Bjoern Bringmann — Princeton University
The goal of these lectures is to study the global well-posedness of singular SPDEs. As a warm-up, we first prove the global well-posedness of the $\Phi^4_2$-model. We then turn to the global well-posedness of the stochastic Abelian-Higgs model, which is a stochastic geometric evolution equation. In this context, we also discuss covariant monotonicity formulas and covariant stochastic objects.
Scott Armstrong — Sorbonne University and CNRS
We study Brownian motion advected by a stationary, divergence-free random drift whose spatial correlations decay slowly. Such long-range dependence is expected to produce {\it superdiffusion}: for a typical realization of the drift, the variance of the displacement of the particle grows faster than linearly in time, with a precise exponent determined by the correlation structure of the drift. This behavior was predicted in the physics literature around 1990 via perturbative renormalization group heuristic arguments. We recast the problem in PDE terms via the infinitesimal generator of the stochastic process, which is a divergence-form drift-diffusion operator, whose random coefficients exhibit an approximate self-similarity across scales. Our main tool is a scale-by-scale coarse-graining (renormalization) scheme. At each scale we compare the operator to an effective Laplacian with a diffusivity that depends on the scale, and we obtain quantitative control of the approximation error. Because of the self-similar structure, this procedure must be iterated across all scales: the operator does not converge to a single homogenized limit; instead, the analysis yields a scale-dependent effective diffusivity. This may be viewed as a rigorous counterpart of the earlier renormalization group predictions. A key analytic input is an {\it anomalous regularization} phenomenon: we prove near-Lipschitz regularity for solutions that are uniform in the microscopic (molecular) diffusivity parameter, allowing quantitative control even in the regime of very weak underlying diffusion. In other words, the chaotic behavior of the random drift produces regularity in the solutions of the equation. The talk is based on joint work with Ahmed Bou-Rabee (UPenn) and Tuomo Kuusi (Helsinki) which is available here: https://arxiv.org/abs/2601.22142
Aurelien Deya — Université de Lorraine and CNRS
We will first present the physical motivations behind this model, then turn to the comparison with its "standard" counterpart, in which the harmonic oscillator (on $\mathbb{R}^3$) is replaced by the Laplacian on the torus. The results, from joint work with Reika Fukuizumi (Tokyo) and Laurent Thomann (Nancy), include the interpretation of the model in the renormalized sense, the global existence and uniqueness of a solution, the existence of an invariant measure, and finally its uniqueness in the weak nonlinearity regime.
Hugo Eulry — ENS Lyon
Invariant measures play a central role in understanding the long-time behavior of dynamical systems, as they describe their statistical equilibrium. Conversely, understanding the dynamics provides valuable information about the corresponding invariant measures. In this talk, we will focus on the dynamical approach to invariant measures. After presenting the general strategy for proving unique ergodicity in case of parabolic SPDEs, we will discuss how these methods can be adapted to a non-translation-invariant framework for dynamics driven by a random operator. We will then show that similar ideas can also be adapted for dispersive equations, such as the stochastic damped wave equation, even when crucial properties such as the strong Feller property fail. This is based on joint works with Antoine Mouzard and Nikolay Tzvetkov.
Yingtong Hou — Université de Lorraine
In this talk, we provide a method to obtain the renormalised measures in quantum field theory by renormalising the cumulant expansion of the original measures. After reviewing the main ingredients of renormalisation: cumulant expansion, Wick renormalisation, Feynman diagrams, we introduce our approach based on BPHZ renormalisation via multi-indices which are combinatorial objects originating from describing scalar-valued singular SPDEs. To automate the renormalisation procedure, we propose a multi-index counterpart of the Hopf-algebraic program initiated by Connes and Kreimer for the renormalisation of Feynman diagrams. This is a joint work with Yvain Bruned.
Elias Nohra — Sorbonne University
In this talk, I will present some recent contributions to the study of the 2D Yang--Mills measure, based on joint works with Nguyen Viet Dang (Strasbourg). I will present results concerning the construction of the 2D Yang–Mills measure as a random distributional connection on general surfaces, using a novel gauge: the Morse gauge. I will describe its appearance as a universal scaling limit of lattice gauge models, and its semiclassical (small-area) limit.
Nikolay Tzvetkov — ENS Lyon
In the first part of the talk, in dimension $d$, we will consider a free energy associated with a fractional Laplacian of order $\alpha/2$ in the presence of a quartic interaction. We will recall that for $\alpha>d/2$ the construction of the Gibbs measure is straightforward that for $3d/8<\alpha\leq d/2$ after a renormalisation the Gibbs measure is absolutely continuous with respect to the Gaussian measure resulting from the free energy and that for $d/4<\alpha\leq 3d/8$ the Gibbs measure is non trivial and singular with respect to the same free Gaussian measure. The critical case is therefore $\alpha=d/4$. The main new result is that in the 4d critical case we can construct a non trivial radial Gibbs measure which is non Gaussian and singular with respect to the corresponding Gaussian measure. As one may expect in such an energy supercritical situation the potential energy is treated a non perturbative way. This result is in sharp contrast with the non radial situation in which a triviality result is known. This is a joint work with Hiro Oh, Leonardo Tolomeo and Yuzhao Wang.