A Self-dual Variational Approach to Stochastic Partial Differential Equations

Abstract

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.