Cauchy law of an infinite-horizon Brownian exponential integral (${\int }_0^{\infty }{e}^{W_s-s/2}d{B}_s\sim \mathrm{Cauchy}(0,1)$)

For independent Brownian motions $W,B$, the strong law for Brownian motion makes ${\int }_0^{\infty }{e}^{2W_s-s}ds$ finite almost surely, so bracket localization gives an almost sure limit of the stochastic integral. The arctangent transform of a two-noise affine diffusion and endpoint convergence of a bounded angle diffusion identify its distribution function as $1/2+\arctan (x)/\pi$. Conditional Gaussianity rules out atoms.