Finite-interval diffusion exit Green kernel ($K(x,v)=\frac{(s(x\wedge v)-s(\ell ))(s(r)-s(x\vee v))}{s(r)-s(\ell )}$)

The expected exit time from $(\ell ,r)$ for a one-dimensional diffusion is ${\int }_{\ell }^{r}K(x,v)m(v)dv$ when finite. Differentiating this expression on each side of $v=x$ gives the equation $\mathcal{L}u=-1$ 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 $\kappa >4$.