Tree algebras over topological vector spaces in rough path theory

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