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04 05 Im NoticiaTítulo: Generalized many-dimensional excited random walk in Bernoulli environment

Palestrante: Rodrigo Barreto Alves (FGV-EMap)
Data: 09/05/2022
Horário: 15h às 16h (Rio de Janeiro local time)
Local: Transmissão online

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Resumo: We study an extension of the generalized excited random walk (GERW) on Z^d introduced in [{\em Ann. Probab. 40 (5), 2012}}] by Menshikov, Popov, Ramírez and Vachkovskaia. 
Our extension consists in studying a version of the GERW where excitation may/may not occur according to a time-dependent probability. Specifically, given a sequence of parameters {p_n}_{n > 0}, with p_n in (0, 1] for all n > 0, whenever the process visits a site at time n for the first time, with probability p_n it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability 1-p_n, it behaves as a d-martingale with zero-mean vector. Whenever the process visits an already-visited site, the process acts again as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in short p_n-GERW. Under the same hypothesis of bounded jumps, uniform ellipticity and with a sequence {p_n\}_{n > 0} which decays polynomially, namelly p_n = Cn^{-\beta} with \beta > 0 and C is a positive constant, we show a series of results for the p_n-GERW depending on the value of \beta and on the dimension. Specifically, for \beta < 1/6 and d > 1, we show that the p_n-GERW has a positive probability of never returning to the origin in the drift direction, for \beta > 1/2, d > 1$ and \beta=1/2 and d=2 we obtain, under certain conditions, a Functional Central Limit Theorem. Finally, for \beta=1/2 and d > 3 we obtain, under suitable conditions, that the p_n-GERW is a tight process, and all sub-sequences converge, in distribution, to a process which is stochastically dominated in the drift direction below and above by a Brownian Motion plus a continuous function.

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