 Planar Brownian motion avoids a fixed point

Planar Brownian motion started away from zero does not hit zero. Stopping the harmonic function $\log |x|$ on an annulus gives probability $(\log R-\log |x|)/(\log R-\log \epsilon )$ of reaching its inner circle before its outer circle. This tends to zero as $\epsilon \downarrow 0$. Exhausting the plane by outer circles proves point avoidance at finite times.

// Target: probability-and-statistics.bigb