I have a question about a proof in Protter. Let $B$ Brownian motion and $u$ a harmonic (subharmonic) function. Then $u(B)$ is a local martingale (submartingale). I was able to show the case of local martingale by myself. The case for subharmonic functino should be similar as Protter suggest. We have by Itô
$u(B_t)=u(B_0)+\int_0^t\nabla u(B_s)dB_S+\frac{1}{2}\int_0^t \Delta u(B_s)ds$
Using subharmonicity of $u$ we have:
$u(B_t)\ge u(B_0)+\int_0^t\nabla u(B_s)dB_S$
How does this imply the submartingale property of $u(B)$?