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

We compute exactly the mean perimeter and the mean area of the convex hull of a two-dimensional isotropic 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 root D/r f(1) (rt) and the mean area is given by < A(t)> = 2 pi D/r f(2)(rt) where the scaling functions f(1)(z) and f(2)(z) are computed explicitly. For large t >> 1/r, the mean perimeter grows extremely slowly as < L(t)> proportional to ln(rt) with time. Likewise, the mean area also grows slowly as < A(t)> proportional to 1n(2) (rt) for t >> 1/r. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times due to the isotropy of the Brownian motion. Numerical simulations are in perfect agreement with our analytical predictions.

Automated summary

This study computed the mean perimeter and mean area of the convex hull of Brownian motion with resetting, showing that the convex hull approaches a circular shape at late times. The analytical predictions were confirmed through numerical simulations.