Title: Probabilistic scaling, propagation of randomness and invariant Gibbs measures.
Abstract: In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will continue by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this context and how it opened the door to unveil the random structures of nonlinear waves that live on high frequencies and fine scales as they propagate. We will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated to nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and of the hyperbolic $\Phi^4_3$ model in the context of constructive quantum field theory. We will end with some open challenges about the long-time propagation of randomness and out-of-equilibrium dynamics.
Title: Well-posedness for the hyperbolic nonlinear Schr\"{o}dinger and Boltzmann equations
Abstract: In this talk, we address two distinct but analytically related topics: the hyperbolic nonlinear Schrödinger equation (HNLS) on $\mathbb{R}\times\mathbb{T}$, and the periodic Boltzmann equation. In the first part, we focus on establishing local and global Strichartz-type estimates for the HNLS. These estimates serve as the foundation for proving local and global well-posedness results. In the second part, we turn our attention to the Boltzmann equation, a fundamental model in the kinetic theory of gases and plasmas. We present a novel Strichartz estimate tailored to the periodic setting, which enables us to prove local well-posedness for the Boltzmann equation. Notably, this result constitutes the first local well-posedness result in the periodic case for regularity thresholds below \(d/2\), which is the limit for energy methods.
Title: Regularization by noise phenomena in nonlinear PDEs with modulated dispersion
Abstract: In this talk, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) with a modulated dispersion. We observe the following regularization-by-noise effects resulting from this modulation:
(i) We establish well-posedness of the modulated KdV on both the circle and real line in the regime where the unmodulated KdV is ill-posed. In particular, we show that the modulated KdV on the circle is locally well-posed in Sobolev spaces of arbitrarily low regularity, provided that the modulation is sufficiently irregular.
(ii) While BO exhibits quasilinear behavior, we show that sufficiently irregular modulations "semilinearize" the equation by proving its local well-posedness via a contraction argument.
(iii) We establish nonlinear smoothing for these modulated equations, where a gain of regularity of the nonlinear part can be (arbitrarily) larger for more irregular modulations.
If time permits, I will discuss more recent results on the stochastic modulated KdV on the circle with multiplicative noise, where we exhibit a new type of regularization-by-noise phenomenon on the stochastic term.
Based on joint works with Khalil Chouk (formerly Edinburgh), Massimiliano Gubinelli (Oxford), Tadahiro Oh (Edinburgh), Guopeng Li (BIT), and Andreia Chapouto (Versailles).
Title: Invariant Gibbs measure for 3D cubic NLW
Abstract: In this talk, we'll present our results about invariant Gibbs measures for the periodic cubic nonlinear wave equation (NLW) in 3D. The interest in this result stems from connections to several areas of mathematical research. At its core, the result concerns a refined understanding of how randomness gets transported by the flow of a nonlinear equation which involves probability theory and partial differential equations. This is joint work with Bjoern Bringmann (Princeton), Yu Deng (UChicago) and Andrea Nahmod (UMass Amherst).
Title: Random averaging operator, random tensor, and fractional NLS
Abstract: In this talk, we will consider the Schrödinger equation with cubic nonlinearity on the circle, with initial data distributed according to the Gibbs measure. We will discuss the challenges and strategies involved in establishing the Poincaré recurrence property with respect to the Gibbs measure in the full dispersive range. This work, using the theory of the random averaging operator developed by Deng-Nahmod-Yue '19, addresses an open question proposed by Sun-Tzvetkov '21. We will also explain why the Gibbs dynamics for the full dispersive range is sharp in some sense. Additionally, we will see how the theory of random tensors works for extending this work to multi-dimensional settings.
Title: A review of "classical" stochastic Schrodinger Equations
Abstract: I will speak about works with Fabian Hornung and Lutz Weis about the existence of a global solution to the stochastic nonlinear Schrödinger equation (SNSL) on a 3-dimensional manifold with multiplicative Gaussian (and jump) noise and the uniqueness for such equations in case of Gaussian bilinear noise. I will lthen speak about works with Benedetta Ferrario and Margherita Zanella about the existence of stationary solutions and the existence and uniqueness of invariant measures for such equations. I will conclude with describing a recent result with BF, MZ and Marion Maurelli for the existence and uniqueness of global strong solutions to such equations driven by a special form of a Winer process. Some of the proof of which is based on novel Strichartz estimates and Littlewood-Paley and decomposition in time.
Title: Critical phenomena in the study of focusing Gibbs measures
Abstract: In this talk, I will discuss several critical models in the study of focusing Gibbs measures. In a seminal paper (1988), Lebowitz-Rose-Speer initiated the study on the (non-)construction of focusing Gibbs measures, which was continued by Bourgain (1994, 1997). Over the last five years, there has been significant progress in the subject. In particular, in recent breakthrough works, Oh-Sosoe-Tolomeo (2022) and Oh-Okamoto-Tolomeo (2025) completed the research program on the $d$-dimensional torus, initiated by Lebowitz-Rose-Speer and Bourgain by exhibiting phase transitions for critical models in the 1-$d$ and 3-$d$ cases.
I will first revisit the two-dimensional case. By considering weakly interacting quartic potentials, I identify a critical threshold for the strength of interaction and thus exhibit a new phase transition, thus answering an open question posed by Brydges-Slade (1996). Then, I will revisit the one-dimensional case, where the phase transition in terms of mass was shown by Lebowitz-Rose-Speer and Oh-Sosoe-Tolomeo. For this problem, we provide an optimal blowup rate of the partition function in the mass critical and supercritical regimes. Lastly, I will consider the mass-critical focusing Gibbs measure with the harmonic potential. By considering weakly interacting potentials, an analogous phase transition has been established.
This talk is based on joint works with Guopeng Li, Rui Liang, Tadahiro Oh, Liying Tao, Leonardo Tolomeo and Yuzhao Wang.
Title: Quasi-invariance and singularity for the cubic Szego equation
Abstract: We shall consider the flow of a family of Gaussian fields under the dynamics of the cubic Szego equation; a toy model for a weakly-dispersive Hamiltonian system. We see that above a critical regularity, the measures are quasi-invariant under the flow (almost-sure properties of the data are preserved), but below this regularity, quasi-invariance fails. In fact, the distribution at almost every time is singular with respect to the initial distribution.
We introduce a method to prove singularity by exhibiting an instantaneous growth of Sobolev norms of the solution, coupled with an abstract argument to show that such a growth cannot occur with positive probability at all times. We will also discuss some heuristics as to when one expects quasi-invariant flow, and how similar methods can be applied to understanding invariant measures for some 2D stochastic Navier-Stokes equations.
Title: Radon measure-valued solutions for compressible Euler equations with applications in discontinuous flow and phase transitions of droplets
Abstract: In this talk, we focus on the construction and applications of Radon measure-valued solutions for the compressible Euler equations. We begin with the Riemann problem for the one-dimensional isentropic Euler equations, where the flux function changes across a flow discontinuity curve x=x(t) separating two different polytropic gases. To capture the concentration of mass and momentum at the discontinuity, we construct Radon measure-valued solutions.
However, in many realistic physical processes, especially in complex systems involving energy exchange, isentropic assumptions are often difficult to satisfy. To extend the study, we consider the non-isentropic case and investigate the motion and mass growth of droplets with phase transitions in a homogeneous medium. Within the Radon measure framework, the gas phase is described by the regular part of the measure, while the droplet is modeled by its atomic part. We prove the existence of local and global solutions for a single droplet and analyze the collision of two droplets.
Due to the non-uniqueness and potential non-solvability of piecewise constant solutions in both isentropic and non-isentropic settings, we turn to piecewise smooth solutions to ensure uniqueness. We further investigate singular Riemann problems with initial mass concentration under various physical assumptions. We establish the existence and uniqueness of local solutions composed of delta shocks and classical waves, and demonstrate their stability under small perturbations of the initial data.