Let $t \in \mathbf{R}_+$. Verify that the function $P_t(f)$ is well defined for $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_t$ is linear on $C^0(\mathbf{R}) \cap CL(\mathbf{R})$, where
$$\forall x \in \mathbf{R}, \quad P_t(f)(x) = \int_{-\infty}^{+\infty} f\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$