## 3 Prehistorical Description of the Optimal Fluctuations
### In the Quasi-Stationary Cases  
Here, we restrict our investigations to the domain \(D\) defined previously. Consider the diffusion that is absorbed in the boundary \(\partial D\). The law of it, represented by  
\[p_{abs}(\mathbf{x},t|\mathbf{x}_{0})=P_{\mathbf{x}_{0}}(\mathbf{x}_{t}^{ \varepsilon}\simeq\mathbf{x},t<\tau_{\partial D}^{\varepsilon}|\mathbf{x}_{0 }),\ \mathbf{x}\in D \tag{56}\]  
satisfies the Kolmogorov forward equation:  
\[ \begin{array}{l}\frac{\partial p_{abs}}{\partial t}=\mathcal{L}_{x}^{ \varepsilon}p_{abs},\ t>0,\ \mathbf{x}\in D\\ p_{abs}(\mathbf{x},t|\mathbf{x}_{0})=0,\ \mathbf{x}\in\partial D\end{array}  \tag{57}\]  
The probability in \(D\) continues to be lost as time evolves, in the sense of  
\[\int_{D}p_{abs}(\mathbf{x},t|\mathbf{x}_{0})\,\mathrm{d}x=P_{\mathbf{x}_{0}}( t<\tau_{\partial D}^{\varepsilon}) \tag{58}\]  
which corresponds to the probability that the stochastic trajectories escape from \(D\). The quasi-stationary distribution is then defined as the invariant probability measure with respect to the law of temporal evolution, conditionally on not having left the domain \(D\) up to the moment \(t\), namely [40]:  
\[ \begin{array}{l}\rho_{qs}(\mathbf{x})\ \ =\ \frac{\int_{D}P_{\mathbf{x}_{0}}( \mathbf{x}_{t}^{\varepsilon}\simeq\mathbf{x}+t<\tau_{\partial D}^{\varepsilon} )\rho_{\mathbf{x}_{0}}(\mathbf{x}_{0})\mathrm{d}\mathbf{x}_{0}}{\int_{D}P_{ \mathbf{x}_{0}}(t<\tau_{\partial D}^{\varepsilon})\rho_{\mathbf{x}_{0}}( \mathbf{x}_{0})\mathrm{d}\mathbf{x}_{0}}\\ \ \ =\ \frac{\int_{D}p_{abs}(\mathbf{x},t|\mathbf{x}_{0})\rho_{\mathbf{x}_{0}}( \mathbf{x}_{0})\mathrm{d}\mathbf{x}_{0}}{\int_{D}P_{\mathbf{x}_{0}}(t<\tau_{ \partial D}^{\varepsilon})\rho_{\mathbf{x}_{0}}(\mathbf{x}_{0})\mathrm{d} \mathbf{x}_{0}}\end{array}  \tag{59}\]  
This is the eigenfunction associated with the principal eigenvalue \(-\lambda_{1}\) for the Fokker-Planck operator \(\mathcal{L}^{*}\) with a homogeneous boundary condition [41; 42]. For any initial distribution \(p_{0}\), the quasi-stationary distribution can be approached by the so-called Yaglom limit of the conditional probability [43]:  
\[\rho_{qs}(\mathbf{x})=\underset{t\to\infty}{\lim}\frac{\int_{D}P_{\mathbf{x} _{0}}(\mathbf{x}_{t}^{\varepsilon}\simeq\mathbf{x},t<\tau_{\partial D}^{ \varepsilon})p_{0}(\mathbf{x}_{0})\mathrm{d}\mathbf{x}_{0}}{\int_{D}P_{ \mathbf{x}_{0}}(t<\tau_{\partial D}^{\varepsilon})p_{0}(\mathbf{x}_{0}) \mathrm{d}\mathbf{x}_{0}} \tag{60}\]Now, consider a timescale in the following range:  
\[1\ll t\ll\exp\left(\frac{V(\mathbf{x}_{eq},\partial D)}{\varepsilon}\right) \tag{61}\]  
A typical example of such a timescale is \(t\sim\exp\bigl{(}\left(V(\mathbf{x}_{eq},\partial D)-\eta\right)/\varepsilon \bigr{)}\), in which the process forgets its initial position and \(P_{\mathbf{x}_{0}}(t<\tau_{\partial D}^{\varepsilon})\) is still close to one. It follows that  
\[p(\mathbf{x},t|\mathbf{x}_{0})=P_{\mathbf{x}_{0}}(\mathbf{x}_{t}^{\varepsilon }\simeq\mathbf{x},t<\tau_{\partial D}^{\varepsilon})+P_{\mathbf{x}_{0}}( \mathbf{x}_{t}^{\varepsilon}\simeq\mathbf{x},t\geq\tau_{\partial D}^{ \varepsilon})=p_{abs}(\mathbf{x},t|\mathbf{x}_{0})+o(1) \tag{62}\]  
\[p_{abs}(\mathbf{x},t|\mathbf{x}_{0})\simeq\rho_{qs}(\mathbf{x})P_{\mathbf{x}_ {0}}(t<\tau_{\partial D}^{\varepsilon}) \tag{63}\]  
As a consequence, during the range of time (Equation (61)), the distribution \(p(\mathbf{x},t|\mathbf{x}_{0})\) is well approximated by the quasi-stationary distribution \(\rho_{qs}(\mathbf{x})\) and, in particular, it does not depend on the time or the initial position \(\mathbf{x}_{0}\). In the definition of the quasi-stationary distribution, a repulsive effect of the boundary is inevitable due to the conditioning with respect to the event \(\left\{t<\tau_{\partial D}^{\varepsilon}\right\}\). In a domain far away from \(\partial D\), this effect disappears.