Let ((B, W)) be a two-dimensional Brownian motion and define [T_{t}=inf left{s geq 0: W_{s}>t ight}] Prove

Let \((B, W)\) be a two-dimensional Brownian motion and define

\[T_{t}=\inf \left\{s \geq 0: W_{s}>t\right\}\]

Prove that \(\left(Y_{t}=B_{T_{t}}, t \geq 0\right)\) is a Cauchy process, i.e., a process with independent and stationary increments, such that \(Y_{t}\) has a Cauchy law with characteristic function \(\exp (-t|u|)\).

\(\mathbb{E}\left(e^{i u B_{T_{t}}}\right)=\int e^{-\frac{1}{2} u^{2} T_{t}(\omega)} \mathbb{P}(d \omega)=\mathbb{E}\left(e^{-\frac{1}{2} u^{2} T_{t}}\right)=e^{-t|u|}\).