Local Measure of Convex Surfaces induced by the Wiener Measure of Paths

TL;DR

This paper investigates a measure on convex surfaces derived from the Wiener measure of Brownian bridges, proposing a local classical action for odd dimensions based on geometric invariants.

Contribution

It introduces a novel local classical action that characterizes the measure on convex surfaces induced by Brownian bridges, supported by numerical evidence.

Findings

The measure is generated by a local classical action for odd dimensions.

The action depends solely on geometric invariants of the surfaces.

Numerical evidence supports the proposed measure and action.

Abstract

The Wiener measure induces a measure of closed, convex, (d-1)-dimensional, Euclidean (hyper-)surfaces that are the convex hulls of closed d-dimensional Brownian bridges. I present arguments and numerical evidence that this measure, for odd d, is generated by a local classical action of length dimension two that depends on geometric invariants of the (d-1)-dimensional surface only.