A geometric view upon the renormalisation of stochastic PDEs: the example of $\Phi^4$

Abstract: In this talk, I wish to present some ideas concerning the well-posedness of the $\Phi^4$ equation, which is a stochastic partial differential equation (SPDE) with a cubic nonlinearity and perturbed by an additive random (and rough) noise. More precisely, we are interested in the range of noises where this SPDE is singular (i.e. is classically ill-posed) but subcritical (i.e. the nonlinearity formally vanishes at small scales). In this range, even giving a meaning to the equation is highly non-trivial and relies on an appropriate procedure of regularisation and renormalisation, as was first understood by Da Prato and Debussche (2003) and later widely generalised by several approaches including Hairer's theory of regularity structures (2014). I will, on the one hand, introduce some of the important insights in the theory of singular SPDEs, and, on the other hand, present some more recent contributions. In particular, I will be describing how taking a geometric viewpoint upon the solution manifold gives rise to a new perspective on what in the theory of regularity structures is called a ``model'' for the equation. If time permits, I will also briefly present a recently-developed ``intrinsic'' approach for the actual solution theory, yielding well-posedness of the equation given this model as input. (Based on joint works with Felix Otto, Rhys Steele and Markus Tempelmayr).

MathematicsPhysics

Audience: researchers in the topic

Comments: The talk is in room 004 on ground floor of SISSA, via Bonomean 265. Zoom access is also provided.

Probability, Statistical Mechanics and Quantum Fields

Series comments: Paweł Duch (EPFL) will give a mini-course on "Singular Stochastic PDEs", in the period March 9-20, 2026.