Expected perimeter and area of the convex hull of planar Brownian motion stopped upon exiting the unit disk

We study the convex hull of planar Brownian motion run until the exit time from the unit disk.

Our primary objectives are to compute the expected perimeter and expected area of this convex hull, thereby complementing recent results on the convex hull of reflecting Brownian motion in confined geometries.

We reduce the problem of computing the expected perimeter to computing the expected value of the Brownian motion's maximum horizontal displacement at the exit time, and then recast this maximum in terms of harmonic measure in a domain we call the truncated disk.

The problem of computing the expected area is reduced to computing the expected value of the difference of squares of the Brownian motion's maximum horizontal displacement at the exit time, and the value of the vertical displacement at the time this maximum horizontal displacement is achieved.

In particular, we obtain exact expressions for both the expected perimeter and the expected area.

We conclude with further results on the expected areas of two related hulls of the Brownian path run until exiting the disk, namely, the star hull and topological hull.