= Stopped inverse radial power martingale classification
{title2=$\|B_{t\wedge\tau_a}\|^{2-d},\quad d\geq3$}
For <Brownian motion> started at radius $x>0$, the displayed process is a <continuous local martingale>. It is a bounded true <martingale> when $0<a\leq x$, but a <strict local martingale> when $a=0$ or $a>x$. In the latter case the boundary is outside the starting point, so stopping does not remove the singularity at the origin. Brownian scaling gives $\mathbb E\|B_t\|^{2-d}=O(t^{-(d-2)/2})$; when $a>x$, the stopped expectation tends to $a^{2-d}<x^{2-d}$.
Back to article page
