A theory of regularity structures View Full Text


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Article Info

DATE

2014-11

AUTHORS

M. Hairer

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 Φ34 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. More... »

PAGES

269-504

References to SciGraph publications

  • 1985-09. On the stochastic quantization of field theory in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2010-08. Hölder-Continuous Rough Paths by Fourier Normal Ordering in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2004-08. Trees, Renormalization and Differential Equations in BIT NUMERICAL MATHEMATICS
  • 2009-02-23. Analysis of Brownian functionals in STOCHASTIC SYSTEMS: MODELING, IDENTIFICATION AND OPTIMIZATION, I
  • 2009-06. Taylor formula for homogenous groups and applications in MATHEMATISCHE ZEITSCHRIFT
  • 2002-01. Stochastic analysis, rough path analysis and fractional Brownian motions in PROBABILITY THEORY AND RELATED FIELDS
  • 1995. The Malliavin Calculus and Related Topics in NONE
  • 1974-03. On the Butcher group and general multi-value methods in COMPUTING
  • 1968-02. Boson fields with the :Φ4: Interaction in three dimensions in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1974-06. The λϕ34 field theory in a finite volume in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2006-11. A note on the notion of geometric rough paths in PROBABILITY THEORY AND RELATED FIELDS
  • 2000-02. Scaling Limits of Wick Ordered KPZ Equation in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1936-12. An inequality of the Hölder type, connected with Stieltjes integration in ACTA MATHEMATICA
  • 2011. Fourier Analysis and Nonlinear Partial Differential Equations in NONE
  • 1990-05. Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1955-12. On the theory of general partial differential operators in ACTA MATHEMATICA
  • 1997-02. Stochastic Burgers and KPZ Equations from Particle Systems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1991-09. Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms in PROBABILITY THEORY AND RELATED FIELDS
  • 2012-02. Singular perturbations to semilinear stochastic heat equations in PROBABILITY THEORY AND RELATED FIELDS
  • 1911-09. Über die Bewegungsgruppen der Euklidischen Räume in MATHEMATISCHE ANNALEN
  • 1982-03. Geometric analysis of φ4 fields and Ising models. Parts I and II in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1997-07. Schauder estimates by scaling in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2000-03. Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1912-09. Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich in MATHEMATISCHE ANNALEN
  • 1996. Stochastic partial differential equations in STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
  • 2001-01. Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2011-12. Approximations to the Stochastic Burgers Equation in JOURNAL OF NONLINEAR SCIENCE
  • Journal

    TITLE

    Inventiones Mathematicae

    ISSUE

    2

    VOLUME

    198

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00222-014-0505-4

    DOI

    http://dx.doi.org/10.1007/s00222-014-0505-4

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1027828712


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