Poster session

Idriss Adjaout (Institut Pascal, Clermont Ferrand)

Stochastic Representations of PDEs for Multiphysics Simulation

Monte Carlo methods are attracting increasing interest for solving systems of PDEs, in particular because of their favorable performance in high dimensions and their ability to provide pointwise estimates of the solution. These approaches rely on probabilistic representations of PDEs, for which a fundamental choice has to be made: whether to use a forward or a backward representation. This choice leads to algorithms with substantially different interpretations and numerical properties.

For nonlinear PDEs, this distinction becomes even more significant. Forward and backward viewpoints give rise to different paradigms, in which nonlinearity may emerge either from interactions between particles or from branching mechanisms. In this poster, we discuss these different approaches, their respective advantages, and the persistent challenges.

Hetranso Ahni (Université Jean-Monnet, Saint-Étienne)
Arrhenius’law and Kramers-type law for a kinetic interacting particle system
The Cucker–Smale model, introduced by Cucker and Smale (2007), is a kinetic interacting particle system that exhibits asymptotic emerging phenomena, where agents tend to align their velocities, leading to a flocking configuration. In the presence of small noise, friction, and a confining potential, excursions out of a positively invariant domain—which represents this emerging phenomenon—become possible, albeit extremely rare. We establish Arrhenius-type and Kramers-type laws to describe the asymptotics of the exit time from this domain, thereby extending the seminal work of Freidlin and Wentzell. This is joint work with Jean-Fran¸cois Jabir and Julian Tugaut. 

Julian Amorim (University of São Paulo)

Scaling Limits of Smoluchowski's particles

We prove a law of large numbers and a functional central limit theorem for the empirical density of a Marcus-Lushnikov model. The limiting density turns out to be the solution of a Smoluchowski equation, and the fluctuations around this limit are shown to be described by an Ornstein-Uhlenbeck process with drift term given by the linearization of the Smoluchowski operator.

Alexandre Bertolino (Sorbonne Université - LJLL)

Derivation of a cross diffusion system from a model of repulsive random walks.

We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species move, reproduce and die with rates sensitive to the local densities of the two species. Quantitative estimates are obtained and convergence is proved soon as the population per site and the number of sites go to infinity. The proofs rely on stability estimates and the development of large deviation estimates for structured population models, which are of independent interest. 

Pilar Branquinho (University of Coimbra, Center for Mathematics)

Regularity theory for free transmission problems with mixed singularities

We examine a degenerate free transmission problem in the presence of mixed singularities. Our contribution is to prove new regularity outcomes for viscosity solutions in Hölder spaces. The methods we develop are robust, in the sense they are heavily inspired by viscosity-type techniques, and we expect they would apply to a broad class of models.

Tomas Blore (University of Oxford)

Some insights from the probabilistic to the supercooled Stefan problem

The supercooled Stefan problem is a free boundary PDE which experiences blow-ups and lack of well-posedness. Recent work uses probabilistic approaches to attempt to resolve these issues. From a simple probabilistic approach, we establish well-posedness for quite general initial data. From an interacting particle system approach, we obtain quantitative error bounds and determine a functional central limit theorem.

Nicoleta Cazacu (CMAP, Ecole polytechnique)

A semigroup approach for quantitative uniform in time PoC for singular interaction kernels

Based on a semigroup approach, we provide quantitative uniform in time propagation of chaos results    for a mollified system of interacting particles with singular interaction kernels of type Lk, k > d and a contractive    Lipschitz confinement force field. The method relies on the regularity properties of the solution of the    associated nonlinear Fokker-Planck equation combined with estimates of the Duhamel formulation of the    mollified empirical measure.

Zhe Chen (MAP5, Université Paris Cité)

A Derivation of the Thick Spray Model

We present a formal derivation of a thick spray model from a binary Enskog–Boltzmann system describing gas–particle mixtures. Unlike classical thin spray models, the Enskog framework accounts for finite-size effects through spatially nonlocal collision operators. Considering a regime with a small mass ratio between gas and particle and a hydrodynamic limit for the gas phase, we show formally that the system converges to an Euler–Vlasov description coupling a compressible gas with a kinetic dispersed phase. The derivation naturally produces the volume-fraction function associated with particle excluded volume and yields finite-volume corrections of order (O(a^3)) in the momentum equations. These corrections include pressure gradient effects analogous to buoyancy forces and higher-order modifications of the friction force. The result provides a mesoscopic level justification of thick spray models commonly used in engineering applications and clarifies the mesoscopic origin of volume fracttion function in dense sprays.

Ruiqi Ding (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

Branching Lévy processes with selection and its asymptotic velocity

Building on the branching L\'evy processes of Bertoin and Mallein, we construct $N$-branching L\'evy processes (N-BLPs) in a setting that allows infinite birth rate and goes beyond the usual exponential-integrability assumptions. The process is obtained as the local limit of truncated $(N,b)$-systems and intuitively speaking, keeps only the $N$ rightmost particles after each reproduction event. Moreover, under suitable integrability assumptions, we prove the existence of an asymptotic velocity $v_{N,\Xs}$ and establish Brunet-Derrida behavior in the form

\[ v_{N,\Xs}\sim -C^*\frac{L(\log N)}{(\log N)^\alpha},\]

where $L$ is slowly varying and $\alpha\in(0,2]$ is the stable index. 

To establish this result, we develop the coupling between N-BLPs and a BLP killed below a linear space barrier and derive the estimate for survival probability of the killed BLP.

Ruairi Garrett (Oxford)

Fluctuations of spatial population models with non-local density dependence

The ancestry of a uniform sample from a population of branching brownian particles on the torus, who compete through their total population size, is shown to converge to the brownian spatial coalescent.

Raphael Gastaldello (CNRS / Ecole des Ponts)

On the diffusive-mean field limit of kinetic interacting particle system
We study the joint diffusive-mean field limit for a system of weakly interacting kinetic Langevin dynamics, extending the results of [Delgadino & al ARMA 2020] to the hypoelliptic/hypocoercive case. We show that, in the absence of phase transitions, the two limits commute, and we calculate the covariance matrix of the limiting Brownian motion using the Green-Kubo/Kipnis-Varadhan formula. However, at low temperatures, and in the presence of phase transitions, the two limits may not commute. We demonstrate our findings by providing a detailed analysis of the diffusive-mean field limit for the $O(2)$ model in a magnetic field. Our analysis is based on the systematic use of recently developed hypocoercivity techniques, together with an appropriate linearization of the mean field McKean--Vlasov--Fokker--Planck partial differential equation.

This is a joint work with Greg A. Pavliotis, Gabriel Stoltz & Urbain Vaes.

Nicholas Graham (University of Bath)

Blow-up and Well-posedness of 2D Keller-Segel PDE perturbed by rough Kraichnan noise

I will present an extension of the Blow-up result found in Mayorcas and Tomasevic's 2023 on the Keller-Segel PDE. I will also discuss the problem of well-posedness on the other side of the blow-up threshold and possible approaches to this question.

Jules Grass (Institut Camille Jordan)

Propagation of chaos in Fisher information
We present a new method for proving sharp local propagation of chaos in Fisher Information for particles in $\mathbb{R}^{d}$ with smooth interaction and drift. We rely on a new Lemma computing the Fisher Information of two diffusion processes with smooth drifts and fine estimates on the hessian of the law of the solution of the McKean-Vlasov equation. It allows us to obtain a new propagation of chaos in Fisher information, generalizing Lacker's seminal work, by using the BBGKY hierarchy to obtain a system of differential inequalities satisfied by both the relative entropy and the Fisher Information of $k$ particles. We also show with a simple Gaussian example that our decay rate is optimal.

Marcel Hudiani (Umeå University)

Singular Diffusion Limit of Tagged Particle in Zero Range Process with Sinai-type Medium

Oihan Joyot (IRIT Toulouse)

Learning the dynamic of cellular condensates

Tamari Kldiashvili (De Vinci Higher Education; University of Graz)

Mean-field limit for Motsch-Tadmor model

Dowan Koo (University of Oxford, Yonsei University)

Large time behavior of hydrodynamic flocking models with interaction potentials
We consider a hydrodynamic model for interacting particle systems of Cucker-smale type with non local interaction in pressureless regime. We are mainly interested in the large time behaviour of the system. We first establish the flocking behaviour of our models with compactly supported communication kernel with convex potential. Then, we quantify the convergence rate to the equilibrium state depending on the singularity of the communication kernel near the origin. Based on a joint work with J.A. Carrillo, Y.-P. Choi, and O. Tse.

Adrienne Le Meur (École Polytechnique)

Quantitative propagation of chaos for particle systems with memory and their long-time behaviour

We study a class of interacting diffusion particle systems with memory and their mean-field limit using coupling methods. We first establish quantitative propagation of chaos estimates for this class of models. Our first main result is a uniform-in-time propagation of chaos estimate, under a suitable contraction condition. We also discuss conditions for existence of stationary solutions and show that long-time convergence to a non-stationary equilibrium may also hold. This requires an asymptotic behaviour for the memory interaction and a memory loss condition. As a consequence, our second main result is that the particle system converges asymptotically to the same equilibrium up to the propagation of chaos error, which shows that the large-particle limit and the large-time limit are exchangeable.

Andrew Nugent (University College London)

Opinion dynamics with continuous age structure

This work extends a model of opinion formation as an interacting particle system to explicitly include an age-structured population. A stochastic differential equation model is first introduced, incorporating ageing dynamics and birth/death processes within a bounded confidence framework. The corresponding mean-field partial differential equation is then derived and analysed, with numerical simulations used to compare the resulting dynamics at microscopic and macroscopic scales. The existence of stationary states in the mean-field model is rigorously established, and these states are shown to be non-unique in general.

Jules Olayé (IMT Toulouse)

An ergodic behaviour of a telomere shortening model
In this work, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality.

Sakari Pirnes (University of Helsinki)

Finite lattice kinetic equations for bosons, fermions, and discrete NLS

We introduce and study finite lattice kinetic equations for bosons, fermions, and discrete NLS. For each model this closed evolution equation provides an approximate description of the appropriate covariance function conjectured to characterize the system. To have such a reference solution should simplify rigorous derivation and control of applicability of kinetic theory.  We consider the well-posedness of the resulting evolution equation up to finite kinetic times. We obtain decay of the solutions and upper bounds depending only on Sobolev type norms of the interaction potential and initial data.  Joint work with Jani Lukkarinen and Aleksis Vuoksenmaa.

Vishnu Sanjay (Gran Sasso Science Institute)

On the weak coupling limit of the periodic quantum Lorentz gas

I will show that for the periodic quantum Lorentz gas, in the weak coupling limit, certain observables are freely transported, while for general observables, the existence of the limit depends on the regularity of a certain phase space object at resonant momenta.

Sihyun Song (Yonsei University)

Strong compactness phenomena arising in nonlinear and degenerate kinetic Fokker--Planck equations

We consider nonlinear Vlasov--Fokker--Planck equations in which the diffusion and drift depend on the hydrodynamic quantities of the solution. Working with only the most physically natural assumptions of finite mass, entropy, and energy, we prove that solutions to this equation enjoy strong compactness in $L^1$, despite the diffusion being degenerate and singular simultaneously. The proof is based upon a level set decomposition of the Fisher information, and exploiting averaging lemmas. As a consequence, we deduce the global-in-time existence of weak solutions to this equation in the Diperna--Lions framework.

Andrea Tedesco (Centre for Mathematics of the University of Coimbra (CMUC))

Second order regularity for supersolutions to fully nonlinear PDE and applications
We establish pointwise second-order regularity estimates for L^p-viscosity supersolutions to fully nonlinear equations driven by the Pucci extremal operator. The central result shows that a one-sided curvature bound, namely the existence of a paraboloid touching the solution from below, propagates to a two-sided control, yielding quadratic growth in both directions. The proof combines a weak Harnack inequality with a generalized maximum principle for semiconvex functions, together with an invariance property of semiconvexity under subtraction of paraboloids. As an application, we examine the impulse control problem and derive the semiconvexity of the value function, recovering the structural assumptions required by the main theorem. As a consequence we establish C^{1,1}-regularity estimates for the impulse optimal control. This is based on joint work with E. Pimentel (Coimbra) and E. Teixeira (Oklahoma).