Polar point for planar Brownian motion (${\mathbb{P}}_z(T_w<\infty )=0(z\ne w)$)

A prescribed point is almost surely never visited by planar Brownian motion started elsewhere. The harmonic function $\log |z-w|$ gives an annular hitting probability that tends to zero as the inner radius decreases to zero. This does not contradict recurrence of planar Brownian motion, which concerns neighbourhoods.