Let \(X\) be a Bessel process with dimension \(\delta<2\), starting at \(x>0\) and \(T_{0}=\inf \left\{t: X_{t}=0\right\}\). Using time reversal theorem, prove that the density of \(T_{0}\) is

\[\frac{1}{t \Gamma(\alpha)}\left(\frac{x^{2}}{2 t}\right)^{\alpha} e^{-x^{2} /(2 t)}\]

where \(\alpha=(4-\delta) / 2-1\), i.e., \(T_{0}\) is a multiple of the reciprocal of a Gamma variable.