= Harmonic functions of Brownian motion
{title2=$u(B_{t\wedge\tau})\text{ is a local martingale if }\Delta u=0$}
For a $C^2$ <harmonic function> $u$ and <Brownian motion> stopped before leaving its domain, <Itô formula> gives $du(B_t)=\nabla u(B_t)\cdot dB_t+\frac12\Delta u(B_t)dt$. The drift vanishes. On a compact subdomain, the stopped process is a bounded <martingale>. <Optional stopping theorem> at an almost-surely finite exit time therefore gives the boundary mean formula. At the centre of a disk, <rotational invariance of planar Brownian motion> makes the exit law uniform on the boundary circle. This proves the <mean value property for harmonic functions> probabilistically.
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