# On nonlinear rough paths

## TL;DR

This paper develops a comprehensive theory of nonlinear rough paths, including their definitions, integrals, and applications to differential equations and rough PDEs, extending classical linear rough path theory.

## Contribution

It introduces the concept of nonlinear rough paths, develops their integration theory, and applies these to rough differential equations and PDEs, advancing the field beyond linear rough paths.

## Key findings

- Established the definition of nonlinear rough paths
- Developed integration techniques for nonlinear rough paths
- Applied the theory to rough partial differential equations

## Abstract

In this paper, we establish the theory of nonlinear rough paths. We give the definition of nonlinear rough paths, and develop the integrals. Then, we study differential equations driven by nonlinear rough paths. Afterwards, we compare the nonlinear rough paths and classic theory of linear rough paths. Finally, we apply this theory to rough partial differential equations.