Approximations of the Wiener sausage and its curvature measures

TL;DR

This paper studies the Wiener sausage, a neighborhood of Brownian motion paths, and develops approximation methods for its curvature measures, providing theoretical convergence results and numerical calculations in two dimensions.

Contribution

It introduces approximation techniques for Wiener sausage curvature measures and proves their convergence, complemented by Monte Carlo simulations for numerical estimation.

Findings

Convergence of mean curvature measures for approximations in 2D and 3D.

Numerical estimation of Wiener sausage curvature measures in 2D.

Explicit approximation formulas for curvature measures.

Abstract

A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.