Let X X be a drifted Brownian motion with positive drift and y

Let $X$ be a drifted Brownian motion with positive drift $\nu$ and ${\Lambda }_y^{\nu }$ its last passage time at level $y$. Prove that

$$P_x({\Lambda }_y^{\left(\nu \right)}\in dt)=\frac{\nu }{\sqrt{2\pi t}}\exp (-\frac{1}{2t}(x-y+\nu t{)}^2)dt$$

and

$$P_x({\Lambda }_y^{\left(\nu \right)}=0)=\{\begin{array}{cc}1-e^{-2\nu (x-y)},& \text{for}x>y\\ 0& \text{for}x<y\end{array}$$

Prove, using time inversion that, for $x=0$,

$${\Lambda }_y^{\left(\nu \right)}\stackrel{\text{law}}{=}\frac{1}{T_{\nu }^{\left(y\right)}}$$

where

$$T_a^{\left(b\right)}=inf\{t:B_t+bt=a\}$$