Título: Conditional propagation of chaos for systems of interacting neurons with random synaptic weights
Palestrante: Eva Löcherbach (Université Paris I)
Data: 08/02/2021
Horário: 15:00 até 16:00h (Horário do Rio de Janeiro)
Local: Transmissão on-line
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Resumo: We study the stochastic system of interacting neurons introduced in De Masi et al 2015, in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1 / \sqrt (N). In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as the number of neurons tends to infinity, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system.
Conditionally on W, the different neurons are independent in the limit system. This is the ``conditional propagation of chaos'' property. We prove the well-posedness of the limit equation by adapting the ideas of Graham 1992 to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.
This is a joint work with Xavier Erny and Dasha Loukianova, both of university of Evry.
Título: Central limit theorems for a driven particle in a random medium with mass aggregation
Palestrante: Pablo Almeida Gomes (IME-USP)
Data: 01/02/2021
Horario: 15:00 até 16:00h (Horário do Rio de Janeiro)
Local: Transmissão online
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Resumo: In this talk we investigate a one-dimensional infinite mechanical particle system, driven by a constant force F. The system consists of one charged particle, together with field-neutral ones. Neutral particles are initially randomly placed in the medium, and can be perfectly elastic or inelastic, according to independent Bernoulli random variables. We establish central limit theorems for the velocity and position of the charged particle.
Based on joint work with Luiz Renato Fontes and Rémy Sanchis.
Título: Non-intersecting Brownian motions with outliers, KPZ fluctuations and random matrices
Palestrante: Daniel Remenik (Universidad de Chile)
Data: 18/01/2020
Horario: 15:00 - 16:00 (Horário do Rio de Janeiro)
Local: Transmissão online.
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Resumo: A well known result implies that the rescaled maximal height of a system of N non-intersecting Brownian bridges starting and ending at the origin converges, as N goes to infinity, to the Tracy-Widom GOE random variable from random matrix theory. In this talk I will focus on the same question in case where the top m paths start and end at arbitrary locations. I will present several related results about the distribution of the limiting maximal height for this system, which provides a deformation of the Tracy-Widom GOE distribution: it can be expressed through a Fredholm determinant formula and in terms of Painlevé transcendents; it corresponds to the asymptotic fluctuations of models in the KPZ universality class with a particular initial condition; and it is connected with two PDEs, the KdV equation and an equation derived by Bloemendal and Virag for spiked random matrices. Based on joint work with Karl Liechty and Gia Bao Nguyen.
Título: An Effective Class of Ballistic Random Walks in Mixing Random Environments
Palestrante: Glauco Valle (IM-UFRJ)
Data: 25/01/2021
Horário: 15:00 - 16:00. (Rio de Janeiro local time)
Local: Transmissão online
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Resumo: We study d-dimensional random walks in strong mixing environments (RWRE), with underlying dimension d>=2. Under a suitable polynomial effective condition, we prove a functional central limit theorem of ballistic type. Specifically, we construct a new effective criterion equivalent to usual ballisticity conditions. This construction allows us to prove, in a mixing framework, the RWRE conjecture regarding the equivalence between ballisticity conditions already proved for iid environments. We then obtain the polynomial effective condition that provides the existence of arbitrary finite moments for approximate regeneration times, yielding the central limit theorem for the RWRE.
Joint work with Maria Eulalia Vares (UFRJ) and Enrique Guerra (PUC-Chile).
All the talks are held in English.
We take the opportunity to inform that the videos of the online seminars held during 2020 are available HERE.
Regarding this year, a few days after each meeting the video should be available HERE.
Título: Brownian modules of continuité and diffusion approximation
Palestrante: Julien Chevallier (LJK, Université Grenoble Alpes)
Data: 11/01/2021
Horário: 15:00h - 16:00h (Rio de Janeiro local time)
Local: Transmissão Online
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Resumo: Garsia–Rodemich–Rumsey (1971) proved an inequality which has been used to upper-bound the Brownian modules of continuity. In turn, this upper-bound was used by T.G. Kurtz (1976) to prove a strong diffusion approximation result for pure jump processes. However, this proof makes the crucial assumption that jump rates are uniformly bounded. The main objective of this talk is to show how to get rid of this assumption starting back from GRR inequality. The second objective is to show that this scheme of proof is robust to time scaling.
All the talks are held in English.
We take the opportunity to inform that the videos of the online seminars held during 2020 are available HERE.
Regarding this year, a few days after each meeting the video should be available HERE.
