The expected area of the Wiener sausage swept by a disc

TL;DR

This paper calculates the expected area of Wiener sausages formed by a disc along a 2D Brownian Bridge, revealing the leading term involves Ramanujan's function within a specific region.

Contribution

It provides explicit formulas for the expected area of Wiener sausages for Brownian bridges and unconditioned processes, linking to Ramanujan's function.

Findings

Leading term of expectation involves Ramanujan's function

Results hold within a parabolic region for the process

Explicit formulas for both conditioned and unconditioned processes

Abstract

The expected areas of the Wiener sausages swept by a disc attached to the two-dimensional Brownian Bridge joining the origin to a point x over a time interval [0,t] are computed. It is proved that the leading term of the expectation is given by Ramanujan's function if (x,t) remains in a parabolic region. The corresponding result for unconditioned process is also obtained.