theory of regularity structures
E926684
The theory of regularity structures is a mathematical framework developed by Martin Hairer to rigorously analyze and solve a broad class of singular stochastic partial differential equations.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
analytical framework
ⓘ
mathematical theory ⓘ |
| appliesTo |
Φ^4_3 stochastic PDE
ⓘ
KPZ equation NERFINISHED ⓘ a broad class of singular SPDEs ⓘ stochastic heat equation with multiplicative noise ⓘ stochastic quantization equations ⓘ |
| basedOn |
ideas from renormalization in quantum field theory
ⓘ
ideas from rough path theory ⓘ |
| characteristic |
generalizes Taylor expansions to irregular functions and distributions
ⓘ
provides a robust solution theory for highly singular equations ⓘ separates analytic and probabilistic aspects of SPDEs ⓘ uses graded structures of modelled distributions ⓘ |
| contributedTo |
rigorous construction of
Φ^4_3 quantum field models via SPDEs
ⓘ
solution of the KPZ equation in one dimension ⓘ |
| coreConcept |
model
ⓘ
modelled distribution ⓘ reconstruction operator ⓘ regularity structure ⓘ renormalization of models ⓘ structure group ⓘ |
| describedIn | article "A theory of regularity structures" NERFINISHED ⓘ |
| developer | Martin Hairer NERFINISHED ⓘ |
| difficulty | advanced graduate and research-level theory ⓘ |
| field |
mathematical physics
ⓘ
partial differential equations ⓘ probability theory ⓘ stochastic analysis ⓘ stochastic partial differential equations ⓘ |
| impact |
provided a unifying framework for many previously ad hoc renormalization techniques
ⓘ
transformed the study of singular SPDEs ⓘ |
| inspiredBy | Taylor polynomials and jets ⓘ |
| introducedBy | Martin Hairer NERFINISHED ⓘ |
| introducedIn | 2014 ⓘ |
| publishedIn | Inventiones Mathematicae NERFINISHED ⓘ |
| purpose |
construction of solutions to singular SPDEs
ⓘ
renormalization of ill-posed stochastic PDEs ⓘ rigorous analysis of singular stochastic partial differential equations ⓘ |
| recognizedBy | Fields Medal citation of Martin Hairer in 2014 ⓘ |
| relatedTo |
paracontrolled distributions
ⓘ
renormalization group methods ⓘ rough path theory ⓘ |
| usesTool |
Banach space techniques
ⓘ
combinatorial Hopf algebras ⓘ multiscale analysis ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.