@article{CML_2012__4_1_A5_0, author = {Unterberger, J\'er\'emie}, title = {Mode d{\textquoteright}emploi de la th\'eorie constructive des champs bosoniques~: avec une application aux chemins rugueux}, journal = {Confluentes Mathematici}, publisher = {World Scientific Publishing Co Pte Ltd}, volume = {4}, number = {1}, year = {2012}, doi = {10.1142/S179374421240004X}, language = {fr}, url = {http://archive.numdam.org/articles/10.1142/S179374421240004X/} }
TY - JOUR AU - Unterberger, Jérémie TI - Mode d’emploi de la théorie constructive des champs bosoniques : avec une application aux chemins rugueux JO - Confluentes Mathematici PY - 2012 VL - 4 IS - 1 PB - World Scientific Publishing Co Pte Ltd UR - http://archive.numdam.org/articles/10.1142/S179374421240004X/ DO - 10.1142/S179374421240004X LA - fr ID - CML_2012__4_1_A5_0 ER -
%0 Journal Article %A Unterberger, Jérémie %T Mode d’emploi de la théorie constructive des champs bosoniques : avec une application aux chemins rugueux %J Confluentes Mathematici %D 2012 %V 4 %N 1 %I World Scientific Publishing Co Pte Ltd %U http://archive.numdam.org/articles/10.1142/S179374421240004X/ %R 10.1142/S179374421240004X %G fr %F CML_2012__4_1_A5_0
Unterberger, Jérémie. Mode d’emploi de la théorie constructive des champs bosoniques : avec une application aux chemins rugueux. Confluentes Mathematici, Tome 4 (2012) no. 1, article no. 1240004. doi : 10.1142/S179374421240004X. http://archive.numdam.org/articles/10.1142/S179374421240004X/
[1] A. Abdesselam, Explicit constructive renormalization, Ph.D. Thesis (1997).
[2] A. Abdesselam and V. Rivasseau, Trees, Forests and Jungles : A Botanical Garden for Cluster Expansions, Lecture Notes in Physics, Vol. 446 (Springer, 1995).
[3] A. Abdesselam and V. Rivasseau, An explicit large versus small field multiscale cluster expansion, Rev. Math. Phys. 9 (1997) 123–199.
[4] A. Abdesselam and V. Rivasseau, Explicit fermionic tree expansions, Lett. Math. Phys. 44 (1998) 77–88.
[5] J. Baez and J. Muniain, Gauge Fields, Knots and Gravity, Series on Knots and Every- thing, Vol. 4 (World Scientific, 1994).
[6] G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicol‘o, E. Olivieri, E. Pressutti and E. Scacciatelli, On the ultraviolet stability in the Euclidean scalar field theories, Com- mun. Math. Phys. 71 (1980) 95–130.
[7] G. Benfatto, G. Gallavotti and V. Mastropietro, Renormalization group and the Fermi surface in the Luttinger model, Phys. Rev. B 45 (1992) 5468–5480.
[8] D. Bernard, K. Gawedzki and A. Kupiainen, Anomalous scaling in the N-point func- tion of passive scalar.
[9] J. Bricmont, K. Gawedzki and A. Kupiainen, KAM theorem and quantum field theory, Commun. Math. Phys. 201 (1999) 699–727.
[10] C. Brouder, Runge–Kutta methods and renormalization, Euro. Phys. J. C 12 (2000) 521–534.
[11] D. C. Brydges and T. Kennedy, Mayer expansion of the Hamilton–Jacobi equation, J. Stat. Phys. 49 (1987) 19–49.
[12] E. Bacry and J. F. Muzy, Log-infinitely divisible multifractal processes, Commun. Math. Phys. 236 (2003) 449–475.
[13] P. Cartier and C. DeWitt-Morette, Brydges’ operator in renormalization theory, in Mathematical Physics and Stochastic Analysis, eds. S. Albeverio et al. (World Scien- tific, 2000), pp. 165–168.
[14] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998) 203–242.
[15] L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brow- nian motions, Probab. Th. Relat. Fields 122 (2002) 108–140.
[16] B. Duplantier and S. Sheffield, Liouville quantum gravity and KPZ, arXiv :0808.1560.
[17] Constructive Quantum Field Theory, Proc. of the 1973 Erice Summer School, eds. G. Velo and A. Wightman, Lecture Notes in Physics, Vol. 25 (Springer, 1973).
[18] G. Falkovich, K. Gawedzki and M. Vergassola, Particles and fields in fluid turbulence.
[19] L. Foissy and J. Unterberger, Ordered forests, permutations and iterated integrals, arXiv :1004.5208.
[20] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, Bounds on completely convergent Euclidean Feynman graphs, Commun. Math. Phys. 98 (1985) 273–288.
[21] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, Bounds on renormalized Feyn- man graphs, Commun. Math. Phys. 100 (1985) 23–55.
[22] J. Feldman, J. Magnen, V. Rivasseau and R. Sénéor, Construction and Borel summa- bility of infrared Φ4 4 by a phase space expansion, Commun. Math. Phys. 109 (1987) 437–480.
[23] J. Feldman, V. Rivasseau, J. Magnen and E. Trubowitz, An infinite volume expansion for many Fermions Green’s functions, Helv. Phys. Acta 65 (1992) 679.
[24] U. Frisch, Turbulence : The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, 1995).
[25] P. Friz and N. Victoir, Multidimensional Dimensional Processes Seen as Rough Paths (Cambridge Univ. Press, 2010).
[26] G. Gallavotti, Invariant tori : A field theoretic point of view on Eliasson’s work, in Advances in Dynamical Systems and Quantum Physics, éd. R. Figari (World Scientific, 1995), pp. 117–132.
[27] G. Gallavotti and K. Nicol‘o, Renormalization theory in 4-dimensional scalar fields, Commun. Math. Phys. 100 (1985) 545–590 ; 101 (1985) 247–282.
[28] K. Gawedzki and A. Kupiainen, Massless lattice ϕ4 4 theory : Rigorous control of a renormalizable asymptotically free model, Commun. Math. Phys. 99 (1985) 197–252.
[29] J. Glimm and A. Jaffe, Quantum Physics, A Functionnal Point of View (Springer, 1987).
[30] J. Glimm and A. Jaffe, Positivity of the ϕ4 3 Hamiltonian, Fortschr. Phys. 21 (1973) 327–376.
[31] J. Glimm, A. Jaffe and T. Spencer, The particle structure of the weakly coupled P(ϕ)2 model and other applications of high temperature expansions : Part II. The cluster expansion, in Constructive Quantum Field Theory (Erice 1973), op. cit. [17].
[32] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992).
[33] M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004) 86–140.
[34] R. Gurau, J. Magnen and V. Rivasseau, Tree quantum field theory, arXiv :0807.4122.
[35] B. Hambly and T. Lyons, Stochastic area for Brownian motion on the Sierpinski gasket, Ann. Probab. 26 (1998) 132–148.
[36] K. Hepp, Proof of the Bogoliubov–Parasiuk theorem on renormalization, Commun. Math. Phys. 2 (1966) 301–326.
[37] C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge Univ. Press, 1989).
[38] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus (Springer, 1991).
[39] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157–216.
[40] A. Kupiainen and P. Muratore-Ginanneschi, Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection.
[41] S. Lando and A. Zvonkin, Graphs on Surfaces and Their Applications (Springer, 2003).
[42] M. Laguës and A. Lesne, Invariance d’échelle. Des changements d’état à la turbulence (Belin, 2003).
[43] M. Le Bellac, Quantum and Statistical Field Theory (Oxford Univ. Press, 1991).
[44] A. Lejay, An Introduction to Rough Paths, Séminaire de Probabilités XXXVII, 1–59, Lecture Notes in Math., Vol. 1832 (Springer, 2003).
[45] A. Lejay, Yet another introduction to rough paths, Sém. Probab. 1979 (2009) 1–101.
[46] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Ibro. 14 (1998) 215–310.
[47] T. Lyons and Z. Qian, System Control and Rough Paths (Oxford Univ. Press, 2002).
[48] T. Lyons and N. Victoir, An extension theorem to rough paths, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 835–847.
[49] J. Magnen and V. Rivasseau, Constructive φ4 -theory without tears, arXiv :0706.2457.
[50] J. Magnen and J. Unterberger, From constructive theory to fractional stochastic cal- culus. (I) An introduction : Rough path theory and perturbative heuristics, Ann. Henri Poincaré 12 (2011) 1199–1226.
[51] J. Magnen and J. Unterberger, From constructive theory to fractional stochastic cal- culus. (II) The rough path for 1 6 < α < 1 4 : Constructive proof of convergence, à paraˆıtre à, Ann. Henri Poincaré, arXiv :1103.1750.
[52] J. Magnen and J. Unterberger, Renormalized rough paths : A stochastic differential equation approach, in preparation.
[53] V. Mastropietro, Non-Perturbative Renormalization (World Scientific, 2008).
[54] J. Magnen and D. Iagolnitzer, Weakly self avoiding polymers in four dimensions, Commun. Math. Phys. 162 (1994) 85–121.
[55] J. Moore, Lectures on Seiberg–Witten Invariants, Lecture Notes in Mathematics, Vol. 1629 (Springer, 1996).
[56] E. Nelson, A quartic interaction in two dimensions, in Mathematical Theory of Ele- mentary Particles, eds. R. Goodman and I. Segal (MIT Press, 1966).
[57] D. Nualart, Stochastic calculus with respect to the fractional Brownian motion and applications, Contemp. Math. 336 (2003) 3–39.
[58] R. Peltier and J. Lévy-Véhel, Multifractional Brownian motion : Definition and pre- liminary results, INRIA research report, RR-2645 (1995).
[59] M. Peskine and D. Schröder, An Introduction to Quantum Field Theory (Addison- Wesley, 1995).
[60] V. Rivasseau, From Perturbative to Constructive Renormalization (Princeton Univ. Press, 1991).
[61] V. Rivasseau, F. Vignes-Tourneret and R. Wulkenhaar, Renormalization of noncom- mutative φ4-theory by multi-scale analysis, Commun. Math. Phys. 262 (2006) 565–594.
[62] D. Ruelle, Statistical Mechanics, Rigorous Results (Benjamin, 1969).
[63] M. Salmhofer, Renormalization : An Introduction (Springer, 1999).
[64] H. Triebel, Spaces of Besov–Hardy–Sobolev Type (Teubner, 1978).
[65] J. Unterberger, Stochastic calculus for fractional Brownian motion with Hurst param- eter H > 1/4 : A rough path method by analytic extension, Ann. Probab. 37 (2009) 565–614.
[66] J. Unterberger, A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index H < 1/4, arXiv :0808.3458.
[67] J. Unterberger, A renormalized rough path over fractional Brownian motion, arXiv :1006.5604.
[68] J. Unterberger, A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering, Stoch. Proc. Appl. 120 (2010) 1444–1472.
[69] J. Unterberger, Hölder-continuous paths by Fourier normal ordering, Commun. Math. Phys. 298 (2010) 1–36.
[70] J. Unterberger, A Lévy area by Fourier normal ordering for multidimensional frac- tional Brownian motion with small Hurst index, arXiv :0906.1416.
[71] F. Vignes-Tourneret, Renormalisation des théories de champs non commutatives, Thèse de doctorat de l’Université Paris 11, arXiv :math-ph/0612014.
[72] A. S. Wightman, Remarks on the present state of affairs in the quantum theory of ele- mentary particles, in Mathematical Theory of Elementary Particles, eds. R. Goodman and I. Segal (MIT Press, 1966).
[73] K. G. Wilson, Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4 (1971) 3174–3184.
[74] K. G. Wilson and J. Kogut, The renormalization group and the ε expansion, Phys. Rep. 12 (1974) 75–200.
[75] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic inte- grals, Ann. Math. Statist. 36 (1965) 1560–1564.
Cité par Sources :