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# YEP XIII “Large Deviations for Interacting Particle Systems and Partial Differential Equations”

## Mar 7, 2016 - Mar 11, 2016

(this YEP workshop is part of the SAM “Probability and Analysis”)

**Summary**

The thirteenth edition of the Young European Probabilists (YEP) workshop focusses on “Large Deviations for Interacting Particle Systems and Partial Differential Equations”.

The program consists of three mini courses in the morning sessions intended for young researchers with a background at the interface between analysis and stochastics, and 30-minute presentations by invited speakers in the afternoon sessions.

The YEP XIII brings together many junior and some senior scientists whose research interests are strongly related to the selected topic. This mix of researchers aims at giving a special, open atmosphere which in the past has proven to be very fruitful for fostering discussions among the participants.

**Sponsors**

**Organizers**

Luca Avena | University of Leiden |

Roberto Fernandez | Utrecht University |

Francesca Nardi | TU Eindhoven |

**Speakers**

Mini courses:

Francesco Caravenna | Univeristy of Milan, Bicocca |

Antoine Gloria | ULB Brussels |

Jan Maas | IST Austria |

Invited Speakers:

Marton Balazs | University of Bristol |

Gioia Carinci | TU Delft |

Giacomo Di Gesù | CERMICS, Ecole des Ponts |

Sören Dobberschütz | University of Copenhagen |

Laure Dumaz | Cambridge University |

Clement Erignoux | CMAP, Ecole Polytechnique |

Max Fathi | Berkeley University |

Patricia Goncalves | PUC, Rio de Janeiro |

Chris Janjigian | Madison, Wisconsin |

Richard Kraaij | TU Delft |

Mauro Mariani | La Sapienza, Rome |

Georg Menz | UCLA |

Chiranjib Mukherjee | NYU |

Pierre Nyquist | Brown University |

Carlos Perez Espigares | University of Modena |

Christophe Poquet | Lyon 1 Univeristy |

Michiel Renger | WIAS Berlin |

Andre Schlichting | University of Bonn |

Marielle Simon | INRIA |

Willem van Zuijlen | Leiden University |

**Abstracts**

**How to initialise a second class particle?****
**Since the beautiful paper of Ferrari and Kipnis we know that the second class particle of simple exclusion chooses a uniform random direction in the rarefaction fan. The extremely elegant proof is based, among other ideas, on the fact that increasing the parameter of a Bernoulli distribution can be done by adding or not adding an extra (that is, second class) particle to a site.

Generalising the argument for other models is non-trivial for two reasons. (1) Increasing the parameter of the marginal of a stationary distribution often cannot be done just by adding or not adding a single second class particle. (2) There are lots of choices to make for starting a second class particle when we can have more than one particles on a site. How should we pick our choice?

I will show how to overcome both these issues at once by introducing a coupling measure of possibly negative weights which nevertheless serves as a proper probability distribution to start a second particle from. This distribution seems to be ”the canonical one” in many ways. In particular it allows to extend Ferrari and Kipnis’ results to a vast class of particle systems. I will also illustrate curious second class particle behaviours via some fun examples.

(joint work with Attila Lszl Nagy)

**PRESENTATION**

**Microscopic models for Free Boundary Problems**

Macroscopic laws of transport are described by PDE’s. Their derivation from microscopic models of interacting particles is a recurrent theme in non-equilibrium statistical mechanics literature. When the microscopic system is open, there are several mechanisms to couple the system with the external forces. In this talk I will present a class of systems where the interaction with the exterior takes place in correspondence of a free boundary.

**Analysis of the relaxation time of a large bistable particle system at low temperature
**A large system of strongly coupled diffusions on unbounded state space moving in a double-well potential is considered. This system can be seen as a spatially discrete approximation of the stochastically perturbed Allen-Cahn equation on the one-dimensional torus, which is a basic and widely studied stochastic partial differential equation.

In the small temperature regime the typical picture of a metastable dynamics emerges: the system quickly reaches a local equilibrium in one of the two wells, depending on its initial condition; this state endures for a very long time, until a sufficiently large stochastic fluctuation enables the system to overcome the energetic barrier separating the two wells and thus to distribute according to the global equilibrium.

I will present some results, obtained in collaboration with Dorian Le Peutrec, which quantify the mentioned slowdown in the relaxation to equilibrium. More specifically, these results concern spectral gap and log-Sobolev constant in regimes of low temperature and large number of particles.

**Boundary conditions at the interface between a liquid and a porous medium****
**The boundary conditions between a free fluid flow and a flow in a porous medium are of theoretical as well as practical importance in a number of disciplines, for example in reservoir engineering, marine biology and soil chemistry. A classical result is the boundary condition of Beavers-Joseph-Saffman, which requires the fluid’s velocity field to be continuous in the direction normal to the interface. In tangential direction, a jump between the velocities in the free fluid and the porous medium occurs. With experimental evidence dating back to the 1960s, this boundary condition has recently been justified mathematically by Jäger, Mikelić and Marciniak-Czochra – but only for the case of a planar interface. In this talk, we present their method of multiscale matched asymptotic expansion and how it can be generalized to also include the case of a curved porous-liquid interface, giving a generalized boundary condition of Beavers-Joseph-Saffman.

**PRESENTATION**

**Beta ensembles at high temperature
**In this talk, I will introduce random operators describing the continuum limit of beta ensembles (by Ramirez, Valko and Virag). I will show how to derive the behavior of the particles at high temperature thanks to this approach.

(joint work with R. Allez)

**Clément Erignoux
**

**Hydrodynamics of a non-gradient model for collective dynamics**

Extensive work has been put in the modelling of animal collective dynamics in the last decades, building on the work of ViscekAl (1995). These empirical approaches have unveiled several interesting phenomenon regarding phase transitions and separations. However, most of the theoretical background in collective dynamics modelling relies on mean-field approximations. I will present a lattice model where interactions between partices happen at a purely microscopic level, and describe some of the challenges in the proof of its hydrodynamic limit.

**PRESENTATION**

**Ricci curvature and functional inequalities for interacting particle systems
**In this talk, I will present a few results on entropic Ricci curvature bounds for interacting particle systems. These curvature bounds can be used to prove functional inequalities, such as spectral gap bounds and modified logarithmic Sobolev inequalities, which measure the rate of convergence to equilibrium for the underlying dynamic.

(joint work with M. Erbar and J. Maas)

**On the asymptotic behavior of slowed exclusion processes
**In this talk I will describe the asymptotic behavior of the symmetric simple exclusion with a slow bond and a particular emphasis will be given on the equilibrium fluctuations of its weakly asymmetric version. Depending on the strength of the asymmetry we see a crossover from the Edwards-Wilkinson universality class to the KPZ universality class.

**PRESENTATION**

**Large deviations for certain inhomogeneous corner growth models
**The corner growth model is a classical model of growth in the plane and is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived.

This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in these models.

(joint work with Elnur Emrah)

**Large deviations for interacting jump processes via solving a set of Hamilton-Jacobi equations
**We revisit the large deviation principle for trajectories of interacting jump processes on a finite state space. We give a new proof based on proving uniqueness of viscosity solutions of a set of associated Hamilton-Jacobi equations. The obtained result extends previously known results obtained via classical change of measure techniques. The method that shows that uniqueness of viscosity solutions leads to the large deviation principle are based on the work by Feng and Kurtz[2006].

**Macroscopic fluctuations for random collisional dynamics
**I will discuss fluctuations for the current of energy transferred by a large system of particles. The dynamics features random elastic collisions among particles, and the system is in contact with boundary wall at different temperatures.

**The log-Sobolev inwquality unbounded spin systems
**The log-Sobolev inequality (LSI) is a very useful tool for analyzing high-dimensional situations. For example, the LSI can be used for deriving hydrodynamic limits, for estimating the error in stochastic

homogenization, for deducing upper bounds on the mixing times of Markov chains, and even in the proof of the Poincaré conjecture by Perelman. For most applications, it is crucial that the constant in the

LSI is uniform in the size of the underlying system. In this talk, we discuss when to expect a uniform LSI in the setting of unbounded spin systems.

**Occupation measures, compactness and large deviations
**In a reasonable topological space, large deviation estimates essentially deal with probabilities of events that are asymptotically (exponentially) small, and in a certain sense, quantify the rate of these decaying probabilities. In such estimates, lower bound for open sets and upper bound for compact sets are essentially local estimates. However, upper bounds for all closed sets often require compactness of the ambient space or stringent technical assumptions (e.g., exponential tightness), which is often absent in many interesting problems which are motivated by questions arising in statistical mechanics (for example, distributions of occupation measures of Brownian motion in the full space Rd). Motivated by problems that carry certain shift-invariant structure, we present a robust theory of “translation-invariant ompactification” of orbits of probability measures in Rd. This enables us to prove a desired large deviation estimates on this “compactified” space. Thanks to the inherent shift- invariance of the underlying problem, we are able to apply this abstract theory painlessly and solve a long standing problem in statistical mechanics, the mean-field polaron problem.

(joint work with S. R. S. Varadhan, Erwin Bolthausen and Wolfgang König)

**PRESENTATION**

**A large deviation analysis of some qualitative properties of parallel tempering and infinite swapping algorithms
**We review the MCMC method knows as parallel tempering and its so-called infinite swapping limit, both which correspond to a collection of interacting particles. Focusing on infinite swapping we then employ a large deviation analysis and methods from stochastic optimal control to discuss certain qualitative properties. In particular we discuss how symmetry properties of the underlying potential landscape may affect convergence properties and how the rate function identifies those parts of the state space where noise due to sampling has the greatest impact on the overall performance of the algorithms.

(joint work with Jim Doll and Paul Dupuis)

**PRESENTATION**

**A weak additivity principle for current statistics in d-dimensions
**The additivity principle (AP) allows to compute the current distribution in many one-dimensional (1d) nonequilibrium systems. Here I will extend this conjecture to general d-dimensional driven diffusive systems, and validate its predictions against both numerical simulations of rare events and microscopic exact calculations of three paradigmatic models of diffusive transport in d = 2. Crucially, the existence of a structured current vector field at the fluctuating level, coupled to the local mobility, turns out to be essential to understand current statistics in d > 1. I will prove that, when compared to the straightforward extension of the AP to high-d, the so-called weak AP always yields a better minimizer of the macroscopic fluctuation theory action for current statistics.

**PRESENTATION**

**Random long time dynamics in the stochastic Kuramoto model
**The stochastic Kuramoto model is a toy model used to study synchronization phenomena. It consists in a population of N rotators with mean field interaction, each rotator being perturbed by a Brownian noise and possessing its own natural frequency of rotation. These frequencies are identically distributed and drawn independently, and correspond to a second source of randomness for the system (in addition to the thermal noise), called disorder.

On finite time intervals [0,T] and in the limit of infinite population the model is described by a PDE of Fokker-Planck type. This limit model undergoes a synchronization type phase transition: when the interaction is strong enough this PDE admits a stable curve M (in fact a circle), corresponding to the synchronization of the rotators around a synchronization center that rotates at constant speed.

When the distribution of the disorder is symmetric, this speed is equal to zero. We will see that in that case, when the size N of the population is large but finite, disorder-induced traveling-waves appear on the time scale N^{1/2}, with a speed given by the asymmetry of the finite-size draw of the disorder.

(joint work with L. Bertini, G. Giacomin and E. Luçon)

**Large deviations for reacting particle systems: the empirical and ensemble process
**We study the empirical measure of particles that can react randomly to form new particles. The resulting Markov processes are typically used as microscopic models for the deterministic reaction rate equation. In a similar fashion, but one level higher, one can study the empirical measure of the empirical measure, which then converges to the solution of a Liouville transport equation as both the number of particle systems and the number of particles in each system go to infinity. We are after large deviations for both processes. It turns out that the spaces of bounded variation provide natural topologies for both processes. In fact, the usage of these topologies in large deviations is not very common and therefore, hopefully, interesting in its own right.

**Discrete gradient flow structures for mean-field systems
**In this talk, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a free energy functional with respect to a certain metric structure, we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity.

We will address work in progress and open questions regarding the displacement convexity of the limit and the N-particle system exemplified for the Curie-Weiss mean-field spin system.

(joint work with M. Erbar (U Bonn), M. Fathi (UC Berkeley), V. Laschos (WIAS Berlin); arXiv: 1601.08098)

**PRESENTATION**

**Equilibrium fluctuations for one-dimensional conservative systems with degenerate rates
**The study of fluctuations for one-dimensional conservative systems (like, for instance, exclusion-type processes) often involves the so-called Boltzmann-Gibbs principle which states that the space-time fluctuations of any local field associated to a conservative mode can be written as a linear functional of the conservative field. A second-order Boltzmann-Gibbs principle has been introduced in 2014 by Gonçalves and Jara in order to investigate the first-order correction of this limit, in which case is given by a quadratic functional of the conservative field. The proof of that result was based on a multiscale analysis assuming that the underlying particle system is of exclusion type and for which a spectral gap inequality holds. In collaboration with O. Blondel and P. Gonçalves, we gave a new proof of that second-order Boltzmann-Gibbs principle in order to fit microscopic dynamics with kinetic constraints, which are not ergodic and provide blocked states.

**PRESENTATION**

**Gibbsianness related to minimisers of a large deviation rate function
**In this talk I discuss dynamical Gibbs-non-Gibbs transitions for mean-field spin systems. Gibbsianness is related to the number of global minimisers of a large deviation rate function. A unique global minimiser implies Gibbsianness while multiple imply non-Gibbsianness. I explain the background of this relation and list possible scenarios when the spins perform independent Brownian motions.

(joint work with Frank den Hollander and Frank Redig)

**PRESENTATION**