Cauchy process ($\mathbb{E}{e}^{iu{X}_t}=e^{-t|u|}$)

The standard symmetric Cauchy process is the Lévy process with Lévy characteristic exponent $\Psi (u)=|u|$. Its time-$t$ distribution for $t>0$ is the Cauchy distribution of location zero and scale $t$. It can be constructed by subordination of a Lévy process: evaluate an independent standard Brownian motion at the Brownian first-passage subordinator, whose Laplace exponent is $\sqrt{2\lambda }$.