If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant?
There is an exercise in R.Durrett's probability book, which gives out a method to prove it by martingale theories, but I still don't know how to do it.
First of all, let $X_1,X_2,\cdots$ be i.i.d. random variables uniform on $B(0,1)$, and define $S_n=S_{n-1}+X_n$ for $n\geq 1$ and $S_0=x$. It's not difficult to see that $f(S_n)$ is a nonnegative supermartingale.
Then, a theorem tells us that all nonnegative supermartingales converge a.s. to a limit. So $f(S_n)$ converges to a certain $Y$.
It seems logical that $f$ should be constant because otherwise $f(S_n)$ cannot converge, but I don't know which property of $f$ can be used to prove this clearly.