On the expected volume of the Wiener sausage

Abstract

We consider the expected volume of the Wiener sausage on the time interval [0,t] associated with a closed ball. Let L(t) be the expected volume minus the volume of the ball. We obtain that L(t) is asymptotically equal to a constant multiple of t^1/2 as t tends to 0 and that it is represented as an absolutely convergent power series of t^1/2 for any t > 0 in the odd dimensional cases. Moreover, the explicit form of L(t) can be given in five and seven dimensional cases.