Speed density of a one-dimensional diffusion ($m(x)=2/(\sigma (x{)}^2{s}^{\prime }(x))$)

For a diffusion with generator $({\sigma }^2/2){\partial }_{xx}+b{\partial }_x$ and increasing scale function of a one-dimensional diffusion $s$, its speed density is $m=2/({\sigma }^2{s}^{\prime })$. Scale records hitting probabilities; speed records expected occupation and exit times. Multiplying scale by a positive constant divides speed by the same constant, leaving their Green-kernel product unchanged.

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 $\kappa >4$.