Tree algebras over topological vector spaces in rough path theory

Abstract

We work with non-planar rooted trees which have a label set given by an arbitrary vector space $V$. By equipping $V$ with a complete locally convex topology, we show how a natural topology is induced on the tree algebra over $V$. In this context, we introduce the Grossman-Larson and Connes-Kreimer topological Hopf algebras over $V$, and prove that they form a dual pair in a certain sense. As an application we define the class of branched rough paths over a general Banach space, and propose a new definition of a solution to a rough differential equation (RDE) driven by one of these branched rough paths. We show equivalence of our definition with a Davie-Friz-Victoir-type definition, a version of which is widely used for RDEs with geometric drivers, and we comment on applications to RDEs with manifold-valued solutions.

This version has been completely rewritten and it consists of the first half of the previous version, which has been substantially extended and generalised. 24 pages

Cited by in corpus (5)

• A Rough Path Perspective on Renormalization
• Planarly branched rough paths and rough differential equations on homogeneous spaces
• An isomorphism between branched and geometric rough paths
• On the definition of a solution to a rough differential equation
• Stochastic flows and rough differential equations on foliated spaces