On the convex hull of a Brownian excursion with parabolic drift

The solutions of Burgers equation with white noise initial velocity are closely connected to the convex hull Ha of a Brownian excursion with parabolic drift s↦es+a2s2. We derive from the law of the minimum σ and the location η of the minimum of s↦2es/s(1−s) a complete description of the convex hull Ha. As an application, we determine the statistical properties of the Burgers turbulence on the circle.