The expected area of the Wiener sausage swept by a disc

The expected areas of the Wiener sausages swept by a disc attached to the two-dimensional Brownian Bridge joining the origin to a point xx over a time interval [0,t][0,t] are computed. It is proved that the leading term of the expectation is given by Ramanujan’s function if |x|=O(t). The second term is also given explicitly when |x|=o(t). The corresponding result for unconditioned process is also obtained.