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A renormalized rough path over fractional Brownian motion. (English) Zbl 1276.60048
Let \(\Gamma_t= (\Gamma_t(1),\dotsc,\Gamma_t(d))\) be a \(d\)-dimensional path not differentiable, but only \(\alpha\)-Hölder for some \(\alpha\in(0, 1)\). Then the iterated integral of \(\Gamma\) is not canonically defined. Rough path theory presents a way to overcome this difficulty. A rough path over \(\Gamma\) is a substitute for iterated integrals \[ \int^t_s d\Gamma_{t_1}(i_1)\dotsm \int^{t_{n-1}}_s d\Gamma_{i_n}(i_n), \] defined by the Hölder-continuity condition and two algebraic conditions called Chen property and Shuffle property. The author remarks that, if \(\Gamma\) is differentiable, the iterated integral has the Chen and Shuffle properties.
The existence of a rough path for a finite \(p\)-variation path was proved by T. Lyons and N. Victoir [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24, No. 5, 835–847 (2007; Zbl 1134.60047)], but this needed the axiom of choice, so it is not canonical.
In this paper, a canonical construction of a rough path over a fractional Brownian motion with arbitrary Hurst index is given (Theorem 0.1).
For this purpose, Hopf algebras \(H\) of decorated rooted trees, and the shuffle algebra \(Sh\) are introduced. A rough path over \(\Gamma\) is a functional \(J^{ts}_\Gamma(i_1,\dotsc, i_n)\), \(n\leq [1/\alpha]\), \(i_1,\dotsc, i_n\in [1,\dotsc, d]\), such that \(J^{ts}_\Gamma(i)= \Gamma_t(i)- \Gamma_s(i)\). Constructing \(J^{ts}_\Gamma\) using the skeleton integral which is a regularization of the iterated integral, it is shown \(J^{ts}_\Gamma\) is a character of \(Sh\) (Proposition 1.4). Then adopting the author’s Fourier normal ordering algorithm [Commun. Math. Phys. 298, No. 1, 1–36 (2010; Zbl 1221.46047)], \(J^{ts}_\Gamma\) is shown to satisfy the Chen and Shuffle properties (Proposition 1.11).
To obtain a Hölder continuous path as a function of \(s\) and \(t\) for any \(\alpha-\varepsilon\), \(\varepsilon> 0\) provided \(1/\alpha\not\in\mathbb{N}\), the skeleton integrals of the harmonic representation of the fractional Brownian motion \[ B_1(i)= (2\pi c_\alpha)^{-{1\over 2}} \int {e^{it\xi}- 1\over i\xi}\,|\xi|^{{1\over 2}-\alpha} W_\xi(i),\quad 1\leq i\leq d, \] are interpreted from the Feynman-integral point of view (Section 2). Then, applying Bogolioubov-Parasiuk-Hepp-Zimmerman renormalization scheme (cf. [K. Hepp, Commun. Math. Phys. 2, No. 4, 301–326 (1966; Zbl 1222.81219)]), a rough path over a fractional Brownian motion is constructed (Section 4 and 5).

MSC:
60G22 Fractional processes, including fractional Brownian motion
60J65 Brownian motion
26A16 Lipschitz (Hölder) classes
58D30 Applications of manifolds of mappings to the sciences
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
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