  The paper introduces and investigates the natural extension to the
path-dependent setup of the usual concept of canonical Markov class introduced
by Dynkin and which is at the basis of the theory of Markov processes. That
extension, indexed by starting paths rather than starting points will be called
path-dependent canonical class. Associated with this is the generalization of
the notions of semi-group and of additive functionals to the path-dependent
framework. A typical example of such family is constituted by the laws
$({\mathbb P}^{s,$\eta$})\_{(s,\eta) \in{\mathbb R} \times \Omega}$ , where for
fixed time s and fixed path $\eta$ defined on [0, s], $({\mathbb
P}^{s,$\eta$})\_{(s,\eta) \in {\mathbb R} \times \Omega}$ being the (unique)
solution of a path-dependent martingale problem or more specifically a weak
solution of a path-dependent SDE with jumps, with initial path $\eta$. In a
companion paper we apply those results to study path-dependent analysis
problems associated with BSDEs.
