# zbMATH — the first resource for mathematics

Ramification of rough paths. (English) Zbl 1315.60065
Summary: The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen’s multiplicative property involves the Dürr-Connes-Kreimer coproduct on rooted trees. This turns out to be the natural setting for a non-geometric theory of rough paths.

##### MSC:
 60H05 Stochastic integrals 16T30 Connections of Hopf algebras with combinatorics
##### Keywords:
rough paths; rooted trees; Hopf algebras; $$B$$-series
Full Text:
##### References:
 [1] Butcher, J.C., Numerical methods for ordinary differential equations, (2003), John Wiley & Sons Ltd. Chichester · Zbl 1032.65512 [2] Butcher, J.C., An algebraic theory of integration methods, Math. comp., 26, 79-106, (1972) · Zbl 0258.65070 [3] Cayley, P., On the analytical forms called trees, Amer. J. math., 4, 1/4, 266-268, (1881) · JFM 13.0867.02 [4] Hairer, E.; Wanner, G., On the Butcher group and general multi-value methods, Computing (arch. elektron. rechnen), 13, 1, 1-15, (1974) · Zbl 0293.65050 [5] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations, I. nonstiff problems, Springer ser. comput. math., vol. 8, (1993), Springer-Verlag Berlin · Zbl 0789.65048 [6] Connes, A.; Kreimer, D., Hopf algebras, renormalization and noncommutative geometry, Comm. math. phys., 199, 1, 203-242, (1998) · Zbl 0932.16038 [7] Connes, A.; Kreimer, D., Renormalization in quantum field theory and the riemann – hilbert problem, I. the Hopf algebra structure of graphs and the main theorem, Comm. math. phys., 210, 1, 249-273, (2000) · Zbl 1032.81026 [8] Connes, A.; Kreimer, D., Renormalization in quantum field theory and the riemann – hilbert problem, II. the β-function, diffeomorphisms and the renormalization group, Comm. math. phys., 216, 1, 215-241, (2001) · Zbl 1042.81059 [9] Brouder, C., Trees, renormalization and differential equations, Bit, 44, 3, 425-438, (2004) · Zbl 1072.16033 [10] Brouder, C., Runge – kutta methods and renormalization, Eur. phys. J. C, 12, 521-534, (2000) [11] Kreimer, D., On the Hopf algebra structure of perturbative quantum field theories, Adv. theor. math. phys., 2, 2, 303-334, (1998) · Zbl 1041.81087 [12] Kreimer, D., Chen’s iterated integral represents the operator product expansion, Adv. theor. math. phys., 3, 3, 627-670, (1999) · Zbl 0971.81093 [13] Dür, A., Möbius functions, incidence algebras and power series representations, Lecture notes in math., vol. 1202, (1986), Springer-Verlag Berlin · Zbl 0592.05006 [14] Sweedler, M.E., Hopf algebras, Math. lecture note ser., (1969), W.A. Benjamin, Inc. New York · Zbl 0194.32901 [15] Hoffman, M.E., Combinatorics of rooted trees and Hopf algebras, Trans. amer. math. soc., 355, 9, 3795-3811, (2003), (electronic) · Zbl 1048.16023 [16] Foissy, L., LES algèbres de Hopf des arbres enracinés décorés, II, Bull. sci. math., 126, 4, 249-288, (2002) · Zbl 1013.16027 [17] Foissy, L., LES algèbres de Hopf des arbres enracinés décorés, I, Bull. sci. math., 126, 3, 193-239, (2002) · Zbl 1013.16026 [18] Chen, K.T., Iterated path integrals, Bull. amer. math. soc., 83, 5, 831-879, (1977) · Zbl 0389.58001 [19] Chen, K.-T., Collected papers of K.-T. Chen, contemporary mathematicians, (2001), Birkhäuser Boston Inc. Boston, MA, edited and with a preface by Philippe Tondeur, and an essay on Chen’s life and work by Richard Hain and Tondeur [20] Lyons, T.J., Differential equations driven by rough signals, Rev. mat. iberoamericana, 14, 2, 215-310, (1998) · Zbl 0923.34056 [21] Lyons, T.; Qian, Z., System control and rough paths, Oxford math. monogr., (2002), Oxford University Press Oxford, Oxford Science Publications · Zbl 1029.93001 [22] Lejay, A., An introduction to rough paths, (), 1-59 · Zbl 1041.60051 [23] Gubinelli, M., Controlling rough paths, J. funct. anal., 216, 1, 86-140, (2004) · Zbl 1058.60037 [24] Feyel, D.; de La Pradelle, A., Curvilinear integrals along enriched paths, Electron. J. probab., 11, 860-892, (2006) · Zbl 1110.60031 [25] Friz, P.; Victoir, N., Approximations of the Brownian rough path with applications to stochastic analysis, Ann. inst. H. Poincaré probab. statist., 41, 4, 703-724, (2005) · Zbl 1080.60021 [26] M. Gubinelli, Rough solutions of the periodic Korteweg – de Vries equation, preprint, 2006 [27] M. Gubinelli, S. Tindel, Rough evolution equations, Ann. Probab. (2010), in press · Zbl 1193.60070 [28] Neuenkirch, A.; Nourdin, I.; Rößler, A.; Tindel, S., Trees and asymptotic developments for fractional stochastic differential equations, Ann. inst. H. Poincaré probab. statist., 45, 1, 157-174, (2009) · Zbl 1172.60017 [29] Rößler, A., Rooted tree analysis for order conditions of stochastic runge – kutta methods for the weak approximation of stochastic differential equations, Stoch. anal. appl., 24, 1, 97-134, (2006) · Zbl 1094.65008 [30] Gubinelli, M., Rooted trees for 3D navier – stokes equation, Dyn. partial differ. equ., 3, 2, 161-172, (2006) · Zbl 1132.35434
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.