HölderContinuous Rough Paths by Fourier Normal Ordering
Abstract
We construct in this article an explicit geometric rough path over arbitrary ddimensional paths with finite 1/ αvariation for any {αin(0,1)}. The method 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 nontrivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and of the Hopf shuffle algebra (in connection with the shuffle or geometric property). Hölder continuity is proved by using Besov norms. The method is wellsuited in particular in view of applications to probability theory (see the companion article [34] for the construction of a rough path over multidimensional fractional Brownian motion with Hurst index α < 1/4, or [35] for a short survey in that case).
 Publication:

Communications in Mathematical Physics
 Pub Date:
 August 2010
 DOI:
 10.1007/s0022001010641
 arXiv:
 arXiv:0903.2716
 Bibcode:
 2010CMaPh.298....1U
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 05C05;
 16W30;
 60F05;
 60G15;
 60G18;
 60H05
 EPrint:
 50 pages, 6 figures