A Self-dual Variational Approach to Stochastic Partial Differential Equations

TL;DR

This paper introduces a novel self-dual variational approach to solving stochastic partial differential equations driven by monotone vector fields, offering an alternative to classical methods and aiming to address unresolved problems.

Contribution

It develops a new variational calculus framework for stochastic PDEs, constructing weak solutions as minima of self-dual energy functionals, expanding the toolkit beyond existing techniques.

Findings

Constructed weak solutions as minima of energy functionals

Applicable to equations with additive and non-additive noise

Provides a new variational perspective for stochastic PDEs

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.