  Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability
measures, where $E$ is a Polish space,defined on the canonical probability
space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a
martingale problem with respect to a time-inhomogeneous generator $a$ is
well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with
generator $a$ for which we introduce a notion of so called {\it decoupled mild}
solution and study the equivalence with the notion of martingale solution
introduced in a companion paper. We also investigate well-posedness for
decoupled mild solutions and their relations with a special class of BSDEs
without driving martingale. The notion of decoupled mild solution is a good
candidate to replace the notion of viscosity solution which is not always
suitable when the map $a$ is not a PDE operator.
