BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.

Summary 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.