Seminário de Probabilidade e Análise Estocástica 

In this talk I will discuss a family of Gibbsian measures on the set of Laguerre tessellations. These measures may be used to provide a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles.

In the context of randomly fluctuating interfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang (KPZ) and the Edwards-Wilkinson (EW). Notoriously, the KPZ equation is known to interpolate between them in that its small-scale statistics are those of EW while its large-scale fluctuations are those of KPZ. In a recent work with M. Hairer, we showed that the universality picture outlined above is not exhaustive and identified a new universality class together with the universal process at its core, the Brownian Castle (BC). After reviewing the origin, construction and characterising properties of BC, the talk will be devoted to show that there exist a huge family of processes that play a role similar to that of the KPZ equation, connecting though the BC and EW universality classes. We called these processes $\nu$-Brownian Castle, for $\nu$ a probability measure on [0,1], and are linked to the Brownian Net and the stochastic flows of kernels of Schertzer, Sun and Swart. Time allowing, we will show that (one of) these processes naturally arise as the limit, under a suitable scaling, of a microscopic model given by a stochastic PDE. This is joint (ongoing) work with M. Hairer, T. Rosati and R. Sun.

The stationary horizon (SH) is a recently constructed cadlag stochastic process whose states are Brownian motions and the process is indexed by the drifts. It is part of the universality picture of the 1+1 dimensional Kardar-Parisi-Zhang (KPZ) class. SH was discovered as a diffusive limit of the Busemann process of the exponential corner growth model (Busani) and simultaneously as the Busemann process of Brownian last-passage percolation (Sorensen and the speaker). This talk is about SH as the Busemann process of the directed landscape, as the stationary distribution of the KPZ fixed point, and as the scaling limit of the TASEP speed process. Joint work with Ofer Busani (Bonn/Edinburgh) and Evan Sorensen (Madison/Columbia).

In this talk I will show how the same algebraic approach, which relies on the $\mathfrak{su}(1,1)$ Lie algebra, can be used to construct two duality results. One is well-known: the two processes involved are the symmetric inclusion process and a Markov diffusion called Brownian Energy process. The other one is a new result which involves a particle system of zero-range type, called harmonic process, and a redistribution model similar to the Kipnis-Marchioro-Presutti model. Despite the similarity, it turns out that the second relation involves integrable models and thus duality can be pushed further. As a consequence, all moments in the stationary nonequilibrium state can be explicitly computed.

We study the Inclusion Process with vanishing diffusion coefficient, which is known to exhibit condensation and metastable dynamics for cluster locations. Here we focus on the dynamics of mass distribution rather than locations, and consider the process on the complete graph in the thermodynamic limit with fixed particle density. We describe the mass distribution for a given configuration by a measure on a suitably scaled mass space and derive a limiting measure-valued process. When the diffusion coefficient scales like the inverse of the system size, the scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative point of view for this well-established dynamics. Testing configurations with size-biased functions, our approach can be generalized to other scaling regimes. This leads to an interesting characterization of the limiting dynamics via duality and provides a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. This is joint work with Simon Gabriel and Paul Chleboun (both Warwick)

For finite size Markov chains, the probability that a time-averaged observable take an anomalous value in the long time limit was quantified in a celebrated result by Donsker and Varadhan. In the study of interacting particle systems, one is interested not only in the large time, but also in the large system size limit. In this second limit, observables of the chain each live at different scales, and one has to understand how scales decouple. In a joint work with Thierry Bodineau, we study a paradigmatic example of out of equilirium interacting particle systems: the one-dimensional symmetric simple exclusion process connected with reservoirs of particles at different density. We focus on the scale of two point correlations and obtain the long time, large system size limits on the probability of observing anomalous correlations. This is done through quantitative, non-asymptotic estimates at the level of the dynamics. The key ingredient is a precise approximation of the dynamics and its invariant measure (not explicitly known), that is of independent interest. The quality of this approximation is controlled through relative entropy bounds, making use of recent results of Jara and Menezes.

In this talk, we consider the asymmetric exclusion process with long jumps, where the transition rate is proportional to $ |x|^{-d-\alpha} $ for some $ \alpha > 0 $. We state the central limit theorems for the tagged particle when the process starts from its stationary measure. When time permitted, we will also outline the proof.

We discuss random growth models in the KPZ universality on a ring. When the time and the size of the ring both tend to infinity in a critical way, the height fluctuation field is expected to converge to a field that interpolate the KPZ fixed point on the line and the Brownian motion. We discuss the convergence of the multi-time, multi-position distributions for the totally asymmetric simple exclusion process. In the second part of the talk, we discuss various deterministic differential equations associated with the KPZ fixed point and its periodic version for the narrow-wedge initial condition.

In the study of scaling limits of reversible particle systems with the property of self-duality, many quantities of interest become easier to manipulate and simplify. For the particular case of fluctuations from the hydrodynamic limit, and in the additional presence of orthogonality, these simplifications have interesting consequences. In this talk, we will briefly discuss some of those consequences. First, we will obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann–Gibbs principle. Second we will introduce what we call the k-th order fluctuation field. We will then explain how these fields can be interpreted as some type of discrete analogue of powers of the well-known density fluctuation field, and show how their scaling limits formally correspond to the SPDE associated with the kth-power of a generalized Ornstein-Uhlenbeck process.

This work takes inspiration from [1] and [2], and it is a joint effort with G. Carinci (Università di Modena e R. Emilia) and Frank Redig (TU Delft).

1. Sigurd Assing, A limit theorem for quadratic fluctuations in symmetric simple exclusion, Stochastic Process. Appl. 117 (2007), no. 6, 766–790.
2. Patrícia Gonçalves and Milton Jara, Quadratic fluctuations of the symmetric simple exclusion, ALEA Lat. Am. J. Probab. Math. Stat. 16 (2019), no. 1, 605–632.

Random intersection graph is a simple random graph that incorporates community structures. To build such a graph, imagine there are $n$ individuals and $m$ potential communities. Each individual joins a community independently with probability $p$. The graph $G(n, m, p)$ has $n$ nodes, corresponding to the $n$ individuals. Each pair of these individuals share an edge between them if they belong to a common community. The critical threshold for the emergence of a giant component emerges turns out to be at $p^2 ~ 1/nm$. I’ll discuss some results that can help us to understand what a large $G(n, m, p)$ looks like at the critical threshold.

The canonical framework for mean-field systems is to consider $N$ particles (diffusions, or dynamics with jumps, etc) that interact on the complete graph in a uniform way, the strength of interaction between two particles being of size $1/N$. The behavior of the system is hence captured by the empirical measure of the system which converges as $N\to\infty$ to the solution of a nonlinear Fokker Planck equation. The motivation of this talk is simple: what can we say if one no longer interacts on the complete graph, i.e. one removes connections between particles ? if the graph is sufficiently close to the complete graph, one expects the same asymptotic behavior. We will address this question at the level of the LLN and fluctuations of the empirical measure of the system. This is based on joint works with G. Giacomin, S. Delattre, F. Coppini and C. Poquet.

Generalized Ray-Knight theorems for edge local times proved to be a very useful tool for studying the limiting behavior of several classes of self-interacting random walks (SIRWs) on integers. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs introduced and studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss new results which resolve an open question posed in Toth’s paper. We show that in the asymptotically free case the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth) while in the polynomially self-repelling case the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of all perturbed Brownian motions. This negative result was somewhat unexpected. The question whether there is convergence in the polynomially self-repelling case and, if yes, then how to describe the limiting process is open. This is joint work with Thomas Mountford, EPFL, and Jonathon Peterson, Purdue University.

In this talk we will review a model that was first introduced by Jonasson, Mossel and Peres. Starting with the usual square lattice on $Z^2$, entire rows (respectively columns) of edges extending along the horizontal (respectively vertical) direction are removed independently at random. On the remaining thinned lattice, Bernoulli bond percolation is performed, giving rise to a percolation model with infinite range dependencies under the annealed law. In 2005, Hoffman solved the main conjecture around this model: proving that this percolation process indeed undergoes a nontrivial phase transition. In this talk, besides reviewing this surprisingly challenging problem, we will present a novel proof, which replaces the dynamic renormalization presented previously by a static version. This makes the proof easier to follow and to extend to other models. We finally present some remarks on the sharpness of Hoffman’s result as well as a list of interesting open problems that we believe can provide a renewed interest in this family of questions. This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

Stein's method is an increasingly popular way to derive quantitative versions of weak convergence theorems, like the central limit theorem. In this talk we use Stein's method to derive a quantitative CLT for Dynkin martingales of Markov chains. Despite its simplicity, we show with some examples that the bounds we obtain in the context of interacting particle systems are surprisingly sharp.

I will talk about a singular non-linear SPDE that was introduced by van Beijeren, Kutner and Spohn (1985) as a continuum version of d-dimensional ASEP. The equation is "supercritical" ($d>3$) or critical ($d=2$) in the SPDE language. We show that the large-scale behavior of the equation is Gaussian in dimension $d$ greater or equal to $3$ (this mirrors analogous results by Landim, Olla, Yau et al for ASEP) and also in dimension $d=2$ (in the so-called weak noise limit, which corresponds to a certain $2-$dimensional WASEP). The scaling is non-trivial in the sense that the non-linearity has a non-vanishing effect on the limit equation. Ongoing work with G. Cannizzaro, L. Haunschmid and M. Gubinelli.

In this talk, first, I give a very brief introduction to the NLS (Nonlinear Schrödinger Equation) and its long-time behavior. Then I introduce a mass-conserving stochastic perturbation of the discrete nonlinear Schrödinger equation that models the action of a heat bath at a given temperature. Afterward, I sketch the fact that the corresponding canonical Gibbs distribution is the unique invariant measure. Finally, as an application, I discuss the one-dimensional cubic focusing case on the torus, where we prove that in the limit for large time, continuous approximation, and low temperature, the solution converges to the steady wave of the continuous equation that minimizes the energy for a given mass. This is based on the following joint work with prof. Stefano Olla (Universite Paris Dauphine-PSL, GSSI, IUF):
Hannani, A., Olla, S. A stochastic thermalization of the Discrete Nonlinear Schrödinger Equation. Stoch PDE: Anal Comp (2022). https://doi.org/10.1007/s40072-022-00263-9

This talk focuses on generalizations of the exclusion process whose hydrodynamic limits are sub-diffusive equations. After recalling some known results in dimension 1, I will present in detail the partial exclusion process in random environment. This is a system of random walks where the random environment is obtained by assigning random maximal occupancies to each site of the Euclidean lattice. I will show that, when assuming that the maximal occupancies are heavy tailed and i.i.d., the hydrodynamic limit of the particle system (in any dimension greater than 1) is the fractional-kinetics equation.

This talk is based on partly ongoing projects in collaboration with A. Chiarini (Padova), F. Redig (TU Delft) and F. Sau (ISTA).

In this talk we will consider two models within Kardar-Parisi-Zhang (KPZ) universality class, and apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point correlation function, the polymer end-point distribution and the second derivative of the variance of the associated height function. Besides that, we will further develop an adaptation of Malliavin-Stein method that quantifies asymptotic independence with respect to the initial data.

We consider a system of N half lines issuing from the origin, on which there is a Poisson process of dust particles initially. We have N^ \alpha workers who clean dust particles according to a greedy algorithm; they move to the closest dust particle and remove it and then wait an exponential time before chosing a new particle. We consider for which values of alpha one can have half lines where two or more workers go to infinity. The talk uses only elementary probability arguments and well known properties of Poisson processes. It should be accessible to all.
Joint with Sergey Foss and Takis Konstantopoulous

The Axelrod model is a spatial stochastic model for the dynamics of cultures which includes two important social components: homophily, the tendency of individuals to interact more frequently with individuals who are more similar, and social influence, the tendency of individuals to become more similar when they interact. Each individual is characterized by a collection of opinions about different issues, and pairs of neighbors interact at a rate equal to the number of issues for which they agree, which results in the interacting pair agreeing on one more issue. This model has been extensively studied during the past 20 years based on numerical simulations and heuristic arguments while there is a lack of analytical results. This talk gives rigorous fluctuation and fixation results for the one-dimensional system that sometimes confirm and sometimes refute some of the conjectures formulated by applied scientists.