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Do we have $Cap_{N}(S(0,r))=P_{x}(T_{S(0,r)}<\infty)$ for $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a ball centered at the origin ?

I know for $x=0$, they are both equal to 1. How can I go about proving or disproving the above equality?

Thanks

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As mentioned in the other answer, equality cannot hold because the hitting probability depends on $x$. There are some inequalities between capacity and hitting probability under additional assumptions. The sharpest estimates I know are obtained when one replaces Newton capacity with a scale-invariant version, Martin capacity, see Benjamini, Pemantle, Peres: Martin capacity for Markov chains, Annals of Prob. 23 (3), 1332-1346.

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Actually, I don't think it does because $Cap_{N}(S(0,r))$ is a constant but $P_{x}(T_{S(0,r)}<\infty)$ changes as x changes.

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