  The random motion of a Brownian particle confined in some finite domain is
considered. Quite generally, the relevant statistical properties involve
infinite series, whose coefficients are related to the eigenvalues of the
diffusion operator. Unfortunately, the latter depend on space dimensionality
and on the particular shape of the domain, and an analytical expression is in
most circumstances not available. In this article, it is shown that the series
may in some circumstances sum up exactly. Explicit calculations are performed
for 2D diffusion restricted to a circular domain and 3D diffusion inside a
sphere. In both cases, the short-time behaviour of the mean square displacement
is obtained.
