Tiến Sĩ A shape theorem for the spread of an infection

Thảo luận trong 'Khoa Học Tự Nhiên' bắt đầu bởi Củ Đậu Đậu, 2/4/14.

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    Abstract
    In [KSb] we studied the following model for the spread of a rumor or infection: There is a “gas” of so-called A-particles, each of which performs a
    continuous time simple random walk onZd, with jump rate DA. We assume
    that “just before the start” the number ofA-particles at x, NA(x,0ư), has a
    meanàAPoisson distribution and that theNA(x,0ư),x∈Zd, are independent. In addition,
    there areB-particles which perform continuous time simple
    random walks with jump rateDB. We start with a finite number ofB-particles
    in the system at time 0. The positions of these initialB-particles are arbitrary,
    but they are nonrandom. TheB-particles move independently of each other.
    The only interaction occurs when aB-particle and anA-particle coincide; the
    latter instantaneously turns into aB-particle. [KSb] gave some basic estimates
    for the growth of the set  B(t):={x∈Zd:aB-particle visits xduring [0,t]}.
    In this article we show that ifDA=DB, then B(t):= B(t)+[ư12,12]d grows
    linearly in time with an asymptotic shape, i.e., there exists a nonrandom set
    B0such that (1/t)B(t)→B0, in a sense which will be made precise.
     

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