Say I have $2$ random walkers that are positioned along a line. One of them will be on the left and the other on the right. Every time they meet, this ordering does not change, namely the two walkers cannot overtake each other, (the left one will remain on the left). Now because of the Markov property, I can just let the two walkers diffuse and at each meeting point, change labels to the walks (the left-most trajectory will always belong to the left walker, etc.).
My question is this: Is there a way to analytically derive the distributions of the two walkers over time? What direction should I take to do that, and eventually scale the problem up to $n$ walkers?
Update: Solution: I essentially want to find the minimum of two random variables that are independent of each other, and both normally distributed in the same way, which is a well-known problem.