Solutions to kinetic-type evolution equations: beyond the boundary case

Abstract

We study the asymptotic behavior as t → ∞ of a time-dependent family (µt) t≥0 of probability measures on R solving the kinetic-type evolution equation ∂tµt + µt = Q(µt) where Q is a smoothing transformation on R. This problem has been investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928-1961, 2012 and Bogus, Buraczewski and Marynych [To appear in Stochastic Process. Appl.]. Combining the refined analysis of the latter paper providing a probabilistic description of the solution µt as the law of a suitable random sum related to a continuous-time branching random walk at time t with recent advances in the analysis of the extremal positions in the branching random walk we are able to solve the remaining case that has not been addressed before.