= Area of a planar Brownian path
{title2=$\lambda(\{B_t:0\leq t\leq1\})=0$}
A <planar Brownian motion> range has zero two-dimensional <Lebesgue measure> almost surely. Assuming finite expected area, <Brownian scaling> gives half that expected area for each half-time range. Inclusion-exclusion then makes their expected overlap zero. Translate their ranges by the midpoint position; the <independent Brownian arms at a deterministic time> make the expected overlap the integral of the squared point-hitting <probabilities> for a half-time range. Those <probabilities> vanish almost everywhere, so the half-time expected area, and hence the full-time expected area, vanish. This argument does not assert zero hitting <probability> for every individual point solely from an almost-everywhere integral conclusion.
Back to article page
