FER University of ZagrebFaculty of Electrical Engineering and Computing

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The convex hull of a Brownian bridge with a random endpoint

Two paths of a planar Gaussian process with their convex hulls shaded green
The same path for two values of the parameter σ, with the corresponding convex hulls.

The paper On the convex hull of a planar Brownian bridge with a random Gaussian endpoint (N. Sandrić, S. Šebek, L. Šimek, arXiv:2606.24485) has been posted on arXiv.

The authors consider a one-parameter family of planar Gaussian processes that interpolates between a Brownian bridge returning to the origin (σ = 0) and standard Brownian motion (σ = 1). The main result is a closed formula for the expected perimeter and area of the convex hull of the path as a function of σ, recovering the known values in both extreme cases.

The key observation is that isotropy reduces the expected perimeter and area to one-dimensional quantities through the support function and Cauchy’s formulas.

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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