Five lectures on regularity structures and SPDEs

This set of five lectures provides an introduction to regularity structures and their use for the study of singular stochastic partial differential equations. Two appendices provide some additional informations that enter in the main text either as some technical results or as some results that deepen the context within which we set these lectures.

💡 Research Summary

This manuscript presents a pedagogical series of five lectures that introduce the theory of regularity structures and their application to singular stochastic partial differential equations (SPDEs). The authors begin by recalling basic functional‑analytic notions: Hölder spaces C^r, the product of distributions, and Bony’s paraproduct theorem, which guarantees that the pointwise product of two functions is well‑defined whenever their regularities satisfy r₁+r₂>0. This classical result underpins the standard stochastic calculus for stochastic differential equations (SDEs) driven by Brownian motion, where the stochastic integral is defined probabilistically and does not require pointwise regularity of the integrand.

The second lecture shifts focus to controlled differential equations and rough path theory. When a control path h has Hölder regularity α<½, the usual Riemann–Stieltjes integral fails. Lyons’ rough path framework remedies this by augmenting the path with its first‑order increment X_{s,t}=h_t−h_s and a second‑order “area” term 𝔛_{s,t}, which satisfy algebraic Chen relations and analytic size estimates. The sewing lemma is presented as the key analytical tool that turns an approximate local expansion into a genuine integral map. This construction is elaborated in Appendix 2, providing a self‑contained introduction to controlled rough paths.

The third lecture reviews classical SPDEs driven by space‑time white noise ξ. The canonical form (∂_t−Δ)u = f(u) ξ + g(u,∇u) is examined. Since ξ is a distribution of regularity roughly −3/2−ε, the product f(u)·ξ is ill‑defined by Bony’s theorem, rendering the equation “singular.” The authors explain why traditional Itô calculus cannot handle such products and motivate the need for a new analytical framework.

In the fourth lecture three emblematic singular SPDEs are discussed: (i) the two‑dimensional parabolic Anderson model (∂_t−Δ)u = u ξ^{space}, where ξ^{space} is a spatial white noise of regularity −1−ε; (ii) the Φ⁴₃ equation (∂_t−Δ)u = −u³ + ξ^{space‑time}, with space‑time white noise of regularity −5/2−ε; and (iii) the KPZ equation (∂_t−Δ)u = (∂_x u)² + ξ^{space‑time}. In each case the nonlinearity involves a product of a distribution with a function of insufficient regularity, so the equation is meaningless in the classical sense.

The fifth lecture introduces Martin Hairer’s theory of regularity structures, which provides an algebraic–analytic machinery to give meaning to such products. A model (Π,Γ) lifts the random noise ξ into an abstract model space T equipped with a structure group Δ. The reconstruction operator ℛ maps modelled distributions back to genuine space‑time distributions, thereby defining a solution u. Crucially, the product u·ξ is replaced by a renormalised expression involving a counterterm c^ε(u,∇u). The central result (Theorem 1) asserts that for a broad class of Gaussian and non‑Gaussian noises, there exists a mollification ξ^ε, an explicit deterministic counterterm c^ε, and a random positive time T(ω) such that the solutions of the renormalised equation

(∂_t−Δ)u^ε = f(u^ε) ξ^ε + g(u^ε,∇u^ε) + c^ε(u^ε,∇u^ε)

converge in probability (in the space C(