  Unlike many deterministic PDEs, stochastic equations are not amenable to the
classical variational theory of Euler-Lagrange. In this paper, we show how
self-dual variational calculus leads to solutions of various stochastic partial
differential equations driven by monotone vector fields. We construct weak
solutions as minima of suitable non-negative and self-dual energy functionals
on It\^o spaces of stochastic processes. We deal with both additive and
non-additive noise. The equations considered in this paper have already been
resolved by other methods, starting with the celebrated thesis of Pardoux, and
many other subsequent works. This paper is about presenting a new variational
approach to this type of problems, hoping it will lead to progress on other
still unresolved situations.
