Planar Brownian motion avoids a fixed point
= 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 $(\log R-\log|x|)/(\log R-\log\varepsilon)$ of reaching its inner circle before its outer circle. This tends to zero as $\varepsilon\downarrow0$. Exhausting the plane by outer circles proves point avoidance at finite times.
// Target: probability-and-statistics.bigb