  We introduce a variational theory for processes adapted to the
multi-dimensional Brownian motion filtration that provides a differential
structure allowing to describe infinitesimal evolution of Wiener functionals at
very small scales. The main novel idea is to compute the "sensitivities" of
processes, namely derivatives of martingale components and a weak notion of
infinitesimal generators, via a finite-dimensional approximation procedure
based on controlled inter-arrival times and approximating martingales. The
theory comes with convergence results that allow to interpret a large class of
Wiener functionals beyond semimartingales as limiting objects of differential
forms which can be computed path wisely over finite-dimensional spaces. The
theory reveals that solutions of BSDEs are minimizers of energy functionals
w.r.t Brownian motion driving noise.
