惰性漂移图谱模型
The Inert Drift Atlas Model
摘要：
考虑一个大质量（惰性）粒子从上面撞击N个布朗粒子，这些粒子在与惰性粒子碰撞时瞬间反射。惰性粒子的速度由于外部牛顿位势（例如重力）的影响而增加，并且与与布朗粒子碰撞的总局部时间成比例地减小。该系统模拟了具有微观杂质的流体中的半透膜（Knight（2001））。我们研究了过程$(V,\mathbf{Z})$的长时间行为，其中$V$是惰性粒子的速度，$\mathbf{Z}$是按相对位置排列的连续粒子之间的间隙矢量。该系统不是亚椭圆的，不可逆的，并具有奇异形式的相互作用。因此，对系统稳定性的研究需要有新的思路。我们表明，这个过程有一个独特的平稳分布，需要一个明确的产品的形式，这是高斯的速度分量和指数的其他组件。我们还表明，收敛在全变差距离平稳分布发生在一个指数率。我们进一步得到了粒子位置和相交局部时的某些大数定律结果。
英文摘要：
Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight (2001)). We study the long-time behavior of the process $(V,\mathbf{Z})$, where $V$ is the velocity of the inert particle and $\mathbf{Z}$ is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and Exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times.
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