I've recently seen the following question on the internet:
If I wanted to randomly find someone in an amusement park, would my odds of finding them be greater if I stood still or roamed around?
Which I formulate more precisely as follows:
Take a closed euclidean space $S$ with billiard-ball mixing and ergodic dynamics of two point-like agents with uniformly random starting positions and directions (but with given velocities). Is the expected meeting time $E[\tau(v_1,v_2)]$, a function of the velocities of agent $1$ and agent $2$ respectively, a decreasing function of $v_2$ for any $v_1$?
Where the meeting time $\tau$ is defined as the minimum time $t>0$ for which the distance among agents $d(p_1,p_2)$ satisfies $d(p_1,p_2)<D$.
It seems like a fairly difficult thing to show or disprove, at least with my limited knowledge. However, the limiting behavior seems very simple:
$$E[\tau(v_1,\infty)] = 0$$
Which follows directly from the fact that the dynamics is ergodic and mixing (at infinite velocity the agent visits every point instantaneously). Also,
$$E[\tau(0,0)] = \infty$$
For sufficiently small $D$. There's also the fact that
$$E[\tau(v_1,v_2)] = E[\tau(v_2,v_1)]$$
Is there are way to conclude from those observations? Any solution is also welcome.