论文标题
通过粒子在紧凑的平面域外扩散边界遇到的统计
Statistics of boundary encounters by a particle diffusing outside a compact planar domain
论文作者
论文摘要
我们考虑在紧凑的平面集外扩散的粒子，并研究其本地时间$ \ ell_t $，即粒子与边界之间的相遇数量$ t $，直到$ t $。在磁盘的情况下，这也是平面中两个扩散的圆形颗粒的（重新恢复）相遇数量。对于这种情况，我们为边界局部时间$ \ ell_t $的概率密度以及给定阈值的首次跨度时间的概率密度得出明确的积分表示。由于相遇之间非常长的扩散偏差，后者的密度显示出非常缓慢的长期衰变。我们简要讨论了这种行为对化学物理和生物学应用的应用。
We consider a particle diffusing outside a compact planar set and investigate its boundary local time $\ell_t$, i.e., the rescaled number of encounters between the particle and the boundary up to time $t$. In the case of a disk, this is also the (rescaled) number of encounters of two diffusing circular particles in the plane. For that case, we derive explicit integral representations for the probability density of the boundary local time $\ell_t$ and for the probability density of the first-crossing time of a given threshold by $\ell_t$. The latter density is shown to exhibit a very slow long-time decay due to extremely long diffusive excursions between encounters. We briefly discuss some practical consequences of this behavior for applications in chemical physics and biology.
