Solutions of kinetic-type equations with perturbed collisions

We study a class of kinetic-type differential equations \(\partial \phi_t/\partial t+\phi_t=\widehat{\mathcal{Q}}\phi_t\), where \(\widehat{\mathcal{Q}}\) is an inhomogeneous smoothing transform and, for every \(t\geq 0\), \(\phi_t\) is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on \(\widehat{\mathcal{Q}}\) the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to \(\widehat{\mathcal{Q}}\). Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as \(t\to\infty\).