Regularity structures and paracontrolled overviews

A regularity structure has a graded space T with homogeneities in a locally finite set bounded below, and a reexpansion group that lowers homogeneity. A model assigns to each base point x a map $\Pi_x$ from symbols to distributions and consistent transport operators $\Gamma_{xy}$. Compatibility requires $\Pi_x\Gamma_{xy}=\Pi_y$ and the group composition law. Analytic model bounds control how a symbol of a given homogeneity behaves against rescaled test functions, with the equation's chosen scaling.

A modelled distribution is an expansion whose coefficients at nearby points agree after transport up to a quantified remainder. In the standard local reconstruction theorem, a positive regularity index gamma, the required model bounds and the modelled-distribution bounds yield a unique distribution approximated by those local expansions. The positivity condition is important for uniqueness in this formulation. Initial-time singularities, weights, integration maps and nonlinear fixed points require additional structure; the reconstruction theorem alone is not a complete SPDE existence result.

Bony's decomposition separates a product into low–high, high–low and resonant frequency interactions. In Hölder–Besov spaces, the resonant term is continuously defined when the two regularity indices sum to a positive number. For a negative-regularity input and a positive but insufficiently regular coefficient, the sum can fail that condition. Special functions may still multiply, but there is no general continuous product supplied by those regularities alone.

A typical paracontrolled ansatz writes $u=u'\prec X+u^\sharp$, with X a rough reference response and a remainder of better regularity. Expanding the troublesome resonance with ξ then identifies an enhanced object such as $X\circ\xi$, together with commutators and more regular remainders. Probability constructs the enhanced resonance through a specified renormalized approximation. The permissible exponents and number of expansion levels depend on the equation; this schematic ansatz is not a universal formula for all singular products.

A typical construction first proves continuity of a deterministic solution map in an enhanced model norm, locally in time or up to a stopping criterion. It then proves convergence in probability of the renormalized enhanced noises in that norm. Reconstruction or paracontrolled interpretation transfers the limit to a distribution-valued solution. The finite counterterm convention is part of the data. These steps explain why convergence of smooth noise approximations in a weak distribution norm alone is usually insufficient.

Local subcriticality means that the singularity hierarchy becomes more regular under the recursive operations relevant to the equation, leaving finitely many problematic symbols below any required cutoff. The elementary polynomial heat power count is one guide, not the full definition for derivative or quasilinear equations. Global bounds, ergodicity, universality and critical models require further results. Neither framework should be advertised as automatically solving arbitrary stochastic PDEs; its assumptions and the particular enhanced stochastic objects remain central to every application.