Examinando por Autor "Duarte, Mauricio"

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    A Lower Bound for the Number of Elastic Collisions
    (Springer New York LLC, 2019-12) Burdzy, Krzysztof; Duarte, Mauricio
    We prove by example that the number of elastic collisions of n balls of equal mass and equal size ind-dimensional space can be greater than n3/27 for n≥ 3 and d≥ 2. The previously known lower bound was of order n2. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
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    Asymptotics for the heat kernel in multicone domains
    (Academic Press Inc., 2016-02) Collet, Pierre; Duarte, Mauricio; Martínez, Servet; Prat-Waldron, Arturo; San Martín, Jaime
    A multicone domain Ω ⊆ Rn is an open, connected set that resembles a finite collection of cones far away from the origin. We study the rate of decay in time of the heat kernel p(t, x, y) of a Brownian motion killed upon exiting Ω, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize limt→∞ t1+αp(t, x, y) in terms of the Martin boundary of Ω at infinity, where α > 0 depends on the geometry of Ω. We next derive an analogous result for tκ/2Px(T >t), with κ = 1 + α − n/2, where T is the exit time from Ω. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.
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    Fermi acceleration in rotating drums
    (American Institute of Physics Inc., 2022-06-01) Burdzy, Krzysztof; Duarte, Mauricio; Gauthier, Carl-Erik; Graham, C. Robin; San Martin, Jaime
    Consider hard balls in a bounded rotating drum. If there is no gravitation, then there is no Fermi acceleration, i.e., the energy of the balls remains bounded forever. If there is gravitation, Fermi acceleration may arise. A number of explicit formulas for the system without gravitation are given. Some of these are based on an explicit realization, which we derive, of the well-known microcanonical ensemble measure. © 2022 Author(s).
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    Gravitation versus Brownian motion
    (Institute of Mathematical, 2019) Banerjee, Sayan; Burdzyb, Krzysztof; Duarte, Mauricio
    We investigate the motion of an inert (massive) particle being impinged from below by a particle performing (reflected) Brownian motion. The velocity of the inert particle increases in proportion to the local time of collisions and decreases according to a constant downward gravitational acceleration. We study fluctuations and strong laws of the motion of the particles. We further show that the joint distribution of the velocity of the inert particle and the gap between the two particles converges in total variation distance to a stationary distribution which has an explicit product form.
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    On the number of hard ball collisions
    (John Wiley and Sons Ltd., 2020-02) Burdzy, Krzysztof; Duarte, Mauricio
    We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are (Formula presented.) balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than (Formula presented.), then the total number of collisions is bounded by (Formula presented.), for any fixed (Formula presented.) and large (Formula presented.). We also show that if there is a number of collisions larger than (Formula presented.) for an appropriate (Formula presented.), then a large number of these collisions occur within a subfamily of balls that form a very tight configuration. © 2019 London Mathematical Society
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    Powers of Brownian Green Potentials
    (Springer Science and Business Media B.V., 2022-02) Dellacherie, Claude; Duarte, Mauricio; Martínez, Servet; Martín, Jaime San; Vandaele, Pierre
    In this article we study stability properties of gO, the standard Green kernel for O an open regular set in ℝd. In d ≥ 3 we show that gOβ is again a Green kernel of a Markov Feller process, for any power β ∈ [1,d/(d − 2)). In dimension d = 2, we show the same result for gOβ, for any β ≥ 1 and for kernels exp(αgO),exp(αgO)−1, for α ∈ (0,2π), when O is an open Greenian regular set whose complement contains a ball. © 2021, Springer Nature B.V.