From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics

TL;DR

This paper introduces rough path theory and perturbative field theory techniques to address the challenge of defining iterated integrals for low-regularity Gaussian processes like fractional Brownian motion with small Hurst index, aiming to develop a stochastic calculus in such rough settings.

Contribution

It presents a heuristic, physics-inspired overview of how constructive field theory tools can be applied to desingularize iterated integrals in rough path theory, bridging stochastic calculus and quantum field theory.

Findings

Heuristic presentation of rough path theory basics for physicists

Proposal of desingularization via weak non-Gaussian perturbations

Use of cluster expansions and renormalization for convergence

Abstract

Let $B=(B_1(t),..,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha\le 1/4$, 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 H\"older 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 {\em 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 in a second time the companion article \cite{MagUnt2} (or a preliminary version arXiv:1006.1255) for the constructive proofs.