  This paper provides a probabilistic proof of the comparison result for
viscosity solutions of path-dependent semilinear PDEs. We consider the notion
of viscosity solutions introduced in \cite{EKTZ} which considers as test
functions all those smooth processes which are tangent in mean. When restricted
to the Markovian case, this definition induces a larger set of test functions,
and reduces to the notion of stochastic viscosity solutions analyzed in
\cite{BayraktarSirbu1,BayraktarSirbu2}. Our main result takes advantage of this
enlargement of the test functions, and provides an easier proof of comparison.
This is most remarkable in the context of the linear path-dependent heat
equation. As a key ingredient for our methodology, we introduce a notion of
punctual differentiation, similar to the corresponding concept in the standard
viscosity solutions \cite{CaffarelliCabre}, and we prove that semimartingales
are almost everywhere punctually differentiable. This smoothness result can be
viewed as the counterpart of the Aleksandroff smoothness result for convex
functions. A similar comparison result was established earlier in \cite{EKTZ}.
The result of this paper is more general and, more importantly, the arguments
that we develop do not rely on any representation of the solution.
