论文标题
惰性漂移图集模型
The Inert Drift Atlas Model
论文作者
论文摘要
考虑一个由n个布朗尼颗粒撞击的巨大（惰性）粒子，这些粒子在与惰性粒子发生碰撞时会立即反射出来。 惰性颗粒的速度由于外部牛顿电位（例如引力）的影响而增加，并且与与布朗颗粒碰撞的总局部时间成比例下降。 该系统在具有微观杂质的流体中建模了一个半渗透膜（Knight（2001））。 我们研究过程$（V，\ Mathbf {z}）$的长期行为，其中$ v $是惰性粒子的速度，$ \ mathbf {z} $是通过其相对位置订购的连续粒子之间间隙的向量。该系统不是低纤维化的，不是可逆的，并且具有奇异的形式相互作用。因此，对系统的稳定行为的研究需要新的想法。我们表明，此过程具有独特的固定分布，该分布采用明确的产品形式，该形式在速度组件中是高斯，而在其他组件中则指数为指数。我们还表明，与固定分布的总变化距离的收敛性以指数率发生。我们进一步获得了粒子位置和当地交点的大量定律。
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.
