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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.

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My solution here math.stackexchange.com/questions/51926/… carries over to the supermartingale case. –  Byron Schmuland Jun 11 '12 at 22:25
    
Thank you Byron. I read your proof, it indeed solves my question, though I'm still thinking why the random walk in my question is recurrent for every point in the plain. The answer should be in some kind of "random walk theorem" in the book that I'm reading... –  Wei Jun 11 '12 at 22:48
    
Yes, the simple symmetric random walk is recurrent in dimensions 1 and 2, but transient in higher dimensions. This result is certainly in Durrett's book but I haven't got my copy handy, so I can't give you a page number. –  Byron Schmuland Jun 11 '12 at 22:50
    
Well, I know that simple symmetric random walk that take integer values are recurrent in dim 2, but the random walk in my question takes all values in B(0,1). Is it still recurrent? –  Wei Jun 11 '12 at 22:54
    
Oh, I hadn't noticed that you have a continuous state space. The argument I give for the other question would have to be modified to handle your case. –  Byron Schmuland Jun 11 '12 at 23:05

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