## A new probabilistic interpretation of Keller-Segel model for chemotaxis, application to 1-d.

Resumen: The Keller Segel (KS) model for chemotaxis is a two-dimensional system of parabolic or elliptic PDEs. Motivated by the study of the fully parabolic model using probabilistic methods, we give rise to a non-linear SDE of McKean-Vlasov type with a highly non standard and singular interaction kernel. In this talk I will briefly introduce the KS model, point out some of the PDE analysis results related to the model and then, in detail, analyze our probabilistic interpretation in the case d=1. This is a joint work with Denis Talay (TOSCA team, INRIA Sophia-Antipolis...

Read More## Limit distributions related to the Euler discretization error of Brownian motion about random times

Resumen: In this talk we study the simulation of barrier-hitting events and extreme events of one-dimensional Brownian motion. We call “barrier-hitting event” an event where the Brownian motion hits for the first time a deterministic “barrier” function; and call “extreme event” an event where the Brownian motion attains a minimum on a given compact time interval or unbounded closed time interval. To sample these events we consider the Euler discretization approach of Brownian motion; that is, simulate the Brownian motion on a discrete and equidistant times...

Read More## Propagation of critical behavior for unitary invariant plus GUE random matrices

Abstract: It is a well known and celebrated fact that the eigenvalues of random Hermitian matrices from a unitary invariant ensemble form a determinantal point process with correlation kernel given in terms of a system of orthogonal polynomials on the real line. It is a much more recent result that the eigenvalues of the sum of such a random matrix with a matrix from the Gaussian unitary ensemble (GUE) also forms a determinantal point process, with the kernel given in terms of the Weierstrass transform of the original kernel. I’ll talk about the case in which the limiting distribution of...

Read More## TWO-VALUED ENSEMBLE OF THE GAUSSIAN FREE FIELD.

ABSTRACT: The goal of this talk is to understand thin local sets of the continuous Gaussian free filed (GFF) in a domain of R^2, whose corresponding harmonic function takes only two values. We give a characterization of these sets and use it to show that in some sense they are maximal in a bigger class of local sets, where we only ask the function to be bounded. Important corollaries of this work are new constructions of the Conformal Loop Ensemble CLE_4 and a new perspective on the two known couplings between CLE_4 and the GFF. Joint work wiht JUHAN ARU and WENDELIN...

Read More## Limiting laws for some integrated processes

Resumen: The study of limiting laws, or penalizations, of a given process may be seen (in some sense) as a way to condition a probability law by an a.s. infinite random variable. The systematic study of such problems started in 2006 with a series of papers by Roynette, Vallois and Yor who looked at Brownian motion perturbed by several examples of functionals. These works were then generalized to many families of processes: random walks, Lévy processes, linear diffusions… We shall present here some examples of penalization of a non-Markov process, i.e. the integrated Brownian motion, by its...

Read More## Phase Transitions on the Long Range Ising Models in presence of an random external field

Resumen: We study the ferromagnetic one-dimensiosnal Random Field Ising Model with (RFIM) in presence of an external random field. The interaction between two spins decays as $d^{\alpha-2}$ where $d$ is the distance between two sites and $\alpha \in [0,1/2)$ is a parameter of the model. We consider an external random field on $\mathbb{Z}$ with independent but not identically distributed random variables. Specifically for each $i \in \mathbb{Z}$, the distribution of $h_i$ is $P[h_i=\pm \theta(1+|i|)^{-\nu/2}]$. This work, whose main goal is the study of the existence of a phase transition at...

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