Fixed points of multivariate smoothing transformations
Final Report Abstract
Subject of the proposal were smoothing equations for probability distributions in finite-dimensional Euclidean space. These equations often characterize the limit distributions of quantities of interest in probabilistic models. Typical examples of such models are branching processes, fragmentation processes, urn models and models for probabilistic algorithms. The first main goal of the project was to solve these equations in finite dimension (the one-dimensional case was covered before) in a setup as general as possible. The second main goal was to understand important properties of the solutions such as existence and smoothness of densities, finiteness of moments and the asymptotic behavior of the tail probabilities. The third goal was to develop a corresponding theory for systems of smoothing equations. These questions were not only addressed within the project but also by other groups of scientists. A collective effort was made in the last years, and great progress was achieved. Some of the milestones were reached within the project. Namely, the first and second goal of the proposal have been achieved to a large extent. The set of all solutions to non-critical smoothing equations has been determined in a setting so general that it covers the vast majority of applications in the literature including the case of complex smoothing equations. The description of solutions given there reveals a lot of structure that simplifies the study of properties of solutions. Not yet solved so far, however, is the case of critical smoothing equations, which is left for future research. On the other hand, some important preparations for addressing this case have been made. In particular, special solutions related to the limits of Biggins’ martingales at critical complex parameters have been constructed. The tail behavior of solutions to critical smoothing equations was derived. Until recently, the existence of solutions of a special form (stopped L´vy processes) complicated the analysis of smoothing equations, but these solutions rarely showed up in applications. It was shown in the project that these solutions arise as limiting distributions in the study of the asymptotic fluctuations of Biggins’ martingales. Finally, the study of the asymptotic behavior of solutions to kinetic-type evolution equations in one dimension was completed and connected to the study of the extremal process in the branching random walk.
Publications
- Convergence of complex martingales in the branching random walk: the boundary. Electron. Commun. Probab. 22 (2017), Paper No. 18, 14 pp
Kolesko, Konrad; Meiners, Matthias
(See online at https://doi.org/10.1214/17-ECP50) - Solutions to complex smoothing equations. Probab. Theory Related Fields, 168 (2017), no. 1-2, 199–268
Meiners, Matthias; Mentemeier, Sebastian
(See online at https://doi.org/10.1007/s00440-016-0709-1) - Absolute continuity of the martingale limit in branching processes in random environment. Electron. Commun. Probab. 24 (2019), Paper No. 42, 13 pp
Damek, Ewa; Gantert, Nina; Kolesko, Konrad
(See online at https://doi.org/10.1214/19-ECP229) - Solutions to kinetictype evolution equations: beyond the boundary case
Dariusz Buraczewski; Konrad Kolesko; Meiners, Matthias
- Stable-like fluctuations of Biggins’ martingales. Stochastic Process. Appl. 129 (2019), no. 11, 4480–4499
Iksanov, Alexander; Kolesko, Konrad; Meiners, Matthias
(See online at https://doi.org/10.1016/j.spa.2018.11.022) - Fluctuations of Biggins’ martingales at complex parameters. Ann. Inst. Henri Poincaré Probab. Stat., 42 pp.
Iksanov, Alexander; Kolesko, Konrad; Meiners, Matthias
(See online at https://doi.org/10.1214/20-AIHP1046)
