We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$
$$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$

We admit in this question the following theorem: Let $L > 0$. If $H$ is a mapping from $C([0,L])$ to $C([0,L])$ satisfying
$$\exists k \in [0,1[, \quad \forall \rho_{1}, \rho_{2} \in C([0,L]), \quad \|H(\rho_{1}) - H(\rho_{2})\|_{\infty} \leq k \|\rho_{1} - \rho_{2}\|_{\infty}$$
then there exists a unique $\rho \in C([0,L])$ such that $H(\rho) = \rho$.

a) Let $L > 0$ and $\tilde{\alpha} \in C([0,L])$. Show that if $L \in ]0, 1/20[$, there exists a unique function $\tilde{\rho} \in C([0,L])$ such that
$$\forall x \in [0,L], \quad \tilde{\rho}(x) = \tilde{\alpha}(x) + \frac{2}{\sqrt{\pi}} \int_{0}^{L} \tilde{\rho}(y) T_{-1}(|x-y|) dy$$

b) Let $L \in ]0, 1/20[$, and $g_{0}$ continuous and integrable on $\mathbb{R}_{+}$. Show that there exists a function $\tilde{g}$ from $[0,L] \times \mathbb{R}$ to $\mathbb{R}$ such that
\begin{itemize}
\item $\forall v \in \mathbb{R}^{*}$, $\tilde{g}(\cdot, v)$ is of class $C^{1}$ on $[0,L]$
\item $\forall x \in [0,L]$, $\tilde{g}(x, \cdot)$ is integrable on $\mathbb{R}_{+}^{*}$ and $\mathbb{R}_{-}^{*}$
\item $\forall x \in [0,L], v \in \mathbb{R}^{*}$, $v \frac{\partial \tilde{g}}{\partial x}(x,v) = \left(\int_{\mathbb{R}_{+}^{*}} \tilde{g}(x,w) dw + \int_{\mathbb{R}_{-}^{*}} \tilde{g}(x,w) dw\right) \frac{2}{\sqrt{\pi}} e^{-v^{2}} - \tilde{g}(x,v)$
\item $\forall v \in \mathbb{R}_{+}^{*}$, $\tilde{g}(0,v) = g_{0}(v)$, $\quad \forall v \in \mathbb{R}_{-}^{*}$, $\tilde{g}(L,v) = 0$
\end{itemize}