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It is known that the harmonic measure on classical one-third Cantor set has Hausdorff dimension strictly less than $\frac{\log 2}{\log 3}$. Even harmonic measure has a close relation with brownian motion. It is still be mysterious to have a better undertanding towards this.

Can any one give a intuitional explanation for this phenomenon?

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    $\begingroup$ What is the Hausdorff dimension of a measure? Harmonic measure needs a domain and a reference point as part of the setup, can you please clarify what these are in your setting? $\endgroup$
    – Christian Remling
    Commented Jul 28, 2016 at 18:46

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Let's say we start a 1-dimensional Brownian motion somewhere in the unit interval $[0,1]$, and look at where it hits the middle-third Cantor set. Certain points have higher probability than others.

In the picture, below there are four pillars that we can call 00, 01, 10, 11. But notice that 01 and 10 are more likely to be first hit than 00 or 11.

enter image description here $$\quad 00\qquad\qquad\qquad 01\qquad\qquad\qquad\qquad\qquad\qquad 10\qquad\qquad\qquad 11$$

So it is closer to a measure concentrated at a single point, and so the dimension is smaller than the Hausdorff dimension of the set, $\log 2/\log 3$.

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  • $\begingroup$ (Not sure about this, though.) $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Jul 28, 2016 at 7:53
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    $\begingroup$ You certainly need a two-dimensional Brownian motion for the usual connection to harmonic measure (a $1$-dimensional BM started in a gap can of course only hit the left or right endpoint of the gap). $\endgroup$
    – Christian Remling
    Commented Jul 28, 2016 at 18:59

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