Abstract
We construct an explicit transitive free action of a Banach space of H\"older functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.
Highlights
The theory of Rough Paths has been introduced by Terry Lyons in the ’90s with the aim of giving an alternative construction of stochastic integration and stochastic differential equations
A rough path can be described as a Hölder function defined on an interval and taking values in a non-linear finite-dimensional Lie group; models of regularity structures are a generalization of this idea
A crucial ingredient of regularity structures is the renormalisation procedure: given a family of regularized models, which fail to converge in an appropriate topology as the regularization is removed, one wants to modify it in a such a way that the algebraic and analytical constraints are still satisfied and the modified version converges
Summary
The theory of Rough Paths has been introduced by Terry Lyons in the ’90s with the aim of giving an alternative construction of stochastic integration and stochastic differential equations. To define an element X P G it suffices to give the values xX, τ y for all trees τ P T; by freeness there is a unique multiplicative extension to all of H This is not at all the case for geometric rough paths: the shuffle algebra T pAq over an alphabet A is not free over the linear span of words so if one is willing to define a character X over T pAq there are additional algebraic constraints that the values of X on words must satisfy. We connect this approach with a recent work by Bruned, Chevyrev, Friz and Preiß [4] in Section 6.1, who borrowed ideas from the theory of Regularity Structures [6, 27] and proposed a renormalisation procedure for geometric and branched rough paths [4] based on pre-Lie morphisms. We mention that some further algebraic aspects of renormalisation in rough paths have been recently developed in [5]
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