Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and let \(\tau_{0}\) be the first hitting time of 0 . Find the "density" of \(\mathbb{P}^{x}\left(B_{t} \in d z, \tau_{0}>tight)\), i.e. find the function \(f_{t, x}(z)\) such that

\[\mathbb{P}^{X}\left(B_{t} \in A, \tau_{0}>tight)=\int_{A} f_{t, x}(z) d z \quad \text { for all } A \in \mathscr{B}(\mathbb{R})\]