  We study the non-stationary Feller process with time varying coefficients. We
obtain the exact probability distribution exemplified by its characteristic
function and cumulants. In some particular cases we exactly invert the
distribution and achieve the probability density function. We show that for
sufficiently long times this density approaches a Gamma distribution with
time-varying shape and scale parameters. Not far from the origin the process
obeys a power law with an exponent dependent of time, thereby concluding that
accessibility to the origin is not static but dynamic. We finally discuss some
possible applications of the process.

Statistical Mechanics