Regularity structure - Alchetron, The Free Social Encyclopedia

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. The framework covers the Kardar–Parisi–Zhang equation , the Φ 3 4 equation and the parabolic Anderson model, all of which requires renormalization in order for it to be well-defined.

Definition

A regularity structure ( A , T , G ) consists of:

a subset A of R d that's bounded from below and that's without accumulation points;

the model space: a graded vector space T = ⊕ α ∈ A T α , where each T α is a Banach space;

the structure group: a group G of continuous operators Γ such that, for each α ∈ A and each τ ∈ T α , we have ( Γ − 1 ) τ ∈ ⊕ β < α T β .

References

Regularity structure Wikipedia

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