On Nonlinear Rough Paths

Abstract

In this paper, we develop the theory of nonlinear rough paths. Following the ideas of Lyons and Gubinelli, we define the nonlinear rough integral R t s W(dr; Yr), where W and Y are only α-Hölder continuous in time with α ∈ (Formula Presented). Also, we study the Kunita-type equation (Formula Presented), obtaining the local and global existence and uniqueness of the solution under suitable sufficient conditions. As an application, we study transport equations with rough vector fields and observe that the classical solution formula for smooth and Young’s cases does not provide a solution to the rough equation. Indeed this formula satisfies a transport equation with additional compensator terms (see (1.7)).