Group Presentations' Day: March 31st
This is a one-day event where we share the on going and lastest research results of our group
Schedule:
10h-10:15h Opening (coffee in the department)
10:15h-10:45h Beatriz Salvador
10:45h-11:15h João Pedro Mangi
11:15h-12h Gerardo Vargas (part 1)
12h-14h Lunch
14:00h-14:45h Léonard Monsaingeon
14:45h-15:15h Maria Chiara Ricciuti
15:15h-16h Jean-Baptiste Casteras
16h-16:45h Gerardo Vargas (part 2)
Titles and Abstracts:
Beatriz Salvador - A duality and random walk approach to control correlations
In this talk I will show how one can use stochastic duality to close evolution equations for time-dependent correlation functions and how to use these equations to obtain $L^\infty$ bounds for these functions through a random walk approach.
I will give some examples of microscopic models for which this technique can be applied and obtain for those the decay of 2-point correlation functions. If time allows, I will explain the main difficulties for the case of k-point correlation functions for k > 2.
Based on joint works with Chiara Franceschini, Patrícia Gonçalves and Milton Jara.
João Pedro Mangi - Stochastic Oscillators Out of Equilibrium: Scaling Limits and Correlation Estimates
We consider a purely harmonic chain of oscillators perturbed by a stochastic noise. Under this perturbation, the system exhibits two conserved quantities: volume and energy. At the hydrodynamic level, in the diffusive time scale, we show that depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or according to a non-linear system of coupled parabolic equations. At the level of fluctuations, we show that, also in diffusive time scale, under any initial measure, the volume fluctuation field converges. The proofs are based on a precise analysis of the two-point correlation function and a uniform fourth moment bound. We also discuss some open problems and the technical issues faced when studying higher order correlation functions in this models of stochastic oscillators. Joint work with Patricia Gonçalves (IST Lisbon) and Kohei Hayashi (University of Osaka).
Gerardo Vargas - (Part 1) Coming down from infinity and sharp convergence to equilibrium
In this presentation, we study an ordinary differential equation with a unique degenerate attractor at the origin, perturbed by the addition of Brownian noise with a small parameter that regulates its magnitude. Under general conditions, for any fixed noise magnitude, the solution to this SDE converges exponentially fast in total variation distance to its unique equilibrium distribution as time goes by.
We suitably accelerate the random dynamics and establish that this convergence occurs in a sharp form. More precisely, the total variation distance between the accelerated dynamics and its equilibrium distribution converges to a non-degenerate limiting profile. This profile corresponds to the total variation distance between the marginal distribution of an appropriately defined SDE, which comes down from infinity, and its associated equilibrium distribution. We point out that the limiting profile that emerges from this convergence is not a step function, which is typically observed in the context of the cut-off phenomenon for random processes.
This talk is based in a paper in SPA 2025 with Conrado da Costa (Durham University, UK) and Milton Jara (IMPA, Brazil).
Léonard Monsaingeon - (Sticky) optimal transport and diffusion
In this talk I will review the 1998 Jordan-Kinderlehrer-Otto interpretation of Fokker-Planck equations as a Wasserstein gradient flow in the space of probability measures, based on optimal transport theory. I will also discuss the corresponding derivation of microscopic-to-macroscopic dissipation, in the language of large deviation and short time limits. This provides a somehow canonical recipee to study vatiational structures for diffusion processes.
If time permits I will also discuss a counter-example to this general idea, based on the so-called (reflected) Sticky Brownian Motion SBM.
Based on joint works with JB Casteras, L. Nenna and F. Santambrogio.
Maria Chiara Ricciuti - Scaling Limits of Weakly Perturbed Random Interface Models
In this talk, we consider a class of random interface models on the one-dimensional discrete torus $T_N$ parametrised by a positive map $\Phi_N$ on $T_N\times\mathbb{R}$. These models share a weak perturbation, namely an asymmetry of order $N^{-\gamma}$ of the direction of growth that switches from up to down and is always towards the direction that reduces the size of the difference of $\Phi_N$ above and below the interface. We specialise to the case of constant $\Phi_N$, so that the asymmetry direction is based on the sign of the area underneath the interface, and study the hydrodynamic limit, stationary correlation functions and equilibrium fluctuations of the interface. We will also discuss the case of a more general $\Phi_N$. Based on joint work with Martin Hairer and Patrícia Gonçalves.
Jean-Baptiste Casteras - Almost sure existence of solutions for cubic higher order Schr\" odinger equations
In this talk, we will discuss the existence of solutions for cubic higher order Schr\" odinger type equation (NLS) on the whole space with rough initial data. Although such a problem is known to be ill-posed, we show that a randomisation of the initial data yields almost sure local well-posedness. Using estimates in directional spaces, we improve and extend known results for the standard Schr\" odinger equation in various directions: higher dimensions, more general operators, weaker regularity assumptions on the initial conditions. We will also discuss the existence of invariant measures in the whole space.
Gerardo Vargas - (Part 2) The Fréchet distribution arising in the asymptotic law of condition number for random circulant matrices
In this presentation we study the limiting distribution of the largest singular value, the smallest singular value, and the so-called condition number for random circulant matrices, where the generating sequence consists of independent and identically distributed (i.i.d.) random elements satisfying the Lyapunov condition.
Under an appropriate normalization, the joint distribution of the extremal (minimum and maximum) singular values converges in distribution, as the matrix dimension approaches infinity, to an independent product of Rayleigh and Gumbel laws. As a consequence, the condition number (properly normalized) converges in distribution to a Fréchet law in the large-dimensional limit. Broadly speaking, the condition number quantifies the sensitivity of the output of a linear system to small perturbations in its input. The proof is based on the celebrated Einmahl–Komlós–Major–Tusnády coupling. This work is based on a joint paper with Paulo Manrique (IPN, Mexico), in Extremes 2022.