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?

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. But what confuses me is that we have

$$\frac{1}{2}Cap_{N}(S(0,r))\leq P_{x}(T_{S(0,r)}<\infty).$$

So probably I misunderstood the estimate (http://www.math.upenn.edu/~pemantle/papers/martin.pdf Proposition 1.1).

Thanks

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

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?

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. But what confuses me is that we have

$$\frac{1}{2}Cap_{N}(S(0,r))\leq P_{x}(T_{S(0,r)}<\infty).$$

So probably I misunderstood the estimate.

Thanks

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?

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. But what confuses me is that we have

$$\frac{1}{2}Cap_{N}(S(0,r))\leq P_{x}(T_{S(0,r)}<\infty).$$

So probably I misunderstood the estimate (http://www.math.upenn.edu/~pemantle/papers/martin.pdf Proposition 1.1).

Thanks