· 2016
A theory of regularity structures
- 816citations
- 2016year
Short summary
A new algebraic framework, 'regularity structures,' is introduced to describe functions and distributions using generalized local Taylor expansions, moving beyond classical polynomial models.
AI-generated from the title and abstract; the full text is not read.
Key points
- Introduces 'regularity structures' for algebraic description of functions/distributions via generalized local Taylor expansions.
- Replaces classical polynomial models with problem-specific models for broader applicability.
- Enables rigorous formulation of fixed-point equations for semilinear PDEs with singular inputs.
- Provides mathematical rigor for previously ill-defined stochastic PDEs arising in physics.
AI-generated from the title and abstract; the full text is not read.
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
The authors' abstract, as published at the source., 2016 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics