Abstract:

It has been said that a drunken man always finds his way home (recurrence) but a drunken bird may not (transience). I will show that this is true only if birds are allowed to go into outerspace. It is because of this recurrence that all bounded harmonic functions corresponding to simple random walks on $mathbb{Z}^d$ are constant (these are functions that satisfy a certain mean-value property). If time permits, I will relate this to the bounded harmonic functions of Complex Analysis.