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Fix a closed ball $\overline{B}(x,r)\subset D$ and let $\tau$ be its first exit time. By the Strong Markov property, conditioning at $\tau$ givesThe orthogonal invariance of Brownian motion implies that $X_{\tau }$ is uniformly distributed on the sphere $\partial B(x,r)$. HenceThus $\varphi$ has the mean value property on every ball compactly contained in $D$. Since $0\le \varphi \le 1$, the mean-value characterization of harmonic functions yieldsThis function is the harmonic measure of $A$ viewed from $x$.

Let $g=g_A$ and define $u(z)=\mathrm{Im}(z-g(z))$. This is harmonic on $\mathbb{H}\setminus A$, has boundary values $\mathrm{Im}z$ on the hull boundary and zero on the real boundary, and tends to zero at infinity. The representation by harmonic measure and optional sampling theorem therefore giveAt $z=iy$, the hydrodynamic expansion givesConsequently the Brownian representation of half-plane capacity is

Write $A_r=[-r,r]\times (0,1]$ and take $r\ge 1$. By the Brownian representation of half-plane capacity,On hitting $A_r$, the exit height is at most one. Moreover, $A_r$ lies in the half-disc of radius $\sqrt{r^2+1}\le 2r$. The harmonic measure of that semicircle as viewed from $iy$ is $O(r/y)$: mapping its exterior to $\mathbb{H}$ by $z\mapsto z+(2r{)}^2/z$ reduces the estimate to the Poisson kernel for the upper half-plane on an interval of length $O(r)$. Consequentlyfor large $y$, and $\mathrm{hcap}(A_r)\le Cr$. This is the half-plane capacity of a low rectangle.

WriteThis is the harmonic measure of $A$ viewed from $z$, and is a bounded harmonic function outside the closed target disc. The disc is contained in $B(0,r_2)$. In the half-plane cut out by a line $L$ through the origin, the probability of reaching $B(0,r_2)$ before $L$, from a point of modulus $R$, is $O(r_2/R)$ uniformly in the direction. One sees this by mapping the half-plane outside the disc with $z\mapsto \log (z/r_2)$ to a half-strip and solving the corresponding Dirichlet problem by a sine series.

Part (b) consequently shows that the angular oscillation of $H_A$ on a large circle tends to zero, uniformly in the Borel set $A$. The same half-strip estimate in the annulus $R-|a|\le |z|\le R+|a|$ shows that shifting the center of that circle by the fixed vector $a$ changes the average of $H_A$ by $o(1)$, uniformly in $A$. Hence, for every $\epsilon >0$, all sufficiently large $R$ satisfyfor every Borel subset $A$ of the target disc.

For $r\ge 1$, let $R_r=[-r,r]\times (0,1]$. The Brownian representation of half-plane capacity givesOn hitting $R_r$ the imaginary part is at most one, while the harmonic measure estimate supplied in the question shows that the probability of reaching a disc of radius $O(r)$ containing $R_r$ is $O(r/y)$. Hence $\mathrm{hcap}(R_r)\le Cr$, the half-plane capacity of a low rectangle bound.

SetThe scaling and translation of half-plane capacity giveswhereas $\mathrm{diam}(A_n)=\sqrt{4+n^{-2}}\to 2$. This supplies the required sequence.

For the complex upper half-plane, the Poisson kernel isIt is the density of harmonic measure on the real boundary as viewed from $x+iy$.

Potential theory studies harmonic functions, subharmonic functions, and the Laplace operator, including harmonic measure, regular boundary points, and the Dirichlet problem. The Brownian representation of the Dirichlet problem connects these analytic objects with exit probability distributions.

If a compact H-hull lies in the unit disc, compare its tail harmonic measure with that of the empty hull and filled half-disc. The Poisson kernel for the upper half-plane turns the inclusion of Brownian exit events into $x\le g_K(x)\le x+1/x$ for $x>1$. Reflection gives $x+1/x\le g_K(x)\le x$ for $x<-1$.