Skip to main content
Log in

A theory of regularity structures

Inventiones mathematicae Aims and scope

Abstract

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the \(\Phi ^4_3\) Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of \(3\)-dimensional ferromagnets near their critical temperature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aizenman, M.: Geometric analysis of \(\varphi ^{4}\) fields and Ising models. I, II. Commun. Math. Phys. 86(1), 1–48 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  4. Albeverio, S., Liang, S., Zegarlinski, B.: Remark on the integration by parts formula for the \(\phi ^4_3\)-quantum field model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(1), 149–154 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Albeverio, S., Röckner, M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Related Fields 89(3), 347–386 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  10. Bernardin, C., Gonçalves, P., Jara, M.: Private communication (2013)

  11. Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume. Math. Ann. 70(3), 297–336 (1911)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann. 72(3), 400–412 (1912)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bényi, Á., Maldonado, D., Naibo, V.: What is \(\ldots \) a paraproduct? Notices Am. Math. Soc. 57(7), 858–860 (2010)

    MATH  Google Scholar 

  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)

    Google Scholar 

  15. Bonfiglioli, A.: Taylor formula for homogeneous groups and applications. Math. Z. 262(2), 255–279 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  17. Brouder, C.: Trees, renormalization and differential equations. BIT 44(3), 425–438 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26, 79–106 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  20. Chan, T.: Scaling limits of Wick ordered KPZ equation. Commun. Math. Phys. 209(3), 671–690 (2000)

    MATH  Google Scholar 

  21. Chen, K.-T.: Iterated integrals and exponential homomorphisms. Proc. Lond. Math. Soc. (3) 4, 502–512 (1954)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The \(\beta \)-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216(1), 215–241 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Carmona, R.A., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108(518), viii+125 (1994)

  25. Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236 (1951)

    MATH  Google Scholar 

  26. Colombeau, J.-F.: A multiplication of distributions. J. Math. Anal. Appl. 94(1), 96–115 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  28. Coutin, L., Qian, Z.: Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122(1), 108–140 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41(7), 909–996 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Google Scholar 

  32. Delamotte, B.: A hint of renormalization. Am. J. Phys. 72(2), 170 (2004)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  34. Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Da Prato, G., Debussche, A., Tubaro, L.: A modified Kardar–Parisi–Zhang model. Electron. Commun. Probab. 12, 442–453 (2007) (electronic)

    Google Scholar 

  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]

  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)

    MathSciNet  Google Scholar 

  38. Feyel, D., de La Pradelle, A.: Curvilinear integrals along enriched paths. Electron. J. Probab. 11(34), 860–892 (2006) (electronic)

    Google Scholar 

  39. Feldman, J.: The \(\lambda \varphi ^{4}_{3}\) field theory in a finite volume. Commun. Math. Phys. 37, 93–120 (1974)

    Article  Google Scholar 

  40. Feldman, J.S., Osterwalder, K.: The Wightman axioms and the mass gap for weakly coupled \((\Phi ^{4})_{3}\) quantum field theories. Ann. Phys. 97(1), 80–135 (1976)

    Article  MathSciNet  Google Scholar 

  41. Fröhlich, J.: On the triviality of \(\lambda \varphi ^{4}_{d}\) theories and the approach to the critical point in \(d{\>}4\) dimensions. Nucl. Phys. B 200(2), 281–296 (1982)

    Article  Google Scholar 

  42. Funaki, T.: A stochastic partial differential equation with values in a manifold. J. Funct. Anal. 109(2), 257–288 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  43. Friz, P., Victoir, N.: A note on the notion of geometric rough paths. Probab. Theory Related Fields 136(3), 395–416 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  44. Friz, P., Victoir, N.: Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46(2), 369–413 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  46. Gubinelli, M., Imkeller, P., Perkowski, N.: Paraproducts, rough paths and controlled distributions. ArXiv e-prints (2012). [arXiv:1210.2684]

  47. Glimm, J., Jaffe, A.: Positivity of the \(\phi ^{4}_{3}\) Hamiltonian. Fortschr. Physik 21, 327–376 (1973)

    Article  MathSciNet  Google Scholar 

  48. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn. Springer, New York (1987)

  49. Glimm, J.: Boson fields with the \({:}\Phi ^{4}{:}\) interaction in three dimensions. Commun. Math. Phys. 10, 1–47 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  51. Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)

    Article  Google Scholar 

  52. Guerra, F., Rosen, L., Simon, B.: The \({ P}(\phi )_{2}\) Euclidean quantum field theory as classical statistical mechanics. I, II. Ann. Math. (2) 101, (1975), 111–189, 191–259

  53. Gubinelli, M., Tindel, S.: Rough evolution equations. Ann. Probab. 38(1), 1–75 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  54. Gubinelli, M.: Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  55. Gubinelli, M.: Ramification of rough paths. J. Differ. Equ. 248(4), 693–721 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  56. Hairer, M.: An Introduction to Stochastic PDEs. ArXiv e-prints (2009). [arXiv:0907.4178]

  57. Hairer, M.: Rough stochastic PDEs. Commun. Pure Appl. Math. 64(11), 1547–1585 (2011)

    MATH  MathSciNet  Google Scholar 

  58. Hairer, M.: Singular perturbations to semilinear stochastic heat equations. Probab. Theory Related Fields 152(1–2), 265–297 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  59. Hairer, M.: Solving the KPZ equation. Ann. Math. (2013, to appear)

  60. Hida, T.: Analysis of Brownian functionals. Carleton Mathematical Lecture. Notes, No. 13. Carleton University, Ottawa (1975)

  61. Hairer, M., Kelly, D.: Geometric versus non-geometric rough paths. Ann. Inst. Henri Poincaré Probab. Stat., ArXiv e-prints (2012). [arXiv:1210.6294]

  62. Hairer, M., Maas, J.: A spatial version of the Itô-Stratonovich correction. Ann. Probab. 40(4), 1675–1714 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  63. Hairer, M., Maas, J., Weber, H.: Approximating rough stochastic PDEs. Commun. Pure Appl. Math., ArXiv e-prints (2012). [arXiv:1202.3094]

  64. Hopf, E.: The partial differential equation \(u_t+uu_x=\mu u_{xx}\). Commun. Pure Appl. Math. 3, 201–230 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  65. Hörmander, L.: On the theory of general partial differential operators. Acta Math. 94, 161–248 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

  68. Hairer, M., Pillai, N.: Regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths. Ann. Probab. (2013, to appear)

  69. Hairer, M., Ryser, M.D., Weber, H.: Triviality of the 2D stochastic Allen–Cahn equation. Electron. J. Probab. 17(39), 14 (2012)

    Google Scholar 

  70. Hairer, M., Voss, J.: Approximations to the stochastic Burgers equation. J. Nonlinear Sci. 21(6), 897–920 (2011)

    Google Scholar 

  71. Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing (Arch. Elektron. Rechnen) 13(1), 1–15 (1974)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  74. Jona-Lasinio, G., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101(3), 409–436 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  75. Klauder, J.R., Ezawa, H.: Remarks on a stochastic quantization of scalar fields. Prog. Theor. Phys. 69(2), 664–673 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  76. Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)

    Article  MATH  Google Scholar 

  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)

  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)

  79. Lyons, T., Qian, Z.: System control and rough paths. Oxford Mathematical Monographs. Oxford University Press, Oxford Science Publications, Oxford (2002)

  80. Larson, R.G., Sweedler, M.E.: An associative orthogonal bilinear form for Hopf algebras. Am. J. Math. 91, 75–94 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  81. Lyons, T., Victoir, N.: An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(5), 835–847 (2007)

    Google Scholar 

  82. Lyons, T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2), 215–310 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  85. Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35(5), 1250–1310 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  86. Nelson, E.: The free Markoff field. J. Funct. Anal. 12, 211–227 (1973)

    Article  MATH  Google Scholar 

  87. Nualart, D.: The Malliavin Calculus and Related Topics. Probability and Its Applications (New York), 2nd edn. Springer, Berlin (2006)

  88. Pinsky, M.A.: Introduction to Fourier analysis and wavelets. Brooks/Cole Series in Advanced Mathematics, Brooks/Cole (2002)

    MATH  Google Scholar 

  89. Parisi, G., Wu, Y.S.: Perturbation theory without gauge fixing. Sci. Sin. 24(4), 483–496 (1981)

    MathSciNet  Google Scholar 

  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)

  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)

  92. Schwartz, L.: Sur l’impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)

    MATH  MathSciNet  Google Scholar 

  93. Simon, L.: Schauder estimates by scaling. Calc. Var. Partial Differ. Equ. 5(5), 391–407 (1997)

    Article  MATH  Google Scholar 

  94. Sweedler, M.E.: Cocommutative Hopf algebras with antipode. Bull. Am. Math. Soc. 73, 126–128 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  95. Sweedler, M.E.: Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin Inc, New York (1969)

    Google Scholar 

  96. Teichmann, J.: Another approach to some rough and stochastic partial differential equations. Stoch. Dyn. 11(2–3), 535–550 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  97. Unterberger, J.: Hölder-continuous rough paths by Fourier normal ordering. Commun. Math. Phys. 298(1), 1–36 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  100. Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67(1), 251–282 (1936)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

I am very grateful to M. Gubinelli and to H. Weber for our numerous discussions on quantum field theory, renormalisation, rough paths, paraproducts, Hopf algebras, etc. These discussions were of enormous help in clarifying the concepts presented in this article. Many other people provided valuable input that helped shaping the theory. In particular, I would like to mention A. Debussche, B. Driver, P. Friz, J. Jones, D. Kelly, X.-M. Li, M. Lewin, T. Lyons, J. Maas, K. Matetski, J.-C. Mourrat, N. Pillai, D. Simon, T. Souganidis, J. Unterberger, and L. Zambotti. Special thanks are due to L. Zambotti for pointing out a mistake in an earlier version of the definition of the class of regularity structures considered in Sect. 8. Financial support was kindly provided by the Royal Society through a Wolfson Research Merit Award and by the Leverhulme Trust through a Philip Leverhulme Prize.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Hairer.

Appendices

Appendix A: A generalised Taylor formula

Classically, Taylor’s formula for functions on \(\mathbf{R}^d\) is obtained by applying the one-dimensional formula to the function obtained by evaluating the original function on a line connecting the start and endpoints. This however does not yield the “right” formula if one is interested in obtaining the correct scaling behaviour when applying it to functions with inhomogeneous scalings. In this section, we provide a version of Taylor’s formula with a remainder term having the correct scaling behaviour for any non-trivial scaling \(\mathfrak {s}\) of \(\mathbf{R}^d\). Although it is hard to believe that this formula isn’t known (see [15] for some formulae with a very similar flavour) it seems difficult to find it in the literature in the form stated here. Furthermore, it is of course very easy to prove, so we provide a complete proof.

In order to formulate our result, we introduce the following kernels on \(\mathbf{R}\):

$$\begin{aligned} \mu _\ell (x,dy) = \mathbf {1}_{[0,x]}(y){(x-y)^{\ell -1}\over (\ell -1)!}\,dy,\quad \mu _\star (x,dy) = \delta _0(dy). \end{aligned}$$

For \(\ell = 0\), we extend this in a natural way by setting \(\mu _0(x,dy) = \delta _x(dy)\). With these notations at hand, any multiindex \(k \in \mathbf{N}^d\) gives rise to a kernel \({\mathcal Q}^k\) on \(\mathbf{R}^d\) by

$$\begin{aligned} {\mathcal Q}^k(x,dy) = \prod _{i=1}^d \mu ^{k}_i(x_i,dy_i), \end{aligned}$$
(11.1)

where we define

$$\begin{aligned} \mu ^{k}_i(z,\cdot ) = \left\{ \begin{array}{ll} \mu _{k_i}(z, \cdot ) &{} \text {if }i \le \mathfrak {m}(k), \\ {z^{k_i} \over k_i !} \mu _{\star }(z, \cdot ) &{} \text {otherwise,} \end{array}\right. \end{aligned}$$

where we defined the quantity

$$\begin{aligned} \mathfrak {m}(k) = \min \{j\,:\, k_j \ne 0\}. \end{aligned}$$

Note that, in any case, one has the identity \(\mu _i^k(z, \mathbf{R}) = {z^{k_i}\over k_i !}\), so that

$$\begin{aligned} {\mathcal Q}^k(x,\mathbf{R}^d) = {x^k \over k!}. \end{aligned}$$

Recall furthermore that \(\mathbf{N}^d\) is endowed with a natural partial order by saying that \(k \le \ell \) if \(k_i \le \ell _i\) for every \(i \in \{1,\ldots ,d\}\). Given \(k \in \mathbf{N}^d\), we use the shorthand \(k_< = \{\ell \ne k\,:\, \ell \le k\}\).

Proposition 11.1

Let \(A \subset \mathbf{N}^d\) be such that \(k\in A \Rightarrow k_{<} \subset A\) and define \(\partial A = \{k \not \in A\,:\, k - e_{\mathfrak {m}(k)} \in A\}\). Then, the identity

$$\begin{aligned} f(x) = \sum _{k \in A} {D^k f(0) \over k!} x^k + \sum _{k \in \partial A} \int _{\mathbf{R}^d} D^k f(y)\, {\mathcal Q}^k(x,dy), \end{aligned}$$
(11.2)

holds for every smooth function \(f\) on \(\mathbf{R}^d\).

Proof

The case \(A = \{0\}\) is straightforward to verify “by hand”. Note then that, for every set \(A\) as in the statement, one can find a sequence \(\{A_n\}\) of sets such as in the statement with \(A_a = \{0\}\), \(A_{|A|} = A\), and \(A_{n+1} = A_n \cup \{k_n\}\) for some \(k_n \in \partial A_n\). It is therefore sufficient to show that if (11.2) holds for some set \(A\), then it also holds for \(\bar{A} = A \cup \{\ell \}\) for any \(\ell \in \partial A\).

Assume from now on that (11.2) holds for some \(A\) and we choose some \(\ell \in \partial A\). Inserting the first-order Taylor expansion (i.e. (11.2) with \(A = \{0\}\)) into the term involving \(D^\ell f\) and using (11.1), we then obtain the identity

$$\begin{aligned} f(x)&= \sum _{k \in \bar{A}} {D^k f(0) \over k!} x^k + \sum _{k \in \partial A \setminus \{\ell \}} \int _{\mathbf{R}^d} D^k f(y)\, {\mathcal Q}^k(x,dy)\\&+ \sum _{i=1}^d \int _{\mathbf{R}^d} D^{\ell + e_i} f(y)\, \bigl ({\mathcal Q}^{e_i}\star {\mathcal Q}^\ell \bigr )(x,dy). \end{aligned}$$

It is straightforward to check that one has the identities

$$\begin{aligned} \mu _m \star \mu _n = \mu _{m+n},\quad \bigl (\mu _ \star \star \mu _{n}\bigr )(x,\cdot ) {=} {x^n \over n!}\mu _\star , \mu _\star \star \mu _\star {=} \mu _\star ,\quad \mu _n \star \mu _\star = 0, \end{aligned}$$

valid for every \(m,n \ge 0\). As a consequence, it follows from the definition of \({\mathcal Q}^k\) that one has the identity

$$\begin{aligned}{\mathcal Q}^{e_i}\star {\mathcal Q}^\ell = \left\{ \begin{array}{ll} {\mathcal Q}^{\ell +e_i} &{} \text {if } i \le \mathfrak {m}(\ell ), \\ 0 &{} \text {otherwise.} \end{array}\right. \end{aligned}$$

The claim now follows from the fact that, by definition, \(\partial \bar{A}\) is precisely given by \((\partial A \setminus \{\ell \}) \cup \{\ell + e_i \,:\, i \le \mathfrak {m}(\ell )\}\). \(\square \)

Appendix B: Symbolic index

In this appendix, we collect the most used symbols of the article, together with their meaning and the page where they were first introduced.

Symbol

Meaning

\(\mathbf {1}\)

Unit element in \(T\)

\(\mathbf {1}^*\)

Projection onto \(\mathbf {1}\)

\(|\cdot |_\mathfrak {s}\)

Scaled degree of a multiindex

\(\Vert \cdot \Vert _\mathfrak {s}\)

Scaled distance on \(\mathbf{R}^d\)

\(|\!|\!|\cdot |\!|\!|_{\gamma ;\mathfrak {K}}\)

“Norm” of a model / modelled distribution

\(\circ \)

Dual of \(\Delta ^+\)

\(\star \)

Generic product on \(T\)

\(*\)

Convolution of distributions on \(\mathbf{R}^d\)

\(A\)

Set of possible homogeneities for \(T\)

\({\mathcal {A}}\)

Antipode of \({\mathcal {H}}_+\)

\({\mathrm {Alg}}({\mathcal {C}})\)

Subalgebra of \({\mathcal {H}}_+\) determined by \({\mathcal {C}}\)

\(\beta \)

Regularity improvement of \(K\)

\({\mathcal {C}}^\alpha _\mathfrak {s}\)

\(\alpha \)-Hölder continuous functions with scaling \(\mathfrak {s}\)

Symbol

Meaning

\(\fancyscript{D}\)

Abstract gradient

\({\mathcal {D}}^\gamma \)

Modelled distributions of regularity \(\gamma \)

\({\mathcal {D}}_P^{\gamma ,\eta }\)

Singular modelled distributions of regularity \(\gamma \)

\(\Delta \)

Comodule structure of \({\mathcal {H}}\) over \({\mathcal {H}}_+\)

\(\Delta ^+\)

Coproduct in \({\mathcal {H}}_+\)

\(\Delta ^{\!M}\)

Action of \(M\) on \(\Pi \)

\({\hat{\Delta }}^{\!M}\)

Action of \(M\) on \(\Gamma \)

\(F\)

Nonlinearity of the SPDE under consideration

\(F_x\)

Factor in \(\Gamma _{xy} = F_x^{-1} F_y\)

\({\mathcal {F}}\)

All formal expressions for the model space

\({\mathcal {F}}_+\)

All formal expressions representing Taylor coefficients

\({\mathcal {F}}_F\)

Subset of \({\mathcal {F}}\) generated by \(F\)

\({\mathcal {F}}_F^+\)

Subset of \({\mathcal {F}}_+\) generated by \(F\)

\(G\)

Structure group

\(\Gamma _g\)

Action of \({\mathcal {H}}_+^*\) onto \({\mathcal {H}}\)

\({\mathcal {H}}_+^*\)

Dual of \({\mathcal {H}}_+\)

\({\mathcal {H}}_+\)

Linear span of \({\mathcal {F}}_+\)

\({\mathcal {H}}_F^+\)

Linear span of \({\mathcal {F}}_F^+\)

\({\mathcal {H}}\)

Linear span of \({\mathcal {F}}\)

\({\mathcal {I}}\)

Abstract integration map for \(K\)

\({\mathcal {I}}_k\)

Abstract integration map for \(D^k K\)

\({\mathcal {J}}(x)\)

Taylor expansion of \(K * \Pi _x\)

\({\mathcal {J}}_k \tau \)

Abstract placeholder for Taylor coefficients of \(\Pi _x \tau \)

\(\mathfrak {K}\)

Generic compact set in \(\mathbf{R}^d\)

\(K\)

Truncated Green’s function of \({\mathcal {L}}\)

\(K_n\)

Contribution of \(K\) at scale \(2^{-n}\)

\(K_{n,xy}^\alpha \)

Remainder of Taylor expansion of \(K_n\)

\({\mathcal {K}},{\mathcal {K}}_\gamma \)

Operator such that \({\mathcal {R}}{\mathcal {K}}f = K* {\mathcal {R}}f\)

\({\mathcal {L}}\)

Linearisation of the SPDE under consideration

\(\Lambda _n^\mathfrak {s}\)

Diadic grid at level \(n\) for scaling \(\mathfrak {s}\)

\({\mathcal {M}}\)

Multiplication operator on \({\mathcal {H}}_+\)

\(M\)

Renormalisation map

\({\hat{M}}\)

Action of \(M\) on \(f\)

\(M_g\)

Action of \(\fancyscript{S}\) on \(T\)

\(\mathfrak {M}_F\)

Basic building blocks of \(F\)

\(\fancyscript{M}_{\fancyscript{T}}\)

All models for \({\fancyscript{T}}\)

\(\fancyscript{M}_F\)

All admissible models associated to \(F\)

\({\mathcal {N}}_\gamma \)

Operator such that \({\mathcal {K}}_\gamma = {\mathcal {I}}+ {\mathcal {J}}+ {\mathcal {N}}_\gamma \)

\((\varvec{\Pi },f)\)

Alternative representation of an admissible model

\((\varvec{\Pi }^M,f^M)\)

Renormalised model

\((\Pi ,\Gamma )\)

Model for a regularity structure

\(\varphi _x^n\)

Scaling function at level \(n\) around \(x\)

\(\psi _x^n\)

Wavelet at level \(n\) around \(x\)

\(P\)

Time \(0\) hyperplane

\({\mathcal {P}}_F\)

Formal expressions required to represent \(\partial u\)

\({\mathcal Q}_\alpha \)

Projection onto \(T_\alpha \)

\({\mathcal Q}_\alpha ^-\)

Projection onto \(T_\alpha ^-\)

Symbol

Meaning

\(\varvec{R}^+\)

Restriction to positive times

\(R\)

Smooth function such that “\(G = K + R\)

\(R_\gamma \)

Convolution by \(R\) on \({\mathcal {D}}^\gamma \)

\({\mathcal {R}}\)

Reconstruction operator

\(\mathfrak {R}\)

Renormalisation group

\(\mathfrak {s}\)

Scaling of \(\mathbf{R}^d\)

\({\mathcal {S}}_{\mathfrak {s},x}^\delta \)

Scaling by \(\delta \) around \(x\)

\({\mathcal {S}}_P^\delta \)

Scaling by \(\delta \) in directions normal to \(P\)

\(\fancyscript{S}\)

Discrete symmetry group

\(T\)

Model space

\(T_\alpha \)

Elements of \(T\) of homogeneity \(\alpha \)

\(T_\alpha ^+\)

Elements of \(T\) of homogeneity \(\alpha \) and higher

\(T_\alpha ^-\)

Elements of \(T\) of homogeneity strictly less than \(\alpha \)

\(T_g\)

Action of \(\fancyscript{S}\) on \(\mathbf{R}^d\)

\({\bar{T}}\)

Abstract Taylor polynomials in \(T\)

\({\fancyscript{T}}\)

Generic regularity structure \({\fancyscript{T}}= (A,T,G)\)

\({\mathcal T}_\beta \)

Classical Taylor expansion of order \(\beta \)

\({\mathcal {U}}_F\)

Formal expressions required to represent \(u\)

\(V,W\)

Generic sector of \(T\)

\(X^k\)

Abstract symbol representing Taylor monomials

\(\Xi \)

Abstract symbol for the noise

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hairer, M. A theory of regularity structures. Invent. math. 198, 269–504 (2014). https://doi.org/10.1007/s00222-014-0505-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-014-0505-4

Mathematics Subject Classification (2000)

Navigation