## 3 Prehistorical Description of the Optimal Fluctuations
### In the Nonstationary Cases  
#### 3.2.1 Evolutionary Law of the Nonstationary PPD  
Here, we consider the transition from a point \(\mathbf{x}_{0}\) to \(\mathbf{x}_{f}\) over the time span \(T\). Similar to the stationary cases, the nonstationary PPD is defined as follows:  
\[p_{h}(\mathbf{x},t|\mathbf{x}_{f},T;\mathbf{x}_{0},0)=\frac{p(\mathbf{x},t| \mathbf{x}_{0},0)p(\mathbf{x}_{f},T|\mathbf{x}_{t},t)}{p(\mathbf{x}_{f},T| \mathbf{x}_{0},0)},\ 0\leq t\leq T \tag{38}\]  
This describes the prehistorical law of the stochastic process conditionally on the initial state \(\mathbf{x}^{\varepsilon}(0)=\mathbf{x}_{0}\) and final state \(\mathbf{x}^{\varepsilon}(T)=\mathbf{x}_{f}\). The functions \(p(\mathbf{x},t|\mathbf{x}_{0},0)\) and \(p(\mathbf{x}_{f},T|\mathbf{x},t)\) obey the Kolmogorov forward equation (Fokker-Planck equation) and the Kolmogorov backward equation, respectively:  
\[\frac{\partial p(\mathbf{x},t|\mathbf{x}_{0},0)}{\partial t}=\mathcal{L}_{x}^ {*}p(\mathbf{x},t|\mathbf{x}_{0},0) \tag{39}\]  
\[-\frac{\partial p(\mathbf{x}_{f},T|\mathbf{x},t)}{\partial t}=\mathcal{L}_{x} p(\mathbf{x}_{f},T|\mathbf{x},t) \tag{40}\]  
It is therefore straightforward to obtain  
\[\frac{\partial p_{h}(\mathbf{x},t|\mathbf{x}_{f},T;\mathbf{x}_{0},0)}{ \partial t}=\frac{p(\mathbf{x}_{f},T|\mathbf{x},t)}{p(\mathbf{x}_{f},T| \mathbf{x}_{0},0)}\frac{\partial p(\mathbf{x},t|\mathbf{x}_{0},0)}{\partial t }+\frac{p(\mathbf{x},t|\mathbf{x}_{0},0)}{p(\mathbf{x}_{f},T|\mathbf{x}_{0},0 )}\frac{\partial p(\mathbf{x}_{f},T|\mathbf{x},t)}{\partial t} \tag{41}\]  
Substituting Equations (39) and (40) into Equation (41) and regrouping the terms yields two distinct expressions of the evolutionary law. The first is as follows:  
\[-\frac{\partial p_{h}}{\partial t}=\frac{\epsilon}{2}\sum_{i,j}\frac{\partial ^{2}}{\partial x_{i}\partial x_{j}}\bigl{(}A_{ij}(\mathbf{x})p_{h}\bigr{)}- \sum_{i}\frac{\partial}{\partial x_{i}}\Bigl{(}\overline{b}_{i}^{*}(\mathbf{x},T-t)p_{h}\Bigr{)},\ 0\leq t\leq T \tag{42}\]\[\overline{b}_{i}^{\epsilon}(\mathbf{x},T-t)=-b_{i}(\mathbf{x})+\sum_{j}\frac{ \epsilon}{p(\mathbf{x},t|\mathbf{x}_{0})}\,\frac{\partial}{\partial x_{j}}\big{(} A_{ij}(\mathbf{x})p(\mathbf{x},t|\mathbf{x}_{0})\big{)} \tag{43}\]  
Whereas the second is as follows:  
\[\frac{\partial p_{h}}{\partial t}=\frac{\epsilon}{2}\sum_{ij}\frac{\partial^{ 2}}{\partial x_{i}\partial x_{j}}\big{(}A_{ij}(\mathbf{x})p_{h}\big{)}-\sum_{ i}\frac{\partial}{\partial x_{i}}\Big{(}\widetilde{b}_{i}^{\epsilon}(\mathbf{x},t)p_{ h}\Big{)},\ 0\leq t\leq T \tag{44}\]  
\[\widetilde{b}_{i}^{\epsilon}(\mathbf{x},t)=b_{i}(\mathbf{x})+\sum_{j}\epsilon A _{ij}(\mathbf{x})\frac{\partial}{\partial x_{j}}\ln p(\mathbf{x}_{j},T| \mathbf{x},t) \tag{45}\]  
The former is related to a new diffusion process \(\mathbf{y}_{t}^{\epsilon}\):  
\[ \begin{array}{ll}\mathbf{d}\mathbf{y}_{t}^{\epsilon}&=\bar{\mathbf{b}}^{ \epsilon}(\mathbf{y}_{t}^{\epsilon},t)\mathrm{d}t+\sqrt{\epsilon}\sigma( \mathbf{y}_{t}^{\epsilon})\mathrm{d}\mathbf{w}(t)\\ &=\left(-b_{i}(\mathbf{y}_{t}^{\epsilon})+\sum\limits_{j}\frac{\epsilon}{p( \mathbf{y}_{t},T-t|\mathbf{x}_{0})}\,\frac{\partial}{\partial y_{j}}\big{(}A_ {ij}(\mathbf{y}_{t}^{\epsilon})p(\mathbf{y}_{t}^{\epsilon},T-t|\mathbf{x}_{0 })\big{)}\right)\mathrm{d}t+\sqrt{\epsilon}\sigma(\mathbf{y}_{t}^{\epsilon}) \mathrm{d}\mathbf{w}(t)\end{array}  \tag{46}\]  
in the sense of:  
\[p_{h}(\mathbf{y},T-t|\mathbf{x}_{f},T;\mathbf{x}_{0},0)=\overline{p}(\mathbf{ y},t|\mathbf{x}_{f},0),\ 0\leq t\leq T \tag{47}\]  
Meanwhile the second equation (Equation (44)) can be related to another diffusion process \(\mathbf{z}_{t}^{\epsilon}\):  
\[