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An explicit rough path construction for continuous paths with arbitrary H\"older exponent by Fourier normal ordering

We construct in this article an explicit geometric rough path over arbitrary d-dimensional paths with finite 1/α-variation for any α ∈ (0, 1). The method is a rather straightforward extension of that used in a previous article [29] for multi-dimensional fractional Brownian motion. It may be coined as ’Fourier normal ordering’ since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the Hopf algebra structure of decorated rooted trees. The algorithm of regularization follows very closely the BPHZ algorithm for the renormalization of Feynmann diagrams in quantum field theory. The new feature here (compared to [29]) is the use of Besov norms to prove Hölder continuity.