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Harmonic Measure

Harmonic measure describes the distribution of the boundary point where Brownian motion first exits a domain. For a bounded domain Omega subset R^n and a starting point x in Omega, it is the probability measure omega_Omega^x on partialOmega given by those exit probabilities. It represents harmonic extensions of continuous boundary data by integration, when those extensions exist.

The geometric size of harmonic measure can be smaller than that of the boundary. Let b_n be n minus the supremum, over all domains in R^n, of the least Hausdorff dimension of a set carrying full harmonic measure. Albert et al. (2026) proved the explicit bound

b_n>=(1/(n-1))^(2n+6.12sqrt(n))

for n>=2. In dimension three this gives b_3>=2^(-(6+6.12sqrt(3)))>10^(-5). Consequently harmonic measure on every domain in R^3 is carried by a set of Hausdorff dimension strictly less than 2.99999. Albert et al. (2026) report AI assistance with symbolic and numerical computations and parameter searches.

See also

Brownian Motion, Dirichlet Problem, Harmonic Function, Hausdorff Dimension

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References

Albert, M.; Badger, M.; and Genschaw, A. "One-Ring Bourgain Alternative and a 5-Decimal Bound on the Dimension of Harmonic Measure." 20 Oct 2026. https://arxiv.org/abs/2610.28037.

Cite this as:

Weisstein, Eric W. "Harmonic Measure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicMeasure.html

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