FER University of ZagrebFaculty of Electrical Engineering and Computing

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Perimeter and area of the Brownian convex hull in the unit disk

Brownian path inside the unit disk with its star hull and convex hull shaded green
The same Brownian path until it exits the disk, with its star hull (left) and convex hull (right).

The paper Expected perimeter and area of the convex hull of planar Brownian motion stopped upon exiting the unit disk (R. Mrazović, H. Panzo, S. Šebek, arXiv:2604.18427) has been posted on arXiv.

The authors follow planar Brownian motion from the centre of the unit disk until it first leaves the disk, and compute the expected perimeter and expected area of the convex hull of this path. Both problems reduce to the maximum horizontal displacement of the motion at the exit time, which is then expressed through harmonic measure in a domain the authors call the truncated disk.

They obtain exact expressions for the expected perimeter and area, and conclude with results on the expected areas of the star hull and topological hull of the same path.

Results on the geometry of random paths are part of the mathematical groundwork of VALOR.

Stjepan Šebek’s research was partly funded by the VALOR project.

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