Schedule, Notes/Handouts & Videos

Synchronization phenomena describe the long-time behavior of systems composed of interacting particles or agents that eventually reach a common state. In this talk, we will present several stochastic models and investigate their synchronization properties under both additive and multiplicative noise. We will also explore an application to networked systems, modeled by stochastic reaction-diffusion equations on a graph.

I will talk about stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. The idea of using Grassmann random variables for fermionic systems goes back to Osterwalder and Schrader, and to implement it we used tools from non-commutative analysis. I will describe some of the stochastic analytical tools such as Grassmann Brownian motion and non-​commutative $L^{p}$ spaces, and mention some of the applications and open questions. Based on the joint work with F. De Vecchi, L. Fresta and M. Gubinelli.

We study the singular stochastic wave equation on $\T^2$, with a cubic nonlinearity and Gaussian rough `Matérn' forcing (a Fourier multiplier of order $\alpha>0$ applied to space-time white noise) and establish local well-posedness for $\alpha < \tfrac{3}{8}$. This extends \cite{GKO18} beyond white noise and strengthens the quadratic-case result \cite{OO21} ($\alpha<\f 12$). Our argument develops new trilinear estimates in Bourgain spaces together with case-specific cubic counting estimates.

We establish a notion of universality for the parabolic An- derson model via an invariance principle for a wide family of parabolic stochastic partial differential equations. We then use this invariance principle in order to provide an asymptotic theory for a wide class of non-linear SPDEs. A novel ingredient of this invariance principle is the dissipativity of the underlying stochastic PDE.

I will discuss recent work which shows global well-posedness of the dynamical sine-Gordon model up to the third threshold, i.e. 6π. The key novelty in the approach is the introduction of the so-called resonant equation, whose solution is entirely deterministic and which completely captures the size of the solution to the dynamical sine-Gordon model. The probabilistic fluctuations in the dynamical sine-Gordon model are then controlled using uniform estimates for modified stochastic objects. Joint with Bjoern Bringmann.

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton. This is joint work with S. Cao.

The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) which was introduced in the eighties by van Beijren, Kutner and Spohn to describe, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with one conserved scalar quantity. Following a Physics heuristics, the non-linearity is usually chosen to be quadratic as this is the first term that cannot be removed via a Galileian transformation and should thus provide the first non-trivial contribution to the dynamics. As shown by Hairer and Quastel in dimension 1 in the so-called weakly asymmetric scaling, such derivation is not fully correct and, when considering a generic non-linearity higher order terms do contribute to the limit. In the present talk, we consider the critical dimension 2 and prove that, under a logarithmically superdiffusive scaling (no weak asymmetry is required), the same is true, meaning that the limiting contribution of the non-linearity comes in via its second order coefficient of its Hermite expansion. We conclude with some remarks in higher dimension, for which instead every term in the expansion does contribute to the limit.  

This is a joint work with Q. Moulard and T. Klose. 

Energy Lagrangian flows are an extension of the Lagrangian flows of Di Perna-Lions and their stochastic counterparts by Le Bris-Lions and Figalli to the setting of singular SPDEs, partly in regimes where classical and pathwise theories break down. A key notion is "probabilistic subcriticality": even for scaling critical or supercritical equations, regularity of the law combined with coercivity of the generator may yield existence and uniqueness. I will outline the construction and well-posedness results, the relation to pathwise constructions, and applications to quantitative convergence theorems. Based on joint works with Ana Djurdjevac, Lukas Gräfner, and Shyam Popat.

In this talk, we will review some recent applications of operad theory to singular SPDEs.

It is an essential tool for characterising the chain rule symmetry in the full subcritical regime which

leads to renormalising quasilinear SPDEs with local counterterms. Also, it provides negative results

about the existence of another combinatorial set lying between multi-indices and decorated trees for

encoding expansions of solutions of singular SPDEs.

We derive the Φ4 3 measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas. To be precise, starting from many-body quantum mechanics, where the problem is linear and regular but involving non commutative operators, we justify the emergence of the Φ4 3 measure as a semiclassical limit which captures the formation of Bose–Einstein condensation just above the critical temperature. We employ and develop several tools from both stochastic quantization and many-body quantum mechanics. Since the quantum problem is typically formulated using a nonlocal interaction potential, our first key step involves approximating the Φ4 3 measure through a Hartree measure with nonlocal interaction, achieved by paracontrolled calculus. The connection between the quantum problem and the Hartree measure emerges through a variational interplay between classical and quantum models.