The geometry of the space of branched rough paths

Branched rough paths are a generalization of T. Lyons' geometric rough paths, introduced in 2010 by M. Gubinelli. They have played an important role in the solution theory of singular SPDEs and were one of the sources of inspiration for M. Hairer's theory of Regularity Structures. The problem of existence of a branched rough path above a fixed vector-valued Hölder path is, of course, important. We propose a solution to this problem based on an explicit form of the Baker-Campbell-Hausdorff formula due to Reutenauer, and on the Hairer-Kelly map. We introduce a new class of rough paths which we call anisotropic geometric rough paths. Our techniques also allow us to give an action of a Banach space of Hölder functions on branched rough paths, hence endowing space of branched rough paths with the structure of a principal homogeneous space. (This is a joint work with L. Zambotti)