  We have obtained the exact expression of the diffusion propagator in the
time-dependent anharmonic potential $V(x,t)={1/2}a(t)x^2+b\ln x$. The
underlying Euclidean metric of the problem allows us to obtain analytical
solutions for a whole family of the elastic parameter a(t), exploiting the
relation between the path integral representation of the short time propagator
and the modified Bessel functions. We have also analyzed the conditions for the
appearance of a non-zero flow of particles through the infinite barrier located
at the origin (b<0).
