In the document PDFs of Time-Homogeneous 1D Diffusions, the drift and diffusion coefficients are not assumed to be continuous or constant. The analysis is formulated in terms of the generalized differential operator [ D_m D_p^+, ] which accommodates a wide class of generators, including those with discontinuous coefficients.

The operator ( D_m D_p^+ ) is constructed from strictly increasing functions ( m(x) ) and ( p(x) ), derived via the transformations: [ B(x) = \int^x \frac{b(y)}{a(y)}\,dy, \qquad dm(x) = \frac{1}{a(x)} e^{B(x)}\,dx, \qquad dp(x) = e^{-B(x)}\,dx, ] where ( a(x) ) and ( b(x) ) are the diffusion and drift coefficients, respectively. These coefficients may be discontinuous, provided the integrals defining ( B(x) ), ( m(x) ), and ( p(x) ) are well-defined.

Existence and uniqueness of solutions to the resolvent equation [ \lambda F - D_m D_p^+ F = f ] are guaranteed under appropriate boundary conditions (inaccessibility of boundary points), not under continuity assumptions for ( a(x) ) or ( b(x) ).