“Regularity structure”的意思、由来-开放百科全书

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Regularity structure

释义

Definition
References

Martin Hairer's theory of regularity structures provides a framework for studying a large class of subcritical parabolic stochastic partial differential equations arising from quantum field theory.^[1] The framework covers the Kardar–Parisi–Zhang equation , the equation and the parabolic Anderson model, all of which require renormalization in order to have a well-defined notion of solution.

Definition

A regularity structure is a triple consisting of:

a subset of that is bounded from below has no accumulation points;
the model space: a graded vector space , where each is a Banach space; and
the structure group: a group of continuous linear operators such that, for each and each , we have .

A further key notion in the theory of regularity structures is that of a model for a regularity structure, which is a concrete way of associating to any and a “Taylor polynomial” based at and represented by , subject to some consistency requirements.

More precisely, a model for on , with consists of two maps

,

.

Thus, assigns to each point a linear map , which is a linear map from into the space of distributions on ; assigns to any two points and a bounded operator , which has the role of converting an expansion based at into one based at . These maps and are required to satisfy the algebraic conditions

,

and the analytic conditions that, given any , any compact set , and any , there exists a constant such that the bounds

,

hold uniformly for all -times continuously differentiable test functions with unit norm, supported in the unit ball about the origin in , for all points , all , and all with . Here denotes the shifted and scaled version of given by

.

References

1. ^{{cite journal|last1=Hairer|first1=Martin|title=A theory of regularity structures|journal=Inventiones Mathematicae|date=2014|doi=10.1007/s00222-014-0505-4|volume=198|pages=269–504|arxiv=1303.5113|bibcode=2014InMat.198..269H}}

3 : Stochastic differential equations|Quantum field theory|Statistical mechanics

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