Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting

Abstract

We compute exactly the mean perimeter and the mean area of the convex hull of a $2$-d Brownian motion of duration $t$ and diffusion constant $D$, in the presence of resetting to the origin at a constant rate $r$. We show that for any $t$, the mean perimeter is given by $⟨L(t)⟩=2\pi \sqrt{\frac{D}{r}}{f}_1(rt)$ and the mean area is given by $⟨A(t)⟩=2\pi \frac{D}{r}{f}_2(rt)$ where the scaling functions $f_1(z)$ and $f_2(z)$ are computed explicitly. For large $t\gg 1/r$, the mean perimeter grows extremely slowly as $⟨L(t)⟩\propto \ln (rt)$ with time. Likewise, the mean area also grows slowly as $⟨A(t)⟩\propto {\ln }^2(rt)$ for $t\gg 1/r$. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times. Numerical simulations are in perfect agreement with our analytical predictions.

16 pages, 6 figures. Comments added about the circular shape of the convex hull at large time

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