Let $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth and $t \in \mathbf{R}_+$. Show that $x \in \mathbf{R} \mapsto P_t(f)(x)$ is of class $C^2$ on $\mathbf{R}$. Also show that $$\forall x \in \mathbf{R}, \quad P_t(f)'(x) = \mathrm{e}^{-t}\int_{-\infty}^{+\infty} f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y$$ and $$\forall x \in \mathbf{R}, \quad P_t(f)''(x) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} f''\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$
Let $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth and $t \in \mathbf{R}_+$. Show that $x \in \mathbf{R} \mapsto P_t(f)(x)$ is of class $C^2$ on $\mathbf{R}$. Also show that
$$\forall x \in \mathbf{R}, \quad P_t(f)'(x) = \mathrm{e}^{-t}\int_{-\infty}^{+\infty} f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y$$
and
$$\forall x \in \mathbf{R}, \quad P_t(f)''(x) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} f''\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$