Solution of the Kolmogorov–Feller equation arising in the model of biological evolution

The Kolmogorov–Feller equation for the probability density of a Markov process on a half-axis, which arises in important problems of biology, is considered. This process consists of random jumps distributed according to Laplace's law and a deterministic return to zero. It is shown that the Green's function for such an equation can be found both in the form of a series and in explicit form for some ratios of the parameters. This allows one to find explicit solutions to the Kolmogorov–Feller equation for many initial data.