Prove that, for \(\lambda>0\), one has

\[\int_{0}^{\infty} e^{-\lambda t} p_{t}(x, y) d t=\frac{1}{\sqrt{2 \lambda}} e^{-|x-y| \sqrt{2 \lambda}}\]

Prove that if \(f\) is a bounded Borel function, and \(\lambda>0\),

\[\mathbb{E}_{x}\left(\int_{0}^{\infty} e^{-\lambda^{2} t / 2} f\left(W_{t}\right) d t\right)=\frac{1}{\lambda} \int_{-\infty}^{\infty} e^{-\lambda|y-x|} f(y) d y\]