= Reciprocal barrier proof of scalar diffusion comparison
{title2=$f_\epsilon(z)=(z+\epsilon)^{-1}$}
Before the difference $Z$ hits zero, its diffusion coefficient has magnitude at most $KZ$ and its drift is at least $-LZ$. Itô's formula bounds the drift of $f_\epsilon(Z)$ by $(L+K^2)f_\epsilon(Z)$. Its stochastic integrand is bounded by $K/(4\epsilon)$, making <expectation> legitimate on finite horizons. The <Gronwall inequality> and the value $f_\epsilon(0)=1/\epsilon$ then exclude finite-time contact from a strictly positive starting difference.
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