Brazilian Journal of Probability and Statistics

Introduction to regularity structures

Martin Hairer

Full-text: Open access

Abstract

These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in the article [Invent. Math. DOI: 10.1007/s00222-014-0505-4]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. We focus on applying the theory to the problem of giving a solution theory to the stochastic quantisation equations for the Euclidean $\Phi^{4}_{3}$ quantum field theory.

Article information

Source
Braz. J. Probab. Stat. Volume 29, Number 2 (2015), 175-210.

Dates
First available in Project Euclid: 15 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1429105588

Digital Object Identifier
doi:10.1214/14-BJPS241

Mathematical Reviews number (MathSciNet)
MR3336866

Zentralblatt MATH identifier
1316.81061

Keywords
Stochastic PDEs regularity structures renormalisation

Citation

Hairer, Martin. Introduction to regularity structures. Braz. J. Probab. Stat. 29 (2015), no. 2, 175--210. doi:10.1214/14-BJPS241. https://projecteuclid.org/euclid.bjps/1429105588.


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References

  • Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89, 347–386.
  • Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183, 571–607.
  • Bourgain, J. and Pavlović, N. (2008). Ill-posedness of the Navier–Stokes equations in a critical space in 3D. J. Funct. Anal. 255, 2233–2247.
  • Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41, 909–996.
  • Da Prato, G. and Debussche, A. (2003). Strong solutions to the stochastic quantization equations. Ann. Probab. 31, 1900–1916.
  • Giacomin, G., Lebowitz, J. L. and Presutti, E. (1999). Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems. In Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr. 64, 107–152. Providence, RI: Amer. Math. Soc.
  • Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216, 86–140.
  • Gubinelli, M. (2010). Ramification of rough paths. J. Differential Equations 248, 693–721.
  • Hairer, M. (2011). Rough stochastic PDEs. Comm. Pure Appl. Math. 64, 1547–1585.
  • Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178, 559–664.
  • Hairer, M. (2014). A theory of regularity structures. Invent. Math. DOI:10.1007/s00222-014-0505-4.
  • Jona-Lasinio, G. and Mitter, P. K. (1985). On the stochastic quantization of field theory. Comm. Math. Phys. 101, 409–436.
  • Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892.
  • Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics 1908. Berlin: Springer. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004. With an introduction concerning the Summer School by Jean Picard.
  • Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14, 215–310.
  • Meyer, Y. (1992). Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge: Cambridge Univ. Press. Translated from the 1990 French original by D. H. Salinger.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. Probability and Its Applications (New York). New York: Springer-Verlag.
  • Parisi, G. and Wu, Y. S. (1981). Perturbation theory without gauge fixing. Sci. Sinica 24, 483–496.
  • Simon, L. (1997). Schauder estimates by scaling. Calc. Var. Partial Differential Equations 5, 391–407.
  • Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282.