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{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motionwith Hurst index $\alpha<1/4$, or more generally a Gaussian process whose pathshave the same local regularity. Defining properly iterated integrals of $B$ isa difficult task because of the low H\"older regularity index of its paths. Yetrough path theory shows it is the key to the construction of a stochasticcalculus with respect to $B$, or to solving differential equations driven by$B$.  We intend to show in a series of papers how to desingularize iteratedintegrals by a weak, singular non-Gaussian perturbation of the Gaussian measuredefined by a limit in law procedure. Convergence is proved by using "standard"tools of constructive field theory, in particular cluster expansions andrenormalization. These powerful tools allow optimal estimates, and call for anextension of Gaussian tools such as for instance the Malliavin calculus.  After a first introductory paper \cite{MagUnt1}, this one concentrates on thedetails of the constructive proof of convergence for second-order iteratedintegrals, also known as L\'evy area.

annales henri poincaré

From constructive field theory to fractional stochastic calculus. (II)  Constructive proof of convergence for the L\'evy area of fractional Brownian  motion with Hurst index $\alpha\in(1/8,1/4)$