In this seminar I will describe the derivation of certain laws that rule the space-time evolution of the conserved quantities of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a model for which we can obtain a collection of (fractional) reaction-diffusion equations given in terms of the regional fractional Laplacian with different types of boundary conditions.

Discrete Markov random fields on graphs, also known as graphical models in the statistical literature, have become popular in recent years due to their flexibility to capture conditional dependency relationships between variables. They have already been applied to many different problems in different fields such as Biology, Social Science, or Neuroscience. Graphical models are, in a sense, finite versions of general random fields or Gibbs distributions, classical models in stochastic processes. This talk will present the problem of estimating the interaction structure (conditional dependencies) between variables by a penalized pseudo-likelihood criterion. First, I will consider this criterion to estimate the interaction neighborhood of a single node, which will later be combined with the other estimated neighborhoods to obtain an estimator of the underlying graph. I will show some recent consistency results for the estimated neighborhood of a node and any finite sub-graph when the number of candidate nodes grows with the sample size. These results do not assume the usual positivity condition for the conditional probabilities of the model as it is usually assumed in the literature of Markov random fields. These results open new possibilities of extending these models to situations with sparsity, where many parameters of the model are null. I will also present some ongoing extensions of these results to processes satisfying mixing type conditions. This is joint work with Iara Frondana, Rodrigo Carvalho and Magno Severino.

In this talk we will give a new proof for a question that was first posed by Jonasson, Mossel and Peres, concerning percolation on a randomly stretched planar lattice. More specifically, we fix a parameter q in (0, 1) and we slash the lattice Z^2 in the following way. For every vertical line that crosses the x axis along an integer value, we toss an independent coin and with probability q we remove all edges along that line. Then we do the same with the horizontal lines that cross the y axis at integer values. We are then left with a graph G that looks like a randomly stretched version of Z^2 and on top of which we would like to perform i.i.d. Bernoulli percolation. The question at hand is whether this percolation features an non-trivial phase transition, or more precisely, whether p_c(G) < 1. Although this question has been previously solved in a seminal article by Hoffman, we present here an alternative solution that greatly simplifies the exposition. We also explain how the presented techniques can be used to prove the existence of a phase transition for other models with minimal changes to the proof.

This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

In this talk, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. BPI-processes form a subclass of branching processes with interactions, which were recently introduced by González Casanova et al. (2021), and includes the so-called logistic branching process which was studied by Lambert (2005).

Here, we provide a series of integral tests that fully explains how competition and cooperation regulates the long-term behaviour of BPI-processes. In particular, we give necessary and sufficient conditions for the events of explosion and extinction, as well as conditions under which the process comes down from infinity. Moreover, we also determine whether the process admits, or not, a stationary distribution. Our arguments use the moment dual of BPI-processes which turns out to be a family of diffusions taking values on \([0,1]\), that we introduce as generalised Wright-Fisher diffusions together with a complete understanding of the nature of their boundaries.

We study a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of biological mechanisms. We focus here on local interactions: upon firing, a given neuron increases the potential of a fixed number of randomly chosen neurons. We show a phase transition in terms of the stationary distribution of the limiting network. Whereas a finite network activity always vanishes in finite time, the infinite network might converge to either a trivial stationary measure or to a nontrivial one. This allows to model the biological phenomena of persistence: we prove that the network may retain neural activity for large times depending on certain global parameters describing the intensity of interconnection.

Joint work with: Maximiliano Altamirano, Roberto Cortez, Lasse Leskela.

Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of r-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non smooth payoffs are analyzed.

I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, as well as some extensions to processes with memory length larger than 1.

Based on joint work with Konstantin Matetski.

The martingale problem introduced by Stroock and Varadhan is an efficient method to prove the convergence of a sequence of stochastic processes which are derived from Markov processes. In this talk we present two examples to illustrate this approach: the density of particles per site of a sequence of condensing zero range processes and the number of sites occupied by a system of coalescing random walks evolving on a transitive finite graph. Both examples exhibit a sort of asymptotic lumpability.