= Solution
For each $x$, the bounded local martingale $u(x+B_t)$ is a true martingale. Taking expectations gives
$$
u(x)=\mathbb E[u(x+B_t)]
=\int_{\mathbb R^2}u(x+z)
\frac{e^{-|z|^2/(2t)}}{2\pi t}\,dz
$$
for every $t>0$. Thus $u$ equals its convolution with every <heat kernel>. The convolution is smooth, so the originally Borel function $u$ is smooth. Differentiating the <heat semigroup> identity at $t=0$ gives $\Delta u=0$, so $u$ is a bounded <harmonic function> on the plane. The <Harmonic Liouville theorem> now implies that $u$ is constant.
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