Stochastic partial differential equations: a rough path view

TL;DR

This paper introduces a rigorous framework for rough partial differential equations (RPDEs), establishing existence, uniqueness, and stability results by viewing them as integral equations within the rough path theory.

Contribution

It provides a novel integral equation-based definition of RPDEs, enabling robust analysis and extending classical SPDE results to the rough path setting.

Findings

Established well-posedness for RPDEs

Connected RPDEs to measure-valued Zakai equations

Developed Feynman-Kac representation for RPDEs

Abstract

We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equation, based on which we are able to obtain existence, uniqueness and stability results. The case of weak "rough" forward equations, may be seen as robustification of the (measure-valued) Zakai equation in the rough path sense. Feynman-Kac representation for RPDEs, in formal analogy to similar classical results in SPDE theory, play an important role.