We consider a particle diffusing outside a compact planar set and investigate its boundary local time ℓ _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 ℓ _t and for the probability density of the first-crossing time of a given threshold by ℓ _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.

我们认为，一个粒子的紧凑平面集外扩散，并调查其边界当地时间ℓ _牛逼，即颗粒和边界长达时间之间遭遇的重标数牛逼。对于磁盘，这也是平面中两个扩散圆形粒子相遇的（重新缩放）次数。对于这种情况，我们得出明确的积分表示为边界本地时间的概率密度ℓ _ŧ和通过给定的阀值的第一交叉时间的概率密度ℓ _ŧ 。由于相遇之间的扩散时间极长，后者的密度显示出非常缓慢的长时间衰减。我们简要讨论了此行为对化学物理和生物学应用的一些实际后果。