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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).

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
Full Text: DOI
[1] Bass, R.F.; Hambly, B.M.; Lyons, T.J., Extending the Wong-zakai theorem to reversible Markov processes, J. Eur. Math. Soc., 4, 237-269, (2002) · Zbl 1010.60070
[2] Brouder, C.; Frabetti, A., QED Hopf algebras on planar binary trees, J. Algebra, 267, 298-322, (2003) · Zbl 1056.16026
[3] Brouder, C.; Frabetti, A.; Krattenthaler, C., Non-commutative Hopf algebra of formal diffeomorphisms, Adv. in Math., 200, 479-524, (2006) · Zbl 1133.16025
[4] Butcher, J.C., An algebraic theory of integration methods, Math. Comp., 26, 79-106, (1972) · Zbl 0258.65070
[5] Calaque, D.; Ebrahimi-Fard, K.; Manchon, D., Two Hopf algebras of trees interacting, Adv. Appl. Math., 47, 282-308, (2011) · Zbl 1235.16032
[6] Chapoton, F., Livernet, M.: Relating two Hopf algebras built from an operad. International Mathematics Research Notices, Vol. 2007, Article ID rnm131 (2007) · Zbl 1144.18006
[7] Connes, A.; Kreimer, D., Hopf algebras, renormalization and non-commutative geometry, Commun. Math. Phys., 199, 203-242, (1998) · Zbl 0932.16038
[8] Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem (I), Commun. Math. Phys., 210, 249-273, (2000) · Zbl 1032.81026
[9] Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem (II), Commun. Math. Phys., 216, 215-241, (2001) · Zbl 1042.81059
[10] Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122, 108-140, (2002) · Zbl 1047.60029
[11] Foissy, L.: Les algèbres de Hopf des arbres enracinés décorés (I). Bull. Sci. Math. 126(3), 193-239, and (II), Bull. Sci. Math. 126(4), 249-288 (2002). · Zbl 1013.16027
[12] Foissy, L., Unterberger, J.: Ordered forests, permutations and iterated integrals. Preprint http://arxiv.orgl/abs/1004.5208v1 [math.co], 2010 · Zbl 1331.16026
[13] Friz, P., Victoir, N.: Multidimensional dimensional processes seen as rough paths. Cambridge: Cambridge University Press, 2010 · Zbl 1193.60053
[14] Garsia, A.: Continuity properties of Gaussian processes with multidimensional time parameter. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Vol. II: Probability theory. Berkeley, CA: Univ. California Press, 1972, pp. 369-374 · Zbl 1221.46047
[15] Gradinaru, M.; Nourdin, I.; Russo, F.; Vallois, P., \(m\)-order integrals and generalized itô’s formula: the case of a fractional Brownian motion with any Hurst index, Ann. Inst. H. Poincaré Probab. Statist., 41, 781-806, (2005) · Zbl 1083.60045
[16] Gubinelli, M., Controlling rough paths, J. Funct. Anal., 216, 86-140, (2004) · Zbl 1058.60037
[17] Gubinelli, M.: Ramification of rough paths. Preprint http://arxiv.org/abs/math/0610300v1 [math.CA], 2006 · Zbl 1315.60065
[18] Hepp, K., Proof of the Bogoliubov-parasiuk theorem on renormalization, Commun. Math. Phys., 2, 301-326, (1966) · Zbl 1222.81219
[19] Hambly, B.; Lyons, T.J., Stochastic area for Brownian motion on the sierpinski basket, Ann. Prob., 26, 132-148, (1998) · Zbl 0936.60073
[20] Kreimer, D., Chen’s iterated integral represents the operator product expansion, Adv. Theor. Math. Phys., 3, 627-670, (1999) · Zbl 0971.81093
[21] Lejay, A.: An introduction to rough paths. Séminaire de probabilités XXXVII. Lecture Notes in Mathematics, Berlin Heidelberg-New York: Springer, 2003 · Zbl 1032.81026
[22] Lyons, T., Qian, Z.: System control and rough paths. Oxford: Oxford University Press, 2002 · Zbl 1029.93001
[23] Lyons, T.; Victoir, N., An extension theorem to rough paths, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24, 835-847, (2007) · Zbl 1134.60047
[24] Magnen, J., Unterberger, J.: From constructive field theory to fractional stochastic calculus. (I) The Lévy area of fractional Brownian motion with Hurst index\({α ∈ (\frac{1}{8}, \frac{1}{4})}\) . Preprint http://arxiv.org.abs/1004.5208v1 [maths.co], 2010 · Zbl 1264.81272
[25] Murua, A.: The shuffle Hopf algebra and the commutative Hopf algebra of labelled rooted trees. Available on http://www.ehu.es/ccwmuura/research/shart1bb.pdf , 2005 · Zbl 1133.16025
[26] Murua, A., The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math., 6, 387-426, (2006) · Zbl 1116.17004
[27] Nualart, D., Stochastic calculus with respect to the fractional Brownian motion and applications, Conty. Math., 336, 3-39, (2003) · Zbl 1063.60080
[28] Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton Series in Physics, Princeton, NJ: Princeton Univ. Press, 1991 · Zbl 1063.60080
[29] Tindel, S.; Unterberger, J., The rough path associated to the multidimensional analytic fBm with any Hurst parameter, Collectanea Math., 62, 197-223, (2011) · Zbl 1220.60022
[30] Unterberger, J., Stochastic calculus for fractional Brownian motion with Hurst parameter \(H\) > 1/4: a rough path method by analytic extension, Ann. Prob., 37, 565-614, (2009) · Zbl 1172.60007
[31] Unterberger, J., A rough path over multi-dimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering, Stoch. Proc. Appl., 120, 1444-1472, (2010) · Zbl 1221.05062
[32] Unterberger, J.: A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index. Preprint http://arxiv.org/labs/0906.1416v1 [math.PR], 2009
[33] Unterberger, J., Hölder-continuous rough paths by Fourier normal ordering, Commun. Math. Phys., 298, 1-36, (2010) · Zbl 1221.46047
[34] Unterberger, J.: Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion. Preprint http://arxiv.org/labs/0905.0782v2 [math.PR], 2009
[35] Vignes-Tourneret, F.: Renormalisation des théories de champs non commutatives,Thèse de doctorat de l’Université Paris 11, 2006, http://arxiv.org/labs/math-ph/0612014v1 , 2006 · Zbl 1222.81219
[36] Waldschmidt, M., Valeurs zêta multiples, Une introduction. J. de Théorie des Nombres de Bordeaux, 12, 581-595, (2000) · Zbl 0976.11037
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