= Cauchy law of an infinite-horizon Brownian exponential integral
{c}
{title2=$\int_0^\infty e^{W_s-s/2}\,dB_s\sim\operatorname{Cauchy}(0,1)$}
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.
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