Show that the function $\left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \times \mathbb{R} \rightarrow \mathbb{R} \\ & (t, x) \mapsto g_{\sqrt{\sigma^{2}+2t}}(x) \end{aligned}\right.$ satisfies conditions i and iii, where: \begin{itemize} \item[i.] the diffusion equation: $\forall(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \frac{\partial f}{\partial t}(t, x) = \frac{\partial^{2} f}{\partial x^{2}}(t, x)$; \item[iii.] the boundary condition: $\forall x \in \mathbb{R},\ \lim_{t \rightarrow 0^{+}} f(t, x) = g_{\sigma}(x)$. \end{itemize}