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 $r>0$.
One should 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?