  In this paper we study time-inhomogeneous versions of one-dimensional
Stochastic Differential Equations (SDE) involving the Local Time of the unknown
process on curves. After proving existence and uniqueness for these SDE under
mild assumptions, we explore their link with Parabolic Differential Equations
(PDE) with transmission conditions. We study the regularity of solutions of
such PDE and ensure the validity of a Feynman-Kac representation formula. These
results are then used to characterize the solutions of these SDE as
time-inhomogeneous Markov Feller processes.
