Exact expression for the diffusion propagator in a family of time-dependent anharmonic potentials

We have obtained the exact expression of the diffusion propagator in the time-dependent anharmonic potential $V(x,t)=\frac{1}{2}{a\left(t\right)x}^2+b\ln x.\mathrm{}$ The underlying Euclidean metric of the problem allows us to obtain analytical solutions for a whole family of the elastic parameter $a\left(t\right),$ 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 nonzero flow of particles through the infinite barrier located at the origin $\mathrm{}(b<\mathrm{}0\mathrm{}).$