2022/05/01 by Ferrucci, Emilio · 1 citation
#60L20 #60L70 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.00582
A branched rough path X consists of a rough integral calculus for X \colon [0, T] → \mathbb Rd which may fail to satisfy integration by parts. Using Kelly's bracket extension [Kel12], we define a notion of pushforward of branched rough paths through smooth maps, which leads naturally to a definition of branched rough path on a smooth manifold. Once a covariant derivative is fixed, we are able to give a canonical, coordinate-free definition of integral against such rough paths. After characterising quasi-geometric rough paths in terms of their bracket extension, we use the same framework to define manifold-valued rough differential equations (RDEs) driven by quasi-geometric rough paths. These results extend previous work on 3 > p-rough paths [ABCRF22], itself a generalisation of the Ito calculus on manifolds developed by Meyer and Emery [Mey81, E89, E90], to the setting of non-geometric rough calculus of arbitrarily low regularity.