  
\[\overline{b}_{i}^{\varepsilon}(\mathbf{y},t)=-b_{i}(\mathbf{y})+\sum_{j} \frac{\varepsilon}{p(\mathbf{y},T-t)}\frac{\partial}{\partial y_{j}}\big{(}A _{ij}(\mathbf{y})p(\mathbf{y},T-t)\big{)} \tag{71}\]  
In other words,  
\[\begin{array}{rl}\frac{\partial p(\mathbf{y},T-t)}{\partial t}&=\overline{ L}_{\mathbf{y},t}^{\ast}p(\mathbf{y},T-t)\\ &=\sum_{i,j}\frac{\varepsilon}{2}\frac{\partial^{2}}{\partial y_{i}\partial y _{j}}\big{(}A_{ij}(\mathbf{y})p(\mathbf{y},T-t)\big{)}+\sum_{i}\frac{\partial }{\partial y_{i}}\Big{(}\overline{b}_{i}^{\varepsilon}(\mathbf{y},t)p( \mathbf{y},T-t)\Big{)}\end{array} \tag{72}\]  
where \(\overline{L}_{\mathbf{y},t}^{\ast}\) is the adjoint operator of \(\overline{L}_{\mathbf{y},t}\). After a time-reversed transformation \(t\to T-t\), we have  
\[\begin{array}{rl}-\frac{\partial p(\mathbf{y},t)}{\partial t}&=\overline{ L}_{\mathbf{y},T-t}^{\ast}p(\mathbf{y},t)\\ &=\sum_{i,j}\frac{\varepsilon}{2}\frac{\partial^{2}}{\partial y_{i}\partial y _{j}}\big{(}A_{ij}(\mathbf{y})p(\mathbf{y},t)\big{)}+\sum_{i}\frac{\partial}{ \partial y_{i}}\Big{(}\overline{b}_{i}^{\varepsilon}(\mathbf{y},T-t)p( \mathbf{y},t)\Big{)}\end{array} \tag{73}\]  
which gives the evolutionary law of the backward process (Equation (69)). It can be seen that the reversed process is generally time-nonhomogeneous, and its evolutionary law depends on the density \(p(\mathbf{x},t)\). Different choices of the initial density \(p(\mathbf{x},0)\) yield different laws \(\{p(\mathbf{x},t),0\leq t\leq T\}\) and, thus, different reversed evolution laws.