So, I was in my car listening to the radio when I was listening to a story about the captured New York prison escapees, and it had me thinking:
In a hypothetical large fixed area of land, let us present a prisoner p and a cop, c. Assuming both move randomly, would a prisoner, on average, be caught quicker if they remained idle in one spot, or kept on moving?
I've taken coursework in mathematical modeling of epidemiology, and I would assume that the prisoner would be caught faster if both the cop and the prisoner were moving based on the following:
in mathematical modeling, contact rates are simplified via a simplified version of the "mass action" principle from chemistry, which states that a contact rate can be defined by the products of two concentrations. If we assume movement is a "concentration", it would imply that if a escaped prisoner and a cop were both moving, that the overall order of the reaction rate be 2nd order, as opposed to a first order system where it was just the cop moving.
Just curious, is there a probabilistic approach to this that would strengthen this argument further or maybe even give a completely different answer? Would it be dependent on their relative rates?