Cauchy process
= Cauchy process
{c}
{title2=$\mathbb E e^{iuX_t}=e^{-t|u|}$}
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 $\sqrt{2\lambda}$.