Overlap of two Brownian trajectories: exact results for scaling functions

We consider two random walkers starting at the same time $t=0$ from different points in space separated by a given distance $R$. We compute the average volume of the space visited by both walkers up to time $t$ as a function of $R$ and $t$ and dimensionality of space $d$. For $d<4$, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable $R/\sqrt{t}$. We provide general integral formulas for scaling functions for arbitrary dimensionality $d<4$. In contrast, we show that no scaling function exists for higher dimensionalities $d \geq 4$