Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados

The KPZ fixed point

Event Date: Oct 16, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

Abstract:    I will describe the construction and main properties of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class. The construction follows from an exact solution of the totally asymmetric exclusion process (TASEP) for arbitrary initial condition. This is joint work with K. Matetski and J. Quastel.

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A glimpse on excursion theory for the two-dimensional continuum Gaussian free field.

Event Date: Aug 14, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

Resumen:   Based on joint work with Juhan Aru, Titus Lupu and Wendelin Werner. Two-dimensional continuum Gaussian free field (GFF) has been one of the main objects of conformal invariant probability theory in the last ten years. The GFF is the two-dimensional analogue of Brownian motion when the time set is replaced by a 2-dimensional domain. Although one cannot make sense of the GFF as a proper function, it can be seen as a “generalized function” (i.e. a Schwartz distribution). The main objective of this talk is to go through recent development in the understanding of the...

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Exit-time of a self-stabilizing diffusion

Event Date: Jun 27, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

Resumen: In this talk, we briefly present some Freidlin and Wentzell results then we give a Kramers’type law satisfied by the McKean-Vlasov diffusion when the confining potential is uniformly strictly convex. We briefly present two previous proofs of this result before giving a third proof which is simpler, more intuitive and less technical.

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A link between the zeta function and stochastic calculus

Event Date: Jun 15, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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.

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

Event Date: Jun 13, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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...

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

Event Date: Mar 28, 2017 in Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

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...

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