  We show that the occupation measure of planar Brownian motion exhibits a
constant height gap of $5/\pi$ across its outer boundary. This property bears
similarities with the celebrated results of Schramm--Sheffield [18] and
Miller--Sheffield [12] concerning the height gap of the Gaussian free field
across SLE$_4$/CLE$_4$ curves. Heuristically, our result can also be thought of
as the $\theta \to 0^+$ limit of the height gap property of a field built out
of a Brownian loop soup with subcritical intensity $\theta>0$, proved in our
recent paper [3]. To obtain the explicit value of the height gap, we rely on
the computation by Garban and Trujillo Ferreras [1] of the expected area of the
domain delimited by the outer boundary of a Brownian bridge.
