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From constructive field theory to fractional stochastic calculus. I: An introduction: Rough path theory and perturbative heuristics. (English) Zbl 1231.81058
The paper under review contains a heuristic description of a new approach to defining iterated stochastic integrals with respect to Gaussian processes with irregular paths - for example a \(d\)-dimensional fractional Brownian motion with Hurst index (corresponding to the Lipschitz constant of the paths) in \((\frac{1}{6}, \frac{1}{4})\). The novelty lies in applying to this problem the renormalization techniques developed within the quantum field theory. After a general introduction in sections 1 and 2 the authors describe fundamental concepts related to the Lévy area, fractional Brownian motion and the theory of the rough paths. The main focus is put on constructive methods of producing rough paths as opposed to abstract existence results; here also the origin of the connection to the quantum field theory is explained. A short introduction to the relevant aspects of the latter is provided in Section 3 and Section 4 contains a sketch of a possible proof of the main existence result for the stochastic integrals, based on a Fourier version of the Wick ordering and evaluation rules for Feynman diagrams. The details, even in the heuristic formulation, are rather technical and the authors refer to the forthcoming companion paper for rigorous reasonings.

MSC:
81T08 Constructive quantum field theory
60G17 Sample path properties
60G22 Fractional processes, including fractional Brownian motion
60H05 Stochastic integrals
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