Another approach to some rough and stochastic partial differential equations

TL;DR

This paper introduces a novel method for solving rough and stochastic partial differential equations using Banach spaces and rough path theory, enabling high-order numerical schemes with proven convergence.

Contribution

It presents a new approach that combines Banach space frameworks with rough path theory to solve RPDEs and SPDEs, providing a basis for high-order numerical schemes.

Findings

Unique solutions constructed via state space transformation

High-order converging numerical schemes developed

Applicable to rough and stochastic PDEs with finite-dimensional noise

Abstract

In this note we introduce a new approach to rough and stochastic partial differential equations (RPDEs and SPDEs): we consider general Banach spaces as state spaces and -- for the sake of simiplicity -- finite dimensional sources of noise, either rough or stochastic. By means of a time-dependent transformation of state space and rough path theory we are able to construct unique solutions of the respective R- and SPDEs. As a consequence of our construction we can apply the pool of results of rough path theory, in particular we obtain strong and weak numerical schemes of high order converging to the solution process.