This is the abstract of the paper
  "One-dimensional Feller semigroups with reflecting barriers"
by M. Campiti and G. Metafune and D. Pallara.

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Abstract: Given the second order ordinary differential operator
$Au=mu''+qu'$ in the real interval $I$, possibly degenerate at the
boundary, we define
$W(x)=\exp\bigl(-\int_{x_0}^x\frac{q(s)}{m(s)}ds\bigr)$
(where $x_0\in I$ is arbitrarily fixed) and the domain
$$
D_N(A)=\Bigl\{u\in C(\overline I)\cap C^2(I):\ Au \in
C(\overline{I}),\ \lim_{x\to \partial I}\frac{u'(x)}{W(x)} = 0\Bigr\}.
$$
Given $f\in D_N(A)$, we consider the initial value problem $u_t=Au$,
$u(0)=f$ and study under which conditions on the coefficients a classical
solution exists, and when $(A,D_N(A))$ is the generator of a strongly
continuous semigroup in the space $C(\overline{I})$.