We analyze multidimensional Markovian integral equations that are formulated
with a time-inhomogeneous progressive Markov process that has Borel measurable
transition probabilities. In the case of a path-dependent diffusion process,
the solutions to these integral equations lead to the concept of mild solutions
to semilinear parabolic path-dependent partial differential equations (PPDEs).
Our goal is to establish uniqueness, stability, existence, and
non-extendibility of solutions among a certain class of maps. By requiring the
Feller property of the Markov process, we give weak conditions under which
solutions become continuous. Moreover, we provide a multidimensional
Feynman-Kac formula and a one-dimensional global existence- and uniqueness
result.