I have a question that goes as follow: Consider a mass that starts to move from a location $c > 0$ and stops at $0$ after the first hitting time $T$. We know $Y(t)=c- m\cdot t+s\cdot W(t)$, where $Y(t)$ is the location of mass at time $t$; $c$ is the initial value. $Y(0) = c$ represents the distance between the origin station and destination. $m$ is the average speed of mass ($m>0$); $s$ is the diffusion coefficient which can be considered as the uncertainty of the mass location $s > 0, \mathrm{Var}[Y(t)] = t\cdot s^2$. Now, imagine another mass $X$ starts to follow (stalk) $Y$ after $u$ time units while maintaining the minimum distance $d$ of $Y$. If $X(t)$ is position of the $X$, at time $t$, we have $X(t)=\min\{X(t-u)+v\cdot u,Y(t-u)-d\}$, where $v$ is the maximum possible speed of $X$. First term indicates the case where $X$ must reduce its speed to keep the minimum distance $d$, while the second term is the free following case. Consider both movements happens on a street line, and the minimum distance should be kept until $Y$ reaches the destination, after which $X$ can also enter the destination. Knowing this, what can we say about moving process of $X$?
Any help would be appreciated. Thanks!