Global solutions to an equation for mutation process in bacteria and its preservation of positive supports

Abstract

An equation describing mutation process in bacteria is considered in the form of a kinetic equation. The existence of a unique global solution to the Fourier-transformed equation is shown in spaces of characteristic functions whose corresponding probability measures possess finite moments. Due to correspondences of such spaces of probability measures and those of characteristic functions, a unique global probability measure solution to the original equation is found in the weak sense. Moreover, it is shown that if the support of the initial data is in R+, then that of the solution is also in R+. The proof is based on a contraction argument applied to the spatially homogeneous Boltzmann equation and the analysis of a solution as a boundary characteristic function.