Abstracts

Below are the titles, speakers, and abstracts of the mini-courses and seminar talks of the workshop.

Mini-course 1

Global well-posedness for singular SPDEs

Hendrik Weber — University of Münster

Initiated in groundbreaking works by Hairer and Gubinelli, the theory of Parabolic Stochastic PDEs has made spectacular progress over the last few years. A systematic solution theory for various interesting and previously intractable equations from Mathematical Physics is now available. Examples include the KPZ equation and the stochastic quantisation equations for the 3D $\Phi^4$ and Yang Mills measures. These solutions have been shown to arise as scaling limits of discrete models of Statistical Mechanics and to display interesting phenomena, such as phase transitions, in their own right.

In these lectures I will review some aspects of these developments. The main focus will be the question of global existence / a priori bounds for solutions. The first works on regularity structures and paracontrolled distributions were mostly concerned with constructing local-in-time solutions using a fixed point argument in a suitable space of distributions. The focus was primarily on the description of solutions on small scales and dealing with "infinite terms" which arise when applying non-linear functions to distributions. The matching a priori bounds were only developed more recently, among others in a series of papers by Mourrat, Moinat, Chandra, Feltes, Esquivel and me. I will explain these results, the main challenges and the arguments that permitted to overcome them.

Mini-course 2

Introduction to the flow equation approach to singular stochastic PDEs

Paweł Duch - EPFL

This course offers a gentle introduction to the flow equation approach to singular stochastic partial differential equations (SPDEs). The method provides a flexible framework for analyzing a wide class of such equations, including those involving fractional Laplacians across the full subcritical regime. Inspired by the Wilsonian renormalization group, the approach focuses on a coarse-grained process that captures the behavior of solutions across spatial scales. Rather than analyzing the original equation directly, one studies an effective equation whose nonlinear component evolves with scale according to the flow equation, analogous to the Polchinski equation in quantum field theory. Within this framework, renormalization amounts to selecting appropriate boundary conditions for the flow equation, which can be implemented through an inductive argument.

Talk 2

Recent Progress on BPHZ Theorems in Regularity Structures

Rhys Steele — Max Planck Institute for Mathematics in the Sciences

Modern pathwise approaches for subcritical singular stochastic PDEs obtain local-in-time solutions by encoding the probabilistic aspects of the problem in a random element of some nonlinear space of distributions. In regularity structures, this non-linear data is called the model. The construction of an appropriate model from a typical driving noise is one of the central challenges in the theory and is the step in which one must perform a suitable renormalisation. In recent years, there has been significant progress on general constructions of renormalised models in regularity structures. In this talk, I will review this progress and discuss ongoing projects building on these new ideas.

Short talk 1

From skew to reflected SPDEs

Thomas Le Guerch — Sorbonne University

Skew Brownian motion (SBM) is a one-dimensional diffusion that behaves like standard Brownian motion away from $0$, but whose crossings at 0 are asymmetric: upon hitting $0$, it resumes on the positive side with probability $p$. It can be characterised as the unique strong solution to a stochastic equation driven by additive Brownian motion with drift given by $2p-1$ times the symmetric local time at $0$ of the solution. When $p=1$, the solution remains nonnegative and the corresponding process is the reflected Brownian motion (RBM). The RBM satisfies a stochastic equation with reflection. In the context of SPDEs, the analogue of this skew equation is highly singular and involves a Dirac delta nonlinearity acting on the solution. In a joint work with C. Labbé and L. Zambotti, we prove that the solution of a skew SPDE converges to the solution of a reflected SPDE.

Short talk 2

A priori estimates for rough ODEs in the full range of regularity, Part 1

Lorenzo Agabiti — Sorbonne University

Since the seminal work of Lyons, rough path theory has emerged as a powerful framework for solving ordinary differential equations (ODEs) driven by signals of low regularity, where classical methods fail. In the first part of this joint talk, we introduce Davie's approach to rough ODEs within the context of branched rough paths. Our primary focus is on deriving robust a priori estimates for the solutions, with the ultimate goal of covering the full spectrum of driver regularity. Achieving this requires establishing a sewing bound on increments and developing an explicit algebraic formula for the remainders, which fundamentally relies on the pre-Lie algebraic structure of forest grafting.

Short talk 3

A priori estimates for rough ODEs in the full range of regularity, Part 2

Alberto Bonicelli — Sorbonne University

Building on the foundational a priori estimates for rough ODEs introduced in Part 1, the second half of this joint talk addresses the challenge of relaxing the boundedness assumptions on the coefficients of the approximated solution. To effectively handle the Lipschitz-continuous case, we require a refined representation of the remainder that exclusively encompasses the gradients of the coefficients. Because manual derivation becomes intractable at arbitrarily high approximation orders, we employ advanced algebraic tools. Specifically, we introduce a novel representation of the Guin-Oudom extension of grafting using binary trees. Finally, we demonstrate how this algebraic remainder formula yields the precise a priori estimates necessary to establish well-posedness across a broad class of equations.

Short talk 4

A Hölder calculus of currents

Gaining regularity from antisymmetry

Thomas Jaffard — Sorbonne University

Distributions model a wide range of physical phenomena, but difficulties arise as soon as nonlinear phenomena come into play — starting with the product of two distributions, which has in general no canonical meaning. For Hölder distributions on $\mathbb{R}^d$, the situation is sharp: a canonical product can be defined as soon as the regularities of the two factors compensate — that is, their sum is positive. A current generalizes such a distribution to a k-dimensional object in $\mathbb{R}^d$: a differential form whose coefficients are distributions, modeling surfaces or more singular geometric objects. For such objects geometry changes the picture: the antisymmetry of the wedge structure can generate cancellations, making certain products canonically meaningful even when the sum of the regularities is nonpositive. We then build a Hölder calculus of currents, where these cancellations and the Reconstruction Theorem combine to gain regularity, and apply it to the Jacobian currents associated to Hölder functions, recovering the classical Young and Züst integrals.

Short talk 4

Critical Stationary Fluctuations for a reaction-diffusion model

Luis Cardoso — IMPA/University of Münster

In this talk, we consider an interacting particle system in dimension 1 whose diffusive scaling limit is given by a reaction-diffusion equation. At criticality, we go beyond this scaling and investigate the system's fluctuations. Building on previous work on the dynamical case, we study the fluctuations with respect to the non-reversible invariant measure. We show that the global density, properly rescaled, exhibits non-Gaussian limiting behavior, characterized explicitly by a fourth-order exponential density. Furthermore, the density field acting on zero-mean spatial modes possesses strictly smaller, Gaussian fluctuations. Consequently, the rescaled spatial field completely projects onto the global density. This establishes a non-Gaussian fluctuation for the stationary state of a short-range system associated with the $\Phi^4$ universality class. This is based on joint work with Landim and Tsunoda.