Let (varphi: mathbb{R}^{+} ightarrow mathbb{R}) be a locally bounded real-valued function, and (L) the local time

Let \(\varphi: \mathbb{R}^{+} \rightarrow \mathbb{R}\) be a locally bounded real-valued function, and \(L\) the local time of the Brownian motion at level 0 . Prove that \(\left(\varphi\left(L_{t}\right) B_{t}, t \geq 0\right)\) is a Brownian motion time changed by \(\int_{0}^{t} \varphi^{2}\left(L_{s}\right) d s\).

Note that for \(h_{s}=\varphi\left(L_{s}\right)\), one has \(h_{s}=h_{g_{s}}\), then use the balayage formula. Note also that one could prove the result first for \(\varphi \in C^{1}\) and then pass to the limit.