On the solution of the Kolmogorov-Feller equation arising in the model of biological evolution

Abstract

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 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 explicitly find solutions to the Kolmogorov-Feller equation for many initial data.