On the convex hull of a planar Brownian bridge with a random Gaussian endpoint

Abstract

We consider a one-parameter family of isotropic planar Gaussian processes \[ X_σ(t) =B_t+σt Z,\qquad 0\le t\le 1,\quad 0\le σ\le 1, \] where $B$ is a standard ($0$-to-$0$) planar Brownian bridge on $[0,1]$, and $Z\sim \mathrm{N}(0,I)$ is a standard Gaussian random vector independent of $B$. The family interpolates between standard planar Brownian bridge ($\sigma =0$) and standard planar Brownian motion ($\sigma =1$). As the main result of the paper we compute the expected perimeter and area of the convex hull of the random set $\left\{X_{\sigma }(t):0\le t\le 1\right\}$ as closed formulas in terms of $\sigma$, and recover the classical Brownian bridge and Brownian motion values at $\sigma =0$ and $\sigma =1$. We also consider the convex hull spanned by multiple independent processes of this type and the possibilities for closed formulas in special cases. The key observation in our argument is that the isotropy property reduces the expected perimeter and area to one-dimensional quantities through the support function and Cauchy's formulas.

17 pages, 1 figure