= Finite-interval diffusion exit Green kernel
{title2=$K(x,v)=\dfrac{(s(x\wedge v)-s(\ell))(s(r)-s(x\vee v))}{s(r)-s(\ell)}$}
The expected exit time from $(\ell,r)$ for a one-dimensional diffusion is $\int_\ell^rK(x,v)m(v)dv$ when finite. Differentiating this expression on each side of $v=x$ gives the equation $\mathcal Lu=-1$ with zero endpoint values. For the <SLE two-boundary-point ratio diffusion>, $s(v)-s(1)$ behaves as $(v-1)^{1-4/\kappa}$ and the <speed density> behaves as $(v-1)^{4/\kappa}$, so the product is integrable at $1$ for $\kappa>4$.
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