Beta integral for strict ordering of coupled Bessel lifetimes ($\mathbb{P}({\sigma }_x<{\sigma }_y)=I_{1-x/y}(4a-1,1-2a)$)

Couple $dX=dB+a{X}^{-1}dt$ and $dY=dB+a{Y}^{-1}dt$ from $0<x<y$, with $1/4<a<1/2$. Suppose their finite lifetimes end at zero and ${\int }_0^{{\sigma }_y}{Y}_t^{-2}dt=\infty$. Their difference is positive and has derivative $-a(Y-X)/(XY)$, so ${\sigma }_x\le {\sigma }_y$. For $\Theta =(Y-X)/Y$, the Itô formula givesThe normalized scale function of a one-dimensional diffusion $I_{\Theta }(4a-1,1-2a)$ is a bounded local martingale. Its finite quadratic variation and the divergent $Y^{-2}$ clock force $\Theta \to 0$ on simultaneous extinction; on strict extinction $\Theta \to 1$. Its terminal value is therefore the strict-extinction indicator, proving the formula by optional sampling theorem. For SLE boundary images, $a=2/\kappa$; this compares strict and simultaneous boundary swallowing times for $4<\kappa <8$.