I am a physist, but this life problem has been bothering me for a long time. Sorry if it is too easy.

Suppose you are in a supermarket with your friend shopping. At some moment you noticed that you lost your friend. The question is what is the fastest strategy to find your friend: 1) search him; or 2) just stay at one place, waiting for him/her.

To make it purely mathematical problem some simplifications are to be made. Your friend is looking for you. Search in the problem is random, it is Brownian motion. In other terms, whether probabilities of collision of the particles in the two strategies are equal? If no, what is the ratio of them? Reformulated strategies;

1) two Brownian particles are absolutely the same - the same displacement every time and the same rate (search strategy);

2) trajectory of one particle must intersect the other

**motionless** one, i.e. point (stay strategy).

I came to answer that 'search' strategy is faster

times than 'stay' one, but I am not sure, because it is just consideration, not rigorous proof.

Thank you