Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be the disk of radius $r$ centered in the origin. Also, let $\nu_{A,x}^{R}$ be the conditional entrance measure to $A$ starting at $x\in B(R)\setminus A$, given that $\tau(A)<\tau(B(R))$. Assume also that $A\subset B(r)$ for some $0<r<R$. One may probably expect a result of the form $$ \Big|\frac{d \nu_{A,x}^{R}}{d\,hm_A}-1\Big| \leq const\frac{r\ln s}{s}, $$ where $s=\|x\|-r$ (because a similar result holds for the SRW). Could anyone provide a reference?
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