Oppositely driven Bessel exit probability ($\mathbb{P}({\zeta }_X<{\zeta }_Y)=I_{y/(x+y)}(1-2a,1-2a)$)

For $0<a<1/2$, couple $dX=dB+a{X}^{-1}dt$ and $dY=-dB+a{Y}^{-1}dt$ from positive $x,y$, killing each at zero. Their sum has no Brownian term and increases, so simultaneous extinction is impossible. The ratio $U=Y/(X+Y)$ has time-changed generator $\frac{1}{2}{\partial }_u^2+a(1-2u)[u(1-u){]}^{-1}{\partial }_u$. Its scale density is $u^{-2a}(1-u{)}^{-2a}$, integrable at both endpoints. Optional stopping of its normalized scale function at the first extinction gives the displayed regularized incomplete beta function. At $a=1/3$ this is Cardy's formula.