通过相互作用Lévy过程演化的竞争粒子系统
Competing particle systems evolving by interacting Lévy processes
摘要：
我们考虑在连续时间中演化的实数线和半线上的有限和无限粒子系统。因此，粒子由i.i.d.驱动。具有秩相关漂移和扩散系数的Lévy过程。在有限系统中，我们表明，在各自的粒子配置的间隙的过程具有唯一的不变分布，并证明了间隙过程的收敛到后者的全变差距离，假设一个界的跳跃的Lévy过程。在无限情形下，我们证明了在适当的初始条件下，所有粒子的漂移和扩散系数相同时，半直线上粒子系统的间隙过程是紧的。这些过程的应用包括金融市场中排名参与者之间的资本分布建模，某些随机存储和存储网络的稳定性以及自旋玻璃的Sherrington-Kirkpatrick模型的研究。
英文摘要：
We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. Lévy processes endowed with rank-dependent drift and diffusion coefficients. In the finite systems we show that the processes of gaps in the respective particle configurations possess unique invariant distributions and prove the convergence of the gap processes to the latter in the total variation distance, assuming a bound on the jumps of the Lévy processes. In the infinite case we show that the gap process of the particle system on the half-line is tight for appropriate initial conditions and same drift and diffusion coefficients for all particles. Applications of such processes include the modeling of capital distributions among the ranked participants in a financial market, the stability of certain stochastic queueing and storage networks and the study of the Sherrington--Kirkpatrick model of spin glasses.
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