= Beta integral for strict ordering of coupled Bessel lifetimes
{title2=$\mathbb P(\sigma_x<\sigma_y)=I_{1-x/y}(4a-1,1-2a)$}
Couple $dX=dB+aX^{-1}dt$ and $dY=dB+aY^{-1}dt$ 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 $\sigma_x\leq\sigma_y$. For $\Theta=(Y-X)/Y$, the <Itô formula> gives
$$
d\Theta=-\Theta Y^{-1}dB+\Theta Y^{-2}\left(1-a-\frac a{1-\Theta}\right)dt.
$$
The 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 $\Theta\to0$ on simultaneous extinction; on strict extinction $\Theta\to1$. 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 $4<\kappa<8$.
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