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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
1.
Hölder continuity;
2.
multiplicative/Chen property; which comes from the fact \(I_n(F)\) measures area, volume and so on;
3.
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).

MSC:
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
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