  We introduce a variational theory for processes adapted to the
multi-dimensional Brownian motion filtration. The theory provides a
differential structure which describes the infinitesimal evolution of Wiener
functionals at very small scales. The Markov property is replaced by a
finite-dimensional approximation procedure based on controlled inter-arrival
times and jumps of approximating martingales. The theory reveals that a large
class of adapted processes follow a differential rule which is similar in
nature to a fundamental theorem of calculus in the context of Wiener
functionals. Null stochastic derivative term turns out to be a non-Markovian
version of the classical second order parabolic operator and connections with
theory of local-times and $(p,q)$-variation regularity are established. The
framework extends the pathwise functional calculus and it opens the way to
obtain variational principles for processes. Applications to semi-linear
variational equations and stochastic variational inequalities are presented. In
particular, we provide a feasible dynamic programming principle for fully
non-Markovian/non-semimartingale optimal stopping problems.
