Regularity structure


'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 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 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