A formula for the expected volume of the Wiener sausage with constant drift

TL;DR

This paper derives an explicit formula for the expected volume of the Wiener sausage with constant drift in multiple dimensions, revealing its relation to the non-drift case and providing asymptotic behavior for large time.

Contribution

It provides a new explicit formula for the expected volume of the Wiener sausage with drift, extending known results to include drift effects and asymptotic analysis.

Findings

Expected volume expressed as sum of non-drift Wiener sausage volumes

Leading term of expected volume is proportional to time with explicit constant

As drift tends to zero, formula converges to known non-drift result

Abstract

We consider the Wiener sausage for a Brownian motion with a constant drift up to time $t$ associated with a closed ball. In the two or more dimensional cases, we obtain the explicit form of the expected volume of the Wiener sausage. The result says that it can be represented by the sum of the mean volumes of the multi-dimensional Wiener sausages without a drift. In addition, we show that the leading term of the expected volume of the Wiener sausage is written as $\kappa t(1+o[1])$ for large $t$ and an explicit form gives the constant $\kappa$. The expression is of a complicated form, but it converges to the known constant as the drift tends to $0$.