Noticias en español
Long time asymptotics for critical birth and death diffusion processes.
Resumen: We study the long time behavior for the distribution of a critical birth and death diffusion process, motivated by population dynamics in changing environment (cf. a recent paper by Calvez, Henry, Méléard, Tran). The birth rates are bounded but death rates are unbounded. Our analysis is based on the spectral properties of the associated Feynman Kac semigroup. We require a standard spectral gap property for this semigroup with a dominant eigenfunction vanishing at infinity. Some examples of diffusions, diffusions with jump, pure jump dynamics are given for which it is true....
Read MoreResumen: We present a few results to get compound poisson distributions for random dynamical systems and for a class of stochastic differential equations. The main tool will be the use of spectral techniques.
Read MoreOn the stationary measures of two variants of the voter model.
Resumen: The voter model is an interacting particle system describing the collective behaviour of voters who constantly update their political opinions on a given graph. This Markov process is dual to a system of coalescing random walks on the graph. This duality relationship makes the model more tractable by analysing the dynamics of the collision of random walks. This presentation is divided into two parts. First, we introduce two variants of the voter model: the voter model on dynamical percolation (in a random environment) and the voter model with stirring (where a stirring parameter...
Read MoreSymmetries in Overparametrized Neural Networks: A Mean-Field View. & Feature Learning with a structured covariance
Orador: Javier Maass (CMM) 15:00 hrs Resumen: We develop a Mean-Field (MF) view of the learning dynamics of overparametrized Artificial Neural Networks (NN) under distributional symmetries of the data w.r.t. the action of a general compact group G. We consider for this a class of generalized shallow NNs given by an ensemble of N multi-layer units, jointly trained using stochastic gradient descent (SGD) and possibly symmetry-leveraging (SL) techniques, such as Data Augmentation (DA), Feature Averaging (FA) or Equivariant Architectures (EA). We introduce the notions of weakly and strongly...
Read MoreA comparison theorem for integrated stochastic Volterra models with application to the modelling of Lagrangian intermittency in turbulence.
Resumen: We introduce a stochastic model for the Lagrangian velocity and dissipation of a turbulent flow, which takes the form of an integrated Volterra process, as already proposed in the litterature. In order to understand how to reproduce the multifractal behaviours predicted by the Kolomogorov refined theory, we propose a way to compare the effects of different Volterra kernels on the statistics of the integrated process. Since Volterra processes are not Markovian, we use the martingale approach and the functional Itô formula from [Viens, Zhang 2019], combined with the path-dependent...
Read MoreSelection principles for the N-BBM and the Fleming-Viot particle system.
Resumen: The selection problem is to show, for a given branching particle system with selection, that the stationary distribution for a large but finite number of particles corresponds to the travelling wave of the associated PDE with minimal wave speed. This had been an open problem for any such particle system. The N-branching Brownian motion with selection (N-BBM) is a particle system consisting of N independent particles that diffuse as Brownian motions in $\mathbb{R}$, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. We...
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