  For any starting point in $\mathbb{R}^d$, we identify the stochastic
differential equation that is satisfied by distorted Brownian motion with
respect to a certain discontinuous Muckenhoupt $A_2$-weight $\psi$. The
discontinuities of $\psi$ typically take place on a sequence of level sets of
the Euclidean norm $D_k:=\{x\in \mathbb{R}^d\, | \;\|x\|=d_k\}$,
$k\in\mathbb{Z}$, where $(d_k)_{k\in\mathbb{Z}}\subset (0,\infty)$ may have
accumulation points and each level set $D_k$ plays the role of a permeable
membrane.
