zbMATH — the first resource for mathematics

Hölder-continuous rough paths by Fourier normal ordering. (English) Zbl 1221.46047
Let \(\Gamma_t= (\Gamma_t(1),\dots,\Gamma_t(d))\) be a \(d\)-dimensional path and \(V_1,\dots, V_d: \mathbb{R}^d\to\mathbb{R}^d\) be smooth vector fields. Then the equation
\[ dy(t)= \sum^d_{i=1} V_i(y(t))\,d\Gamma_t(i)\tag{1} \]
is solved by iterated integrals if \(\Gamma\) is smooth.
If \(\Gamma\) is only \(\alpha\)-Hölder continuous, \(0<\alpha< 1\), to solve (1), one has to give a meaning to the (formal) iterated integral
\[ I_n(\Gamma)_{ts}= \int^t_s d\Gamma_{t_1}(i_1) \int^{t_1}_s d\Gamma_{t_2}(i_2)\cdots \int^{t_{n-1}}_s d\Gamma_{t_n}(i_n). \]
The theory of rough paths implies the possibility to solve (1) by a redefinition of the integration along \(\Gamma\) [T. Lyons and Q. Qian, System control and rough paths, Oxford Mathematical Monographs. Oxford: Clarendon Press (2002; Zbl 1029.93001)]. A rough path \(\Gamma= (\Gamma^1,\dots, \Gamma^N)\), \(N= [1/\alpha]\), over \(\Gamma\) is an abstraction of the canonical lift \(\Gamma^{\text{cano},n}(i_1,\dots, i_n)= I_n(\Gamma)_{ts}\) when \(\Gamma\) is smooth and plays the role of a substitute of the iterated integral for \(\Gamma\).
A general construction of a rough path over an \(\alpha\)-Hölder continuous \(\Gamma\) has been given by using the axiom of choice [T. Lyons and N. Victoir, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24, No. 5, 835–847 (2007; Zbl 1134.60047); P. Friz and N. Victor, Multidimensional processes seen as rough paths (Cambridge Stud. Adv. Math. 120) (2010; Zbl 1193.60053)].
In this paper, the Fourier normal ordering, a constructive method of construction of rough paths when \(\Gamma\) is a compact supported \(\alpha\)-Hölder continuous path, is presented as a functional \(\Gamma\to{\mathcal R}\Gamma\). It uses the tree encoding of iterated integrals [J. C. Butcher, Math. Comput. 26, 79–106 (1972; Zbl 0258.65070); D. Kreimer, Adv. Theor. Math. Phys. 3, No. 3, 627–670 (1999; Zbl 0971.81093)] and adopts the Hopf algebra interpretation of renormalization of A. Connes and D. Kreimer [Commun. Math. Phys. 199, No. 1, 203–242 (1998; Zbl 0932.16038); Commun. Math. Phys. 210, No. 1, 249–273 (2000; Zbl 1032.81026), ibid. 216, No. 1, 215–241 (2001; Zbl 1042.81059)] via a Fourier transform of \(\Gamma\). The main theorem says that \({\mathcal R}\Gamma\) is a rough path over \(\Gamma\) if \(1/\alpha\) is not an integer. Hence the existence of rough paths is proved without using axiom of choice.
The author says Fourier normal ordering is also relevant for random paths and can show existence of rough paths over a fractional Brownian motion with arbitrary Hurst index [J. Unterberger, Stochastic Processes Appl. 120, No. 8, 1444–1472 (2010; Zbl 1221.05062)]. Previously, existence of rough paths over a fractional Brownian motion was proved assuming that the Hurst index \(\alpha> 1/4\) [cf. J. Unterberger, Ann. Probab. 37, No. 2, 565–614 (2009; Zbl 1172.60007)].
An essential step of the Fourier normal ordering is rewriting of iterated integrals by permuting the order of itegration, which is well formulated by using the tree encoding of iterated integrals. These are exposed in §1 assuming \(\Gamma\) is smooth. If \(\Gamma\) is non-smooth, but a compactly supported \(\alpha\)-Hölder continuous path, then by using Besov decomposition \(\sum_{k\in\mathbb{Z}} D(\phi)\Gamma\), where \((\phi_k)_{k\in\mathbb{Z}}\) is a dyadic partition of unity and \(D(\phi_k)\Gamma={\mathcal F}^{-1}(\phi_k\cdot{\mathcal F}\Gamma)\) (Besov decomposition is explained in the Appendix; cf. [H. Triebel, Spaces of Besov-Hardy-Sobolev type (Teubner-Texte zur Mathematik; Leipzig: B. G. Teubner) (1978; Zbl 0408.46024)]), definitions in §1 again make sense. In this rewriting, diverging series appear, but each term is well defined. Then adopting the Bogolioubov-Hepp-Parasiuk-Zimmerman procedure for renormalization [cf. K. Hepp, “Proof of the Bogoliubov-Parasiuk theorem on renormalization,” Commun. Math. Phys. 2, No. 4, 301–326 (1966; Zbl 1222.81219)] according to Connes-Kreimer, the functional \({\mathcal R}(\Gamma)\) is defined in §2.
Rough paths need to satisfy 5mm
Hölder continuity;
multiplicative/Chen property; which comes from the fact \(I_n(F)\) measures area, volume and so on;
geometric/shuffle property, which comes from Fubini’s theorem.
The rest of the paper proves these properties for \({\mathcal R}\Gamma\). During the proof of the Geometric and Multiplicative properties (§3), alternative abstract, but compact definitions of the regularization algorithm \({\mathcal R}\) are given (Lemma 3.5, Definition 3.7).

46G05 Derivatives of functions in infinite-dimensional spaces
46N30 Applications of functional analysis in probability theory and statistics
34A36 Discontinuous ordinary differential equations
60J65 Brownian motion
Full Text: DOI arXiv
[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] Benassi A., Jaffard S., Roux D.: Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13(1), 19–90 (1997) · Zbl 0880.60053
[3] Brouder C., Frabetti A.: QED Hopf algebras on planar binary trees. J. Alg. 267, 298–322 (2003) · Zbl 1056.16026
[4] Brouder C., Frabetti A., Krattenthaler C.: Non-commutative Hopf algebra of formal diffeomorphisms. Adv. in Math. 200, 479–524 (2006) · Zbl 1133.16025
[5] Butcher J.C.: An algebraic theory of integration methods. Math. Comp. 26, 79–106 (1972) · Zbl 0258.65070
[6] Calaque, D., Ebrahimi-Fard, K., Manchon, D.: Two Hopf algebras of trees interacting. Preprint http://arxiv.org/abs/0806.2238v3[math.co] , 2009 · Zbl 1235.16032
[7] Chapoton, F., Livernet, M.: Relating two Hopf algebras built from an operad, International Mathematics Research Notices, Vol. 2007, Article ID rnm131 · Zbl 1144.18006
[8] Connes A., Kreimer D.: Hopf algebras, renormalization and non-commutative geometry. Commun. Math. Phys. 199(1), 203–242 (1998) · Zbl 0932.16038
[9] Connes A., Kreimer D.: Renormalization in quantum field theory and the Riemann-Hilbert problem (I). Commun. Math. Phys. 210(1), 249–273 (2000) · Zbl 1032.81026
[10] Connes A., Kreimer D.: Renormalization in quantum field theory and the Riemann-Hilbert problem (II). Commun. Math. Phys. 216(1), 215–241 (2001) · Zbl 1042.81059
[11] Coutin L., Qian Z.: Stochastic analysis, rough path analysis and fractional Brownian motions. Prob. Th. Rel. Fields 122(1), 108–140 (2002) · Zbl 1047.60029
[12] Darses, S., Nourdin, I., Nualart, D.: Limit theorems for nonlinear functionals of Volterra processes via white-noise analysis. http://arxiv.org/abs/0904.1401v1[math.PR] , 2009 · Zbl 1225.60041
[13] 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 1013.16026
[14] Friz, P., Victoir, N.: Multidimensional dimensional processes seen as rough paths. Cambridge studies in Adv. Math. 120, Cambridge: Cambridge University Press, 2010 · Zbl 1193.60053
[15] Gubinelli M.: Controlling rough paths. J. Funct. Anal. 216, 86–140 (2004) · Zbl 1058.60037
[16] Gubinelli, M.: Ramification of rough paths. Preprint available on http://arxiv.org/abs/math/0306433v2[math.PR] , 2003
[17] Hepp K.: Proof of the Bogoliubov-Parasiuk theorem on renormalization. Commun. Math. Phys. 2(4), 301–326 (1966) · Zbl 1222.81219
[18] Hambly B., Lyons T.J.: Stochastic area for Brownian motion on the Sierpinski basket. Ann. Prob. 26(1), 132–148 (1998) · Zbl 0936.60073
[19] Kahane, J.-P.: Some random series of functions. Cambridge studies in advanced mathematics 5, Cambridge: Cambridge Univ. Press, 1985
[20] Kreimer D.: Chen’s iterated integral represents the operator product expansion. Adv. Theor. Math. Phys. 3(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-NewYork: Springer, 2003
[22] Lyons T., Qian Z.: System control and rough paths. Oxford University Press, Oxford (2002) · Zbl 1029.93001
[23] Lyons T., Victoir N.: An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(5), 835–847 (2007) · Zbl 1134.60047
[24] Murua, A.: The shuffle Hopf algebra and the commutative Hopf algebra of labelled rooted trees. Available on www.ehu.es/ccwmuura/research/shart1bb.pdf , 2005
[25] Murua A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6(4), 387–426 (2006) · Zbl 1116.17004
[26] Nualart D.: Stochastic calculus with respect to the fractional Brownian motion and applications. Contemporary Mathematics 336, 3–39 (2003) · Zbl 1063.60080
[27] Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton Series in Physics, Princeton, NJ: Princeton Univ. Press, 1991
[28] Tindel, S., Unterberger, J.: The rough path associated to the multidimensional analytic fBm with any Hurst parameter. Preprint available at http://arxiv.org/abs/0810.1408[math.PR] , 2008 · Zbl 1220.60022
[29] Treves, F.: Introduction to pseudodifferential and Fourier integral operators. Vol. 1. Pseudodifferential operators, The University Series in Mathematics, New York-London: Plenum Press, 1980
[30] Triebel H.: Spaces of Besov-Hardy-Sobolev type. Leipzig, Teubner (1978) · Zbl 0408.46024
[31] Triebel, H.: Theory of function spaces. II. Monographs in Mathematics, 84, Basel: Birkhäuser, 1992 · Zbl 0763.46025
[32] Unterberger J.: Stochastic calculus for fractional Brownian motion with Hurst parameter H > 1/4; a rough path method by analytic extension. Ann. Prob. 37(2), 565–614 (2009) · Zbl 1172.60007
[33] Unterberger, J.: A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index H < 1/4. Preprint available at http://arxiv.org/abs/0808.3458v2[math.PR] , 2008
[34] Unterberger, J.: A rough path over multi-dimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering. Preprint available at http://arxiv.org/abs/0901.4771v2[math.PR] , 2009
[35] Unterberger, J.: A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index. Preprint available at http://arxiv.org/abs/0906.1416v1[math.PR] , 2009
[36] Waldschmidt M.: Valeurs zêta multiples. Une introduction. Journal de Théorie des Nombres de Bordeaux 12(2), 581–595 (2000) · Zbl 0976.11037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.