caa a22 4500 445320214 CHVBK 20180317142701.0 cr unu---uuuuu 170323e20110901xx s 000 0 eng 10.1007/s00023-011-0106-3 doi (NATIONALLICENCE)springer-10.1007/s00023-011-0106-3 From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics [Elektronische Daten] [Jacques Magnen, Jérémie Unterberger] Let B=(B 1(t), . . . , B d (t)) be a d-dimensional fractional Brownian motion with Hurst index α≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard” tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (II) The rough path for $${\frac{1}{6} < \alpha < \frac{1}{4}}$$ : constructive proof of convergence, 2011, preprint) for the constructive proofs. Springer Basel AG, 2011 Magnen Jacques Centre de Physique Théorique, CNRS UMR 7644, École Polytechnique, 91128, Palaiseau Cedex, France aut Unterberger Jérémie Institut Élie Cartan Nancy, Université Henri Poincaré, BP 239, 54506, Vandoeuvre-lès-Nancy Cedex, France aut Annales Henri Poincaré SP Birkhäuser Verlag Basel 12/6(2011-09-01), 1199-1226 1424-0637 12:6<1199 2011 12 23 https://doi.org/10.1007/s00023-011-0106-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00023-011-0106-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Magnen Jacques Centre de Physique Théorique, CNRS UMR 7644, École Polytechnique, 91128, Palaiseau Cedex, France aut NATIONALLICENCE 700 1- Unterberger Jérémie Institut Élie Cartan Nancy, Université Henri Poincaré, BP 239, 54506, Vandoeuvre-lès-Nancy Cedex, France aut NATIONALLICENCE 773 0- Annales Henri Poincaré SP Birkhäuser Verlag Basel 12/6(2011-09-01), 1199-1226 1424-0637 12:6<1199 2011 12 23 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer