# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### 06/05/2022, 14:00 — 15:30 — Room P3.10, Mathematics Building Clement Erignoux, INRIA Lille

For several decades, entropy methods have provided robust tools to derive the hydrodynamic limit of various interacting lattice gases. In the diffusive case, however, these tools depend on the so-called gradient condition, that ensures the generator of the process acts as a local laplacian. Whenever this condition is not satisfied, the local distribution of the process is perturbed, which induces changes to the macroscopic behavior of the system. In this mini-course, I will try and explain, without going through all the technical details, Varadhan's non-gradient method to derive the scaling limit of non-gradient diffusive lattice gases. The focus will be on the structure of the proof and the main ideas and concepts that make the proof work, while skipping some of the more technical aspects.

### 05/05/2022, 14:00 — 15:30 — Room P3.10, Mathematics Building Clement Erignoux, INRIA Lille

For several decades, entropy methods have provided robust tools to derive the hydrodynamic limit of various interacting lattice gases. In the diffusive case, however, these tools depend on the so-called gradient condition, that ensures the generator of the process acts as a local laplacian. Whenever this condition is not satisfied, the local distribution of the process is perturbed, which induces changes to the macroscopic behavior of the system. In this mini-course, I will try and explain, without going through all the technical details, Varadhan's non-gradient method to derive the scaling limit of non-gradient diffusive lattice gases. The focus will be on the structure of the proof and the main ideas and concepts that make the proof work, while skipping some of the more technical aspects.

### 03/05/2022, 14:00 — 15:30 — Room P3.31, Mathematics Building Clement Erignoux, INRIA Lille

For several decades, entropy methods have provided robust tools to derive the hydrodynamic limit of various interacting lattice gases. In the diffusive case, however, these tools depend on the so-called gradient condition, that ensures the generator of the process acts as a local laplacian. Whenever this condition is not satisfied, the local distribution of the process is perturbed, which induces changes to the macroscopic behavior of the system. In this mini-course, I will try and explain, without going through all the technical details, Varadhan's non-gradient method to derive the scaling limit of non-gradient diffusive lattice gases. The focus will be on the structure of the proof and the main ideas and concepts that make the proof work, while skipping some of the more technical aspects.

### 02/05/2022, 14:00 — 15:30 — Room P3.10, Mathematics Building Clement Erignoux, INRIA Lille

For several decades, entropy methods have provided robust tools to derive the hydrodynamic limit of various interacting lattice gases. In the diffusive case, however, these tools depend on the so-called gradient condition, that ensures the generator of the process acts as a local laplacian. Whenever this condition is not satisfied, the local distribution of the process is perturbed, which induces changes to the macroscopic behavior of the system. In this mini-course, I will try and explain, without going through all the technical details, Varadhan's non-gradient method to derive the scaling limit of non-gradient diffusive lattice gases. The focus will be on the structure of the proof and the main ideas and concepts that make the proof work, while skipping some of the more technical aspects.

### Minimisers of the Canham-Helfrich functional in the space of generalised Gauss graphs

The Canham-Helfrich functional is the most widely used functional to study the equilibrium of biological membranes as a result of the competition between mean curvature and Gaussian curvature. In this talk, we review some approaches to the minimisation problem for this functional and present novel results in the setting of generalised Gauss graphs.

This is joint work with Anna Kubin and Luca Lussardi.

### One-dimensional chain of oscillators with random flips of velocities (III)

Towards the rigorous derivations of Boltzmann equations (kinetic limit) and a coupled diffusive system for volume and energy (hydrodynamic limit).

### One-dimensional chain of oscillators with random flips of velocities (II)

The Wigner distributions as an important tool to separate macroscopic form microscopic scale and localize in space the energy modes.

### One-dimensional chain of oscillators with random flips of velocities (I)

Introduction to the model, and a few results in very basic cases.

### Hydrodynamic behavior of long-range symmetric exclusion process with a slow barrier

We discuss the hydrodynamical behavior of the long jumps symmetric exclusion process with a slow barrier. When jumps occur between a negative site and a non-negative site, the rates are slowed down by a factor of $αn^{ −β}$, where α > 0 and β ≥ 0. The jump rates are given by a symmetric transition probability p(·). We obtain diverse partial differential equations given in terms of the usual Laplacian (when p(·) has finite variance), and in terms of the regional fractional Laplacian (when p(·) has infinite variance), with different boundary conditions.

### Boundary renormalisation of stochastic PDEs

We discuss solution theories of singular SPDEs endowed with various boundary conditions. In several equations like the KPZ, continuum PAM, or dynamical Phi^4, nontrivial boundary effects arise and another layer of renormalisation at the boundary is required. As a corollary, we obtain an example for a boundary triviality result, where an approximation procedure with Neumann solutions converges to a Dirichlet solution. This is a joint work with M. Hairer.

### An interacting particle system interpolating the Symmetric Simple Exclusion Process and the Porous Media Model

The Porous Media Model (PMM) is an interacting particle system of exclusion-type (two particles cannot occupy the same site) belonging to the class of kinetically constrained lattice gases (KCLG), which have gained a lot of attention as simple models for the liquid/glass transition. In particular, the PMM was first introduced to derive the Porous Media Equation (PME), which can be seen as a generalisation of the heat equation with a non-linear diffusion coefficient, $D(u)=mu^{m-1}$, with $m\gt 1$.

As it was defined, the PMM allows one to derive the PME when the diffusion is a positive integer power of the solution by constraining nearest neighbour jumps with occupation conditions on a neighbourhood of where the jumps take place. This model has not been directly extended yet to encompass the case where the diffusion is a non-integer power. In this talk we present a direct extension of the PMM for $m\in (1,2)$ via the use of the generalised binomial coefficients. The resulting dynamics is very simple and interpolates continuously the Symmetric Simple Exclusion Process, which allows one to derive the Heat Equation, and the PMM with $m=2$, while being of gradient and exclusion types.

If there is still time, we also discuss the difficulties of deriving the Fast Diffusion Equation ($m\lt 1$) and the extension to $m\gt 2$.

### Sharp convergence of Markov chains (IV)

We show how to use entropy methods to obtain sharp estimates on the convergence of finite-state Markov chains to their stationary states.

Fourth session of a minicourse of 4 sessions.

### Sharp convergence of Markov chains (III)

We show how to use entropy methods to obtain sharp estimates on the convergence of finite-state Markov chains to their stationary states.

Third session of a minicourse of 4 sessions.

### Sharp convergence of Markov chains (II)

We show how to use entropy methods to obtain sharp estimates on the convergence of finite-state Markov chains to their stationary states.

Second session of a minicourse of 4 sessions.

### Sharp convergence of Markov chains (I)

We show how to use entropy methods to obtain sharp estimate on the convergence of finite-state Markov chains to their stationary states.

First session of a minicourse of 4 sessions.

### The Mathematics of making a mess (an introduction to random walk on groups)

How many random transpositions does it take to mix up $n$ cards? This is a typical question of random walk on finite groups. The answer is $(1/2)n \log n+Cn$ and there is a sharp phase transition from order to chaos as $C$ varies. The techniques involve Fourier analysis on non-commutative groups (which I will try to explain for non specialists). As you change the group or change the walk, new analytic and algebraic tools are required. The subject has wide applications (people still shuffle cards, but there are applications in physics, chemistry,biology and computer science — even for random transpositions). Extending to compact or more general groups opens up many problems. This was the first problem where the ‘cutoff phenomenon’ was observed and this has become a healthy research area.

Projecto FCT UIDB/04459/2020.

### Hydrodynamics of the simple exclusion process in random environment: a mild solution approach

In this talk, we discuss an alternative point of view, initiated by the works of Nagy (2001) and Faggionato (2007), on the hydrodynamic limit of the exclusion process. We show how this method based on duality and a mild solution representation of the empirical measures carries over to the case of random environment - both static and dynamic. In conclusion, we discuss some recent refinements of this method which allow, for instance, to obtain tightness of the sequence of empirical measures.

Joint work with S. Floreani (TU Delft), F. Redig (TU Delft) and E. Saada (Paris V Descartes).

Projecto FCT UIDB/04459/2020.

### Noise Impact on Finite Dimensional Dynamical Systems

Dynamical systems are often subjected to noise perturbations either from external sources or from their own intrinsic uncertainties. While it is well believed that noises can have dramatic effects on the stability of a deterministic system at both local and global levels, mechanisms behind noise surviving or robust dynamics have not been well understood especially from distribution perspectives. This talk attempts to outline a mathematical theory for making a fundamental understanding of these mechanisms in white noise perturbed systems of ordinary differential equations, based on the study of stationary measures of the corresponding Fokker-Planck equations. New existence and non-existence results of stationary measures will be presented by relaxing the notion of Lyapunov functions. Limiting behaviors of stationary measures as noises vanish will be discussed in connection to important issues such as stochastic stability and bifurcations.

Projecto FCT UIDB/04459/2020.

### From the porous medium model to the porous medium equation

The aim of this seminar is to present an overview of the porous medium model and its hydrodynamic equation, the porous medium equation. We will focus on exploring the main characteristics of this equation and how we can see it from the particle system's point of view.

Projecto FCT UID/MAT/04459/2019.

### Equilibrium fluctuations for symmetric exclusion with long jumps and infinitely extended reservoirs

The aim of this work is the analysis of fluctuations around equilibrium for a diffusive and symmetric exclusion process with long jumps and infinitely extended reservoirs (introduced in [BGJO]). In particular we study how the parameters characterizing the model change the behavior of the stochastic fluctuations of the macroscopic density of particles around the equilibrium. We will see how the SPDE involved will pass from the one which solution is a generalized Ornstein-Uhlenbeck process when the reservoirs are weak, to an SPDE without diffusive term when the reservoirs are so strong that the fluctuations are only caused by their action.

Joint work with C. Bernardin, P. Gonçalves and M. Jara.

### Reference

[BGJO] Bernardin, C., Gonçalves, P. and Oviedo Jimenez, B. Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps.

Projecto FCT UID/MAT/04459/2019.

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