Noticias en español
A link between the zeta function and stochastic calculus
Abstract: The study of the zeros of the Riemann zeta function constitutes one of the most challenging problems in mathematics. A large literature in devoted to the study of the behavior of the zeta zeros. We will discuss how tools from stochastic analysis, and in particular from Malliavin calculus (multiple integrals, Wiener chaos, Stein method etc) can be used in the study of some aspects of the behavior of the zeta function.
Read MoreA 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 MoreLimit 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 MorePropagation 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 MoreTWO-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 WERNER.
Read MoreLimiting 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...
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