Let (X) be a transient diffusion, such that [begin{aligned}mathbb{P}_{x}left(T_{0} 0 mathbb{P}_{x}left(lim _{t ightarrow infty} X_{t}=infty ight)

Let \(X\) be a transient diffusion, such that

\[\begin{aligned}\mathbb{P}_{x}\left(T_{0}<\infty\right) & =0, x>0 \\\mathbb{P}_{x}\left(\lim _{t \rightarrow \infty} X_{t}=\infty\right) & =1, x>0\end{aligned}\]

and note \(s\) the scale function satisfying \(s\left(0^{+}\right)=-\infty, s(\infty)=0\). Prove that for all \(x, t>0\),

\[\mathbb{P}_{x}\left(G_{y} \in d t\right)=\frac{-1}{2 s(y)} p_{t}^{(m)}(x, y) d t\]

where \(p^{(m)}\) is the density transition w.r.t. the speed measure \(m\).