A theory of regularity structures

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. 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 ${\Phi }_3^4$ 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.

180 pages; accepted version with symbolic index

References in corpus (1)

• Homogenization of a singular random one-dimensional PDE

Cited by in corpus (190)

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• Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
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• Multiplicative SHE limit of random walks in space-time random environments
• The Allen-Cahn equation with weakly critical random initial datum
• The ${\Phi }_3^4$ measure has sub-Gaussian tails
• Approximations of dispersive PDEs in the presence of low-regularity randomness
• Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus
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• Well-posedness of Dean-Kawasaki Equation with Singular Interactions
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• Local well-posedness of subcritical non-linear heat equations with Gaussian initial data
• On a Microlocal Version of Young's Product Theorem
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• Singular SPDEs on Homogeneous Lie Groups
• Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs
• Temporal increments of the KPZ equation with general initial data
• Recursive formulae in regularity structures
• Corrigendum to "Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions"
• Anderson Hamiltonians with singular potentials
• Construction and spectrum of the Anderson Hamiltonian with white noise potential on ${\mathbf{R}}^2$ and ${\mathbf{R}}^3$
• Flows driven by multi-indices Rough Paths
• Weak Error of Dean-Kawasaki Equation with Smooth Mean-Field Interactions
• BPHZ renormalisation and vanishing subcriticality asymptotics of the fractional ${\Phi }_d^3$ model
• Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime
• Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems
• Eyring--Kramers law for the hyperbolic ${\varphi }^4$ model
• The critical disordered pinning measure
• Singular limits for stochastic equations
• Rough path theory
• Well-Posedness of the generalised Dean-Kawasaki Equation with correlated noise on bounded domains
• A Stochastic Reconstruction Theorem on Rectangular Increments with an Application to a Mixed Hyperbolic SPDE
• 1D stochastic pressure equation with log-correlated Gaussian coefficients
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• Exponential ergodicity of stochastic heat equations with Hölder coefficients
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