# Stochastic processes and applications

Organizadores: Joaquín Fontbona (fontbona@dim.uchile.cl), Juan Carlos Pardo (jcpardo@cimat.mx), Daniel Remenik (dremenik@dim.uchile.cl), Víctor Rivero (rivero@cimat.mx)

• #### Lunes 13

15:00 - 15:45

#### Sharp bounds for consensus-based optimization of convex functions

Roberto Imbuzeiro Oliveira (Instituto de Matemática Pura e Aplicada, Brasil)

Suppose one wants to optimize a function $$f:{\bf R}^d\to{\bf R}$$. However, the function is not directly available, and all one can do is to use a black box" that on a selected input $$x$$ produces the output $$f(x)$$. Consensus-based optimization is a collection of techniques for optimizing $$f$$ in this black box setting which is based on mean-field particle systems. We study a variant of this general method in the setting where $$f$$ is strongly convex and smooth, and obtain sharp convergence bounds. This is work in progress with Dyego Araújo (IMPA).

15:45 - 16:30

#### PageRank Nibble on directed stochastic block models

Mariana Olvera-Cravioto (University of North Carolina at Chapel Hill, EEUU)

This talk will focus on the probabilistic analysis of the PageRank algorithm on directed stochastic block models with K communities. Specifically, we show that for sparse graphs, i.e., with stochastically bounded degrees, the distribution of a randomly chosen vertex within a given community converges in a Wasserstein metric as the number of vertices grows to infinity. Moreover, the set of limiting distributions for typical vertices in each of the K communities can be characterized via a system of branching stochastic fixed-point equations (SFPEs). We then show how this characterization via SFPEs can be used to analyze the PageRank Nibble algorithm, which is a version of personalized PageRank that can be used for community detection provided one has a seed set of vertices known to belong to the community of interest. Our approach provides a good simple heuristic for choosing the optimal damping factor and provides computable bounds for the probability of misclassification.

16:45 - 17:30

#### Condensating zero-range process with condensation

Inés Armendáriz (Universidad de Buenos Aires, Argentina)

Zero-range processes are interacting particle models where particles jump between different sites according to rates that only depend on the number of particles at the current position. In this talk we will focus on the case when the rates decrease with the number of particles, inducing a condensation phenomenon where a macroscopic number of particles occupies the same position. I will present recent results that identify the coarsening dynamics of the process as fluid limit. These techniques can be applied to derive fluid limits for Jackson networks with rates that depend on the number of customers in the queue. Joint work with Johel Beltrán, Daniela Cuesta, Milton Jara and Matthieu Jonckheere.

17:30 - 18:15

#### Lattice trees in high dimensions

Manuel Cabezas (Pontificia Universidad Católica, Chile)

Lattice trees is a probabilistic model for random subtrees of $$\mathbb{Z}^d$$. In this talk we are going to review some previous results about the convergence of lattice trees to the "Super-Brownian motion" in the high-dimensional setting. Then, we are going to show some new theorems which strengthen the topology of said convergence. Finally, if time permits, we will discuss the applications of these results to the study of random walks on lattice trees.

Joint work with A. Fribergh, M. Holmes and E. Perkins.

• #### Martes 14

15:00 - 15:45

#### Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance

Michael Hoegele (Universidad de los Andes, Colombia)

This talk presents recent results on the cutoff phenomenon for a general class of asymptotically exponentially stable Ornstein-Uhlenbeck systems under ε-small additive Lévy noise. The driving noise processes include Brownian motion, α-stable Lévy flights, finite intensity compound Poisson processes and red noises and may be highly degenerate. Window cutoff thermalization is shown under generic mild assumptions, that is, we see an asymptotically sharp ∞/0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure με along a time window centered in a precise ε-dependent time scale tε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of the drift matrix Q. With this piece of theory at hand we provide a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to ε-small Brownian motion or α-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

Joint work with Gerardo Barrera (University of Helsinki, Finland) and Juan Carlos Pardo (CIMAT, México).

15:45 - 16:30

#### Procesos de Lévy Libres

Octavio Arizmendi (Centro de Investigación en Matemáticas, México)

En esta charla daremos un resumen panorámico sobre Procesos de Lévy Libres incluyendo unimodalidad y comportamiento en tiempos cercanos a 0 y en infinito. Concluiremos hablando de resultados recientes encontrados con Hasebe para el caso multiplicativo en R+ y en el círculo unitario.

16:45 - 17:30

#### Large deviations for the exclusion process with slow boundary

The discrete Coulomb gas is a model where charged particles are put on the $$d$$-dimensional grid. In this talk, I will discuss the basic properties of the Coulomb gas, its connection with other statistical physics models and the scaling limit of the potential of the discrete Coulomb gas at high enough temperature. Joint work with Christophe Garban.