The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations

Abstract

We give an overview of the recent approach to the integration of rough paths that reduces the problem to classical Young integration [13]. As an application, we extend an argument of Schwartz [11] to rough differential equations, and prove the existence, uniqueness and continuity of the solution, which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.

An Algebraic Approach to Integration of Geometric Rough Paths

“…In particular, the lifting of an effect to a geometric rough path is continuous. In [16] effects (dominated paths) are employed to give a short proof of the unique solvability and stability of the solution to differential equations driven by rough paths, and differences between adjacent Picard iterations decay factorially in operator norm.…”

Integration of time-varying cocyclic one-forms against rough paths

“…The enhancement into a groupvalued path is a continuous operation in the space of one-forms (based on (4.2) (4.10)) and integration is a continuous operation from one-forms to paths (based on (3.13)), so if the one-forms associated with a sequence of dominated paths converge then their group-valued enhancements also converge. In [27], we give an accessible overview of the one-form approach developed here. As an application, we extend an argument of Schwartz [29] to rough differential equations, and give a short proof of the global unique existence and continuity of the solution by using one-forms.…”

“…0<u1<u2<T δx k 0,u1 ⊗ δx 1 0,u2 , x k := π k (g), and I ′ is sufficient and necessary to define rough integration (Lemma 11 [27]). In the same manner, the mapping I encodes the integration of a monomial against another monomial (not only the degree-one monomial):…”

“…The approach is employed in [27] to extend an argument of Schwartz [29] to rough differential equations, and give a short proof of the global unique solvability and stability of the solution that is applicable to geometric rough paths and branched rough paths. Consider the rough differential equation…”

“…dg and β n+1 − β n n decay factorially in operator norm (Theorem 22 [27]). More explicitly, there is a constant C = C(p, γ, f Lip(γ) , ω (0, T )) such that, with…”