A link between the zeta function and stochastic calculus

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.

CMM, 15:30

jun / 2017
15

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

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 Mediterranee).

Sala John Von Neumann, 7mo piso, CMM, U. Chile. 16:00 hrs.

jun / 2017
13

Stable Langevin model with diffusive-reflective boundary conditions.

We consider a one-dimensional stable Langevin process confined in the upper half-plane and submitted to a diffusive-reflective boundary condition whenever the particle position hits 0. We show that different regimes appear according to the value of the chosen parameters. We then use this study to construct the law of a (free) stable Langevin process conditioned to stay positive, thus extending earlier works on the integrated Brownian motion. Such construction finally enables us to improve some recent persistence probability estimates. This is a joint work with Jean-François Jabir.

abr / 2017
25

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

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 mesh, e.g., {0, 1/n, 2/n, …}. With this, for each of the aforementioned events we study the discretization error between the actual time the event occurs versus the time the event occurs on the discretized path, and also the discretization error on the position of the Brownian motion at these times.

Our main results are threefold.

First, we show limits in distribution for the discretization errors normalized by their convergence rate, and give closed-form analytic expressions for the limiting random variables.

Second, we use the previous limits in distribution to derive new limits in the theory of Diffusion Approximation, which studies approximating random walks by using diffusion processes. More precisely, we obtain limits that use Brownian motion to approximate the asymptotic behavior of Gaussian random walks in the following situations: (1.) the overshoot of a Gaussian walk above a barrier that goes to infinity; (2.) the minimum of a Gaussian walk over a time horizon that goes to infinity; and (3.) the global minimum of a Gaussian walk having positive drift decreasing to zero.

Third, and perhaps more important, we provide a unified framework that relates several papers since the 1960’s where the constant -zeta(1/2)/\sqrt(2*pi) has appeared, where zeta is the Riemann zeta function. Up to now, how these works are precisely connected has been considered an open question. We show that this constant is the mean of some of the limiting distributions we obtain, and claim that each of these papers is directly connected to a result we derive.

mar / 2017
28

Propagation of critical behavior for unitary invariant plus GUE random matrices

nov / 2016
24

Pontificia Universidad Católica de Chile (PUC-Chile)

Av. Vicuña Mackenna 4860, Macul,

Santiago – Chile

(+56 2) 2354 5779

Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

Beauchef 851, Edificio Norte, Piso 7,

Santiago – Chile