Prove, using Theorem 3.1.1.2, that the joint law of the pair (left(left|B_{t} ight|, L_{t}^{0} ight)) is [mathbb{P}left(left|B_{t}

Prove, using Theorem 3.1.1.2, that the joint law of the pair \(\left(\left|B_{t}\right|, L_{t}^{0}\right)\) is

\[\mathbb{P}\left(\left|B_{t}\right| \in d x, L_{t}^{0} \in d \ell\right)=\mathbb{1}_{\{x \geq 0\}} \mathbb{1}_{\{\ell \geq 0\}} \frac{2(x+\ell)}{\sqrt{2 \pi t^{3}}} \exp \left(-\frac{(x+\ell)^{2}}{2 t}\right) d x d \ell .\]