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

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 by a constant κ. The expression for κ is of a complicated form, but it converges to the known constant as the drift tends to 0.