zbMATH — the first resource for mathematics

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.

81T08 Constructive quantum field theory
60G17 Sample path properties
60G22 Fractional processes, including fractional Brownian motion
60H05 Stochastic integrals
Full Text: DOI arXiv
[1] Abdesselam, A.: Explicit constructive renormalization. Ph.D. thesis (1997)
[2] Bernard, D.; Gawedzki, K.; Kupiainen, A., Anomalous scaling in the N-point function of passive scalar, Phys. Rev. E., 54, 3, 2564 (1996)
[3] Bacry, E.; Muzy, J. F., Log-infinitely divisible multifractal processes, Commun. Math. Phys., 236, 3, 449-475 (2003) · Zbl 1032.60046
[4] Cass, T.; Friz, P., Densities for rough differential equations under Hörmander’s condition, Ann. Math., 171, 3, 2115-2141 (2010) · Zbl 1205.60105
[5] Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Relat. Fields, 122, 1, 108-140 (2002) · Zbl 1047.60029
[6] Duplantier, B., Sheffield, S.: Liouville quantum gravity and KPZ. Preprint. arXiv:0808.1560 · Zbl 1226.81241
[7] Falkovich, G.; Gawedzki, K.; Vergassola, M., Particles and fields in fluid turbulence, Rev. Mod. Phys., 73, 913-975 (2001) · Zbl 1205.76133
[8] Foissy, L., Unterberger, J.: Ordered forests, permutations and iterated integrals. Preprint. arXiv:1004.5208 (2010) · Zbl 1331.16026
[9] Feldman, J.; Magnen, J.; Rivasseau, V.; Sénéor, R., Construction and Borel summability of infrared \({\Phi^4_4}\) by a phase space expansion, Commun. Math. Phys., 109, 437-480 (1987)
[10] Friz, P.; Victoir, N., Multidimensional Dimensional Processes Seen as Rough Paths (2010), Cambridge: Cambridge University Press, Cambridge
[11] Garsia, A.: Continuity properties of Gaussian processes with multidimensional time parameter. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Probability Theory, vol. II, pp. 369-374. University of California Press (1972) · Zbl 0272.60034
[12] Gawedzki, K.; Kupiainen, A., Massless lattice \({\varphi^4_4}\) theory: rigorous control of a renormalizable asymptotically free model, Commun. Math. Phys., 99, 197-252 (1985)
[13] Glimm, J.; Jaffe, A., Quantum Physics: A Functionnal Point of View (1987), New York: Springer, New York
[14] Goldenfeld, N., Lectures on Phase Transitions and the Renormalization Group (1992), New York: Addison-Wesley, New York
[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] Gross, D. J.; Wilczek, F., Ultraviolet behavior of non-abeilan gauge theories, Phys. Rev. Lett., 30, 1343-1346 (1973)
[18] Hairer, M.; Pillai, N. S., Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statis., 47, 601-628 (2011) · Zbl 1221.60083
[19] Hambly, B.; Lyons, T., Stochastic area for Brownian motion on the Sierpinski gasket, Ann. Probab., 26, 1, 132-148 (1998) · Zbl 0936.60073
[20] Ivashkevich, E. V., Symmetries of the stochastic Burgers equation, J. Phys. A, 30, 15, 525-535 (1997) · Zbl 0924.35202
[21] Karatzas, I.; Shreve, S., Brownian Motion and Stochastic Calculus (1991), New York: Springer, New York · Zbl 0734.60060
[22] Kupiainen, A.; Muratore-Ginanneschi, P., Scaling, renormalization and statiscal conservation laws in the Kraichnan model of turbulent advection J, Stat. Phys., 126, 669-724 (2007) · Zbl 1116.82023
[23] Laguës, M.; Lesne, A., Invariance d’échelle (2003), Berlin: Des changements d’état à la turbulence, Berlin
[24] Le Bellac, M., Quantum and Statistical Field Theory (1991), Oxford: Oxford Science, Oxford
[25] Lejay, A.: An introduction to rough paths. Séminaire de Probabilités XXXVII, pp. 1-59. Lecture Notes in Mathematics, vol. 1832 (2003) · Zbl 1041.60051
[26] Lejay, A., Yet another introduction to rough paths, Séminaire de Probabilités, 1979, 1-101 (2009) · Zbl 1198.60002
[27] Lyons, T., Differential equations driven by rough signals, Rev. Mat. Ibroamericana, 14, 2, 215-310 (1998) · Zbl 0923.34056
[28] Lyons, T.; Qian, Z., System Control and Rough Paths (2002), Oxford: Oxford University Press, Oxford · Zbl 1029.93001
[29] 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
[30] Magnen, J.; Iagolnitzer, D., Weakly self avoiding polymers in four dimensions, Commun. Math. Phys., 162, 85-121 (1994) · Zbl 0796.60105
[31] Magnen, J., Unterberger, J.: From constructive theory to fractional stochastic calculus. (II) The rough path for \({\frac{1}{6}<\alpha<\frac{1}{4}} \): constructive proof of convergence. (2011, preprint) · Zbl 1231.81058
[32] Mastropietro, V., Non-Perturbative Renormalization (2008), Singapore: World Scientific, Singapore · Zbl 1159.81005
[33] Nualart, D., Stochastic calculus with respect to the fractional Brownian motion and applications, Contemp. Math., 336, 3-39 (2003) · Zbl 1063.60080
[34] Politzer, H. D., Reliable perturbative results for strong interactions, Phys. Rev. Lett., 30, 1346-1349 (1973)
[35] Peltier, R., Lévy-Véhel, J.: Multifractional Brownian motion: definition and preliminary results. INRIA research report, RR-2645 (1995)
[36] Peskine, M. E., Schröder, D. V.: An Introduction to Quantum Field Theory. Addison-Wesley (1995)
[37] Rivasseau, V.: From perturbative to constructive renormalization. Princeton Series in Physics (1991)
[38] Unterberger, J., Stochastic calculus for fractional Brownian motion with Hurst parameter H > 1/4: a rough path method by analytic extension, Ann. Probab., 37, 2, 565-614 (2009) · Zbl 1172.60007
[39] Unterberger, J.: A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index H < 1/4. Preprint. arXiv:0808.3458
[40] Unterberger, J.: A renormalized rough path over fractional Brownian motion. Preprint. arXiv:1006.5604 · Zbl 1276.60048
[41] Unterberger, J., A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering, Stoch. Process. Appl., 120, 1444-1472 (2010) · Zbl 1221.05062
[42] Unterberger, J., Hölder-continuous paths by Fourier normal ordering, Commun. Math. Phys., 298, 1, 1-36 (2010) · Zbl 1221.46047
[43] Unterberger, J.: A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index. Preprint. arXiv:0906.1416 · Zbl 1221.05062
[44] Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland Personal Library (2007) · Zbl 0974.60020
[45] Vignes-Tourneret, F.: Renormalisation des théories de champs non commutatives. Thèse de doctorat de l’Université Paris 11. arXiv:math-ph/0612014 · Zbl 1185.81116
[46] Wightman, A. S.; Goodman, R.; Segal, I., Remarks on the present state of affairs in the quantum theory of elementary particles, Mathematical Theory of Elementary Particles (1966), Cambridge: MIT Press, Cambridge · Zbl 0173.53903
[47] Wong, E.; Zakai, M., On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36, 1560-1564 (1965) · Zbl 0138.11201
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.