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A theory of regularity structures. (English) Zbl 1332.60093

The author presents a theory in which stochastic partial differential equations (typically with polynomial non-linearities), which are ill-posed due to low regularity of the initial data or roughness of the driving noise, can be reformulated, solved and analysed. The theory consists of an algebraic framework and a calculus for standard operations such as multiplication, composition with smooth functions, integration etc. The theory is consistent with the classical concept of SPDEs as solutions obtained under the theory are limits of classical solutions to suitably regularised problems and also existing results on singular SPDEs (KPZ, stochastic quantisation, Burgers) can be recovered within the new theory. The results can be applied for instance to the continuous parabolic Anderson model in higher dimensions, the stochastic quantisation of \(\Phi^4\) quantum field theory in dimension \(3\), KPZ-type equations or the Navier-Stokes equation with singular forcing. The author links the theory with the theories of rough paths, white noise analysis, Bony’s paraproduct and Colombeau’s generalised functions.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
81S20 Stochastic quantization
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
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[1] Albeverio, S., Cruzeiro, A.B.: Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids. Commun. Math. Phys. 129(3), 431-444 (1990) · Zbl 0702.76041 · doi:10.1007/BF02097100
[2] Albeverio, S., Ferrario, B.: Uniqueness of solutions of the stochastic Navier-Stokes equation with invariant measure given by the enstrophy. Ann. Probab. 32(2), 1632-1649 (2004) · Zbl 1065.60073 · doi:10.1214/009117904000000379
[3] Aizenman, M.: Geometric analysis of \[\varphi^4\] φ4 fields and Ising models. I, II. Commun. Math. Phys. 86(1), 1-48 (1982) · Zbl 0533.58034 · doi:10.1007/BF01205659
[4] Albeverio, S., Liang, S., Zegarlinski, B.: Remark on the integration by parts formula for the \[\phi^4_3\] ϕ34-quantum field model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(1), 149-154 (2006) · Zbl 1091.81064 · doi:10.1142/S0219025706002275
[5] Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Related Fields 89(3), 347-386 (1991) · Zbl 0725.60055 · doi:10.1007/BF01198791
[6] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2011) · Zbl 1227.35004
[7] Benth, F.E., Deck, T., Potthoff, J.: A white noise approach to a class of non-linear stochastic heat equations. J. Funct. Anal. 146(2), 382-415 (1997) · Zbl 0894.60039 · doi:10.1006/jfan.1996.3048
[8] Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571-607 (1997) · Zbl 0874.60059 · doi:10.1007/s002200050044
[9] Brzeźniak, Z., Goldys, B., Jegaraj, T., Weak solutions of a stochastic Landau-Lifshitz-Gilbert equation. Appl. Math. Res. Express. AMRX, : 2012. Art. ID abs 009, 33 (2012) · Zbl 1272.60041
[10] Bernardin, C., Gonçalves, P., Jara, M.: Private communication (2013) · Zbl 0894.60039
[11] Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume. Math. Ann. 70(3), 297-336 (1911) · JFM 42.0144.02 · doi:10.1007/BF01564500
[12] Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann. 72(3), 400-412 (1912) · JFM 43.0186.01 · doi:10.1007/BF01456724
[13] Bényi, Á., Maldonado, D., Naibo, V.: What is \[\ldots \]… a paraproduct? Notices Am. Math. Soc. 57(7), 858-860 (2010) · Zbl 1205.47001
[14] Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14(2), 209-246 (1981) · Zbl 0495.35024
[15] Bonfiglioli, A.: Taylor formula for homogeneous groups and applications. Math. Z. 262(2), 255-279 (2009) · Zbl 1286.43009 · doi:10.1007/s00209-008-0372-z
[16] Bertini, L., Presutti, E., Rüdiger, B., Saada, E.: Dynamical fluctuations at the critical point: convergence to a nonlinear stochastic PDE. Teor. Veroyatnost. i Primenen. 38(4), 689-741 (1993) · Zbl 0819.60070
[17] Brouder, C.: Trees, renormalization and differential equations. BIT 44(3), 425-438 (2004) · Zbl 1072.16033 · doi:10.1023/B:BITN.0000046809.66837.cc
[18] Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26, 79-106 (1972) · Zbl 0258.65070 · doi:10.1090/S0025-5718-1972-0305608-0
[19] Caruana, M., Friz, P.K., Oberhauser, H.: A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(1), 27-46 (2011) · Zbl 1219.60061
[20] Chan, T.: Scaling limits of Wick ordered KPZ equation. Commun. Math. Phys. 209(3), 671-690 (2000) · Zbl 0956.60077
[21] Chen, K.-T.: Iterated integrals and exponential homomorphisms. Proc. Lond. Math. Soc. (3) 4, 502-512 (1954) · Zbl 0058.25603
[22] 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. Commun. Math. Phys. 210(1), 249-273 (2000) · Zbl 1032.81026 · doi:10.1007/s002200050779
[23] Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The \[\beta\] β-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216(1), 215-241 (2001) · Zbl 1042.81059 · doi:10.1007/PL00005547
[24] Carmona, R.A., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108(518), viii+125 (1994) · Zbl 0925.35074
[25] Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225-236 (1951) · Zbl 0043.09902
[26] Colombeau, J.-F.: A multiplication of distributions. J. Math. Anal. Appl. 94(1), 96-115 (1983) · Zbl 0519.46045 · doi:10.1016/0022-247X(83)90007-0
[27] Colombeau, J.-F.: New generalized functions and multiplication of distributions, vol. 84 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam. Notas de Matemática [Mathematical Notes], vol. 90 (1984) · Zbl 0946.35017
[28] Coutin, L., Qian, Z.: Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122(1), 108-140 (2002) · Zbl 1047.60029 · doi:10.1007/s004400100158
[29] Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41(7), 909-996 (1988) · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[30] Daubechies, I.: Ten lectures on wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992) · Zbl 0776.42018
[31] Davie, A.M.: Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. Express. AMRX. Art. ID abm 009, 40 (2008) · Zbl 1163.34005
[32] Delamotte, B.: A hint of renormalization. Am. J. Phys. 72(2), 170 (2004) · doi:10.1119/1.1624112
[33] Da Prato, G., Debussche, A.: Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196(1), 180-210 (2002) · Zbl 1013.60051 · doi:10.1006/jfan.2002.3919
[34] Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900-1916 (2003) · Zbl 1071.81070 · doi:10.1214/aop/1068646370
[35] Da Prato, G., Debussche, A., Tubaro, L.: A modified Kardar-Parisi-Zhang model. Electron. Commun. Probab. 12, 442-453 (2007) (electronic) · Zbl 1136.60043
[36] Weinan, E., Jentzen, A., Shen, H.: Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. ArXiv e-prints (2013). [arXiv:1302.5930] · Zbl 1342.60094
[37] Eckmann, J.-P., Osterwalder, K.: On the uniqueness of the Hamiltionian and of the representation of the CCR for the quartic boson interaction in three dimensions. Helv. Phys. Acta 44, 884-909 (1971)
[38] Feyel, D., de La Pradelle, A.: Curvilinear integrals along enriched paths. Electron. J. Probab. 11(34), 860-892 (2006) (electronic) · Zbl 1110.60031
[39] Feldman, J.: The \[\lambda \varphi^4_3\] λφ34 field theory in a finite volume. Commun. Math. Phys. 37, 93-120 (1974) · doi:10.1007/BF01646205
[40] Feldman, J.S., Osterwalder, K.: The Wightman axioms and the mass gap for weakly coupled \[(\Phi^4)_3\](Φ4)3 quantum field theories. Ann. Phys. 97(1), 80-135 (1976) · doi:10.1016/0003-4916(76)90223-2
[41] Fröhlich, J.: On the triviality of \[\lambda \varphi^4_d\] λφd4 theories and the approach to the critical point in \[d{\>}4\] d4 dimensions. Nucl. Phys. B 200(2), 281-296 (1982) · doi:10.1016/0550-3213(82)90088-8
[42] Funaki, T.: A stochastic partial differential equation with values in a manifold. J. Funct. Anal. 109(2), 257-288 (1992) · Zbl 0768.60055 · doi:10.1016/0022-1236(92)90019-F
[43] Friz, P., Victoir, N.: A note on the notion of geometric rough paths. Probab. Theory Related Fields 136(3), 395-416 (2006) · Zbl 1108.34052 · doi:10.1007/s00440-005-0487-7
[44] Friz, P., Victoir, N.: Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46(2), 369-413 (2010) · Zbl 1202.60058 · doi:10.1214/09-AIHP202
[45] Friz, P.K., Victoir, N.B.: Multidimensional stochastic processes as rough paths, vol. 120 of Cambridge Studies in Advanced Mathematics, Theory and applications. Cambridge University Press, Cambridge (2010) · Zbl 1193.60053
[46] Gubinelli, M., Imkeller, P., Perkowski, N.: Paraproducts, rough paths and controlled distributions. ArXiv e-prints (2012). [arXiv:1210.2684] · Zbl 0173.03101
[47] Glimm, J., Jaffe, A.: Positivity of the \[\phi^4_3\] ϕ34 Hamiltonian. Fortschr. Physik 21, 327-376 (1973) · doi:10.1002/prop.19730210702
[48] Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn. Springer, New York (1987) · Zbl 0461.46051
[49] Glimm, J.: Boson fields with the \[{:}\Phi^4{:}\]:Φ4: interaction in three dimensions. Commun. Math. Phys. 10, 1-47 (1968) · Zbl 0175.24702 · doi:10.1007/BF01654131
[50] Giacomin, G., Lebowitz, J.L., Presutti, E.: Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems. In: Stochastic Partial Differential Equations: Six Perspectives, vol. 64 of Math. Surveys Monogr., pp. 107-152. Amer. Math. Soc., Providence (1999) · Zbl 0927.60060
[51] Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061-1083 (1975) · Zbl 0318.46049 · doi:10.2307/2373688
[52] Guerra, F., Rosen, L., Simon, B.: The \[{ P}(\phi )_2P\](ϕ)2 Euclidean quantum field theory as classical statistical mechanics. I, II. Ann. Math. (2) 101, (1975), 111-189, 191-259 · Zbl 1495.82015
[53] Gubinelli, M., Tindel, S.: Rough evolution equations. Ann. Probab. 38(1), 1-75 (2010) · Zbl 1193.60070 · doi:10.1214/08-AOP437
[54] Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216(1), 86-140 (2004) · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002
[55] Gubinelli, M.: Ramification of rough paths. J. Differ. Equ. 248(4), 693-721 (2010) · Zbl 1315.60065 · doi:10.1016/j.jde.2009.11.015
[56] Hairer, M.: An Introduction to Stochastic PDEs. ArXiv e-prints (2009). [arXiv:0907.4178] · Zbl 1032.81026
[57] Hairer, M.: Rough stochastic PDEs. Commun. Pure Appl. Math. 64(11), 1547-1585 (2011) · Zbl 1229.60079
[58] Hairer, M.: Singular perturbations to semilinear stochastic heat equations. Probab. Theory Related Fields 152(1-2), 265-297 (2012) · Zbl 1251.60052 · doi:10.1007/s00440-010-0322-7
[59] Hairer, M.: Solving the KPZ equation. Ann. Math. (2013, to appear) · Zbl 1281.60060
[60] Hida, T.: Analysis of Brownian functionals. Carleton Mathematical Lecture. Notes, No. 13. Carleton University, Ottawa (1975) · Zbl 1089.60522
[61] Hairer, M., Kelly, D.: Geometric versus non-geometric rough paths. Ann. Inst. Henri Poincaré Probab. Stat., ArXiv e-prints (2012). [arXiv:1210.6294] · Zbl 1314.60115
[62] Hairer, M., Maas, J.: A spatial version of the Itô-Stratonovich correction. Ann. Probab. 40(4), 1675-1714 (2012) · Zbl 1262.60060 · doi:10.1214/11-AOP662
[63] Hairer, M., Maas, J., Weber, H.: Approximating rough stochastic PDEs. Commun. Pure Appl. Math., ArXiv e-prints (2012). [arXiv:1202.3094] · Zbl 1302.60095
[64] Hopf, E.: The partial differential equation \[u_t+uu_x=\mu u_{xx}\] ut+uux=μuxx. Commun. Pure Appl. Math. 3, 201-230 (1950) · Zbl 0039.10403 · doi:10.1002/cpa.3160030302
[65] Hörmander, L.: On the theory of general partial differential operators. Acta Math. 94, 161-248 (1955) · Zbl 0067.32201 · doi:10.1007/BF02392492
[66] Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, 2nd edn. Universitext. Springer, New York (2010) · Zbl 1198.60005
[67] Hida, T., Potthoff, J.: White noise analysis—an overview. In: White Noise Analysis (Bielefeld, 1989), pp. 140-165. World Sci. Publ., River Edge (1990) · Zbl 0819.60041
[68] Hairer, M., Pillai, N.: Regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths. Ann. Probab. (2013, to appear) · Zbl 1288.60068
[69] Hairer, M., Ryser, M.D., Weber, H.: Triviality of the 2D stochastic Allen-Cahn equation. Electron. J. Probab. 17(39), 14 (2012) · Zbl 1245.60063
[70] Hairer, M., Voss, J.: Approximations to the stochastic Burgers equation. J. Nonlinear Sci. 21(6), 897-920 (2011) · Zbl 1273.60004
[71] 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
[72] Hairer, M., Weber, H.: Rough Burgers-like equations with multiplicative noise. Probab. Theory Related Fields, 1-56
[73] Iftimie, B., Pardoux, É., Piatnitski, A.: Homogenization of a singular random one-dimensional PDE. Ann. Inst. Henri Poincaré Probab. Stat. 44(3), 519-543 (2008) · Zbl 1172.74043 · doi:10.1214/07-AIHP134
[74] Jona-Lasinio, G., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101(3), 409-436 (1985) · Zbl 0588.60054 · doi:10.1007/BF01216097
[75] Klauder, J.R., Ezawa, H.: Remarks on a stochastic quantization of scalar fields. Prog. Theor. Phys. 69(2), 664-673 (1983) · Zbl 1098.81582 · doi:10.1143/PTP.69.664
[76] Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889-892 (1986) · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889
[77] Krylov, N.V.: Lectures on elliptic and parabolic equations in Sobolev spaces, vol. 96 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2008) · Zbl 1147.35001
[78] Lyons, T.J., Caruana, M., Lévy, T.: Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. Springer, Berlin, 2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, With an introduction concerning the Summer School by Jean Picard (2004) · Zbl 1176.60002
[79] Lyons, T., Qian, Z.: System control and rough paths. Oxford Mathematical Monographs. Oxford University Press, Oxford Science Publications, Oxford (2002) · Zbl 1029.93001
[80] Larson, R.G., Sweedler, M.E.: An associative orthogonal bilinear form for Hopf algebras. Am. J. Math. 91, 75-94 (1969) · Zbl 0179.05803 · doi:10.2307/2373270
[81] 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
[82] Lyons, T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2), 215-310 (1998) · Zbl 0923.34056 · doi:10.4171/RMI/240
[83] Meyer, Y.: Wavelets and operators, vol. 37 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. Translated from the 1990 French original by D. H. Salinger (1992)
[84] Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. (2) 81, 211-264 (1965) · Zbl 0163.28202
[85] Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250-1310 (2004) · Zbl 1062.60061 · doi:10.1137/S0036141002409167
[86] Nelson, E.: The free Markoff field. J. Funct. Anal. 12, 211-227 (1973) · Zbl 0273.60079 · doi:10.1016/0022-1236(73)90025-6
[87] Nualart, D.: The Malliavin Calculus and Related Topics. Probability and Its Applications (New York), 2nd edn. Springer, Berlin (2006) · Zbl 1099.60003
[88] Pinsky, M.A.: Introduction to Fourier analysis and wavelets. Brooks/Cole Series in Advanced Mathematics, Brooks/Cole (2002) · Zbl 1065.42001
[89] Parisi, G., Wu, Y.S.: Perturbation theory without gauge fixing. Sci. Sin. 24(4), 483-496 (1981) · Zbl 1480.81051
[90] Reutenauer, C.: Free Lie algebras, vol. 7 of London Mathematical Society Monographs. New Series. The Clarendon Press Oxford University Press, Oxford Science Publications, New York (1993) · Zbl 0798.17001
[91] Revuz, D., Yor, M.: Continuous martingales and Brownian motion, vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1991) · Zbl 0731.60002
[92] Schwartz, L.: Sur l’impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847-848 (1954) · Zbl 0056.10602
[93] Simon, L.: Schauder estimates by scaling. Calc. Var. Partial Differ. Equ. 5(5), 391-407 (1997) · Zbl 0946.35017 · doi:10.1007/s005260050072
[94] Sweedler, M.E.: Cocommutative Hopf algebras with antipode. Bull. Am. Math. Soc. 73, 126-128 (1967) · Zbl 0173.03101 · doi:10.1090/S0002-9904-1967-11677-X
[95] Sweedler, M.E.: Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin Inc, New York (1969) · Zbl 0194.32901
[96] Teichmann, J.: Another approach to some rough and stochastic partial differential equations. Stoch. Dyn. 11(2-3), 535-550 (2011) · Zbl 1234.35330 · doi:10.1142/S0219493711003437
[97] Unterberger, J.: Hölder-continuous rough paths by Fourier normal ordering. Commun. Math. Phys. 298(1), 1-36 (2010) · Zbl 1221.46047 · doi:10.1007/s00220-010-1064-1
[98] Walsh, J.B.: An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV-1984, vol. 1180 of Lecture Notes in Math., pp. 265-439. Springer, Berlin (1986)
[99] Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897-936 (1938) · JFM 64.0887.02 · doi:10.2307/2371268
[100] Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67(1), 251-282 (1936) · Zbl 0016.10404 · doi:10.1007/BF02401743
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