财富再分配的粒子系统方法
Particle system approach to wealth redistribution
摘要：
我们研究了一个随机$N$ -粒子系统，代表人口中的经济主体随机交换他们的钱，这是与一类一维动力学方程建模的财富分布在一个简单的市场经济的演变，介绍了Matthes和Toscani \cite{matthes-toscani2008}。我们证明了，除非经济交换满足某些精确的守恒条件，粒子的$p$矩随时间发散的所有$p>1$，并收敛到0的$0<p<1$。这建立了与动力学方程的定性差异，已知动力学方程的解对于所有小于平衡分布的帕累托指数的$p$具有有界的$p$ -矩。另一方面，严格保守经济的情况下，充分处理：使用概率耦合技术，我们得到的粒子系统的稳定性结果，如传播的时刻，指数平衡，和均匀（在时间上）传播的混沌秩序的明确率$N^{-1/3}$。
英文摘要：
We study a stochastic $N$-particle system representing economic agents in a population randomly exchanging their money, which is associated to a class of one-dimensional kinetic equations modelling the evolution of the distribution of wealth in a simple market economy, introduced by Matthes and Toscani \cite{matthes-toscani2008}. We show that, unless the economic exchanges satisfy some exact conservation condition, the $p$-moments of the particles diverge with time for all $p>1$, and converge to 0 for $0<p<1$. This establishes a qualitative difference with the kinetic equation, whose solution is known to have bounded $p$-moments, for all $p$ smaller than the Pareto index of the equilibrium distribution. On the other hand, the case of strictly conservative economies is fully treated: using probabilistic coupling techniques, we obtain stability results for the particle system, such as propagation of moments, exponential equilibration, and uniform (in time) propagation of chaos with explicit rate of order $N^{-1/3}$.
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