  Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct
explicit rough paths over arbitrary H\"older-continuous multidimensional paths.
We apply in this article the Fourier normal ordering ordering algorithm to the
construction of an explicit rough path over multi-dimensional fractional
Brownian motion $B$ with arbitrary Hurst index $\alpha$ (in particular, for
$\alpha\le 1/4$, which was till now an open problem) by regularizing the
iterated integrals of the analytic approximation of $B$ defined in
\cite{Unt08}. The regularization procedure is applied to 'Fourier normal
ordered' iterated integrals obtained by permuting the order of integration so
that innermost integrals have highest Fourier modes. The algebraic properties
of this rough path are best understood using two Hopf algebras: the Hopf
algebra of decorated rooted trees \cite{ConKre98} for the multiplicative or
Chen property, and the shuffle algebra for the geometric or shuffle property.
The rough path lives in Gaussian chaos of integer orders and is shown to have
finite moments. As well-known, the construction of a rough path is the key to
defining a stochastic calculus and solve stochastic differential equations
driven by $B$. The article \cite{Unt09ter} gives a quick overview of the
method.
