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Let $D$ be a disc in $\mathbb{C}\cong\mathbb{R}^2 $ and $z_0$ a fixed point of $D$. Is the harmonic measure for $V=D\setminus\{z_0\}$ known? Any reference would also be welcome.

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    $\begingroup$ should be same as harmonic measure for $D$ $\endgroup$
    – mathworker21
    Commented Jul 20, 2021 at 17:57
  • $\begingroup$ Would you please explain. I don't see why? $\endgroup$
    – M. Rahmat
    Commented Jul 20, 2021 at 18:57
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    $\begingroup$ brownian motion has probability $0$ of hitting any particular point. look up brownian motion interpretation of harmonic measure and 'polar set'. $\endgroup$
    – mathworker21
    Commented Jul 20, 2021 at 19:15
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    $\begingroup$ And in general, irregular boundary points do not influence the harmonic measure. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 20, 2021 at 19:49

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