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 $C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$.
Recall that $\forall x \in \mathbf{R}, \quad P_{t}(f)(x) = \int_{-\infty}^{+\infty} f\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$