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 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 | = O ( t ) . The second term is also given explicitly when | x | = o ( t ) . The corresponding result for unconditioned process is also obtained.