Solutions of kinetic-type equations with perturbed collisions

We study a class of kinetic-type differential equations $\mathrm{\partial }{\phi }_t\mathrm{/}\mathrm{\partial }t+{\phi }_t=\widehat{\mathcal{Q}}{\phi }_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\ge 0$, ${\phi }_t$ 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 $t\to \mathrm{\infty }$.