Self-similar solutions to kinetic-type evolution equations: beyond the boundary case

We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023), perturbations take the form 1/θ log X/E where θ is a positive parameter, X has an arbitrary distribution μ and E is exponential with parameter 1, independent of X. Working under finite mean assumption for μ, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when θ does not exceed certain critical parameter θ_0 . This paper complements their work by providing weak centered asymptotics for the case when θ > θ_0 and presenting the results to μ with regularly varying tails. We prove almost sure convergence of the rightmost position and identify the appropriate centering for the weak convergence, which is of form αn + c log n, with constants α, c depending on the ratio of θ and θ_0. We describe the limiting distribution and provide explicitly the constants appearing in the centering.

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Self-similar solutions to kinetic-type evolution equations: beyond the boundary case

Cited by 3 publications

References 16 publications

Solutions of kinetic-type equations with perturbed collisions

Solutions of kinetic-type equations with perturbed collisions

Central limit theorem in uniform metrics for generalized Kac equations

Branching random walks with regularly varying perturbations

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