Asymptotic analysis of multiscale Lévy-driven stochastic Cucker-Smale and non-linear friction models
Final Report Abstract
Various real-world phenomena, such as the flocking/dispersion of animal populations or the dissipation effects in mechanical or climate systems, can be realistically described using randomly perturbed non-linear Newtonian equations of motion. The qualitative behavior of such systems is often determined by the non-linear and position-dependent dissipative friction force. In this project, we considered a totally coupled slow-fast stochastic system with a dissipative ´ fast component. The model was subjected to weak Levy perturbations operating on the microscopic time scale. The goal of the project was to analyze the limit dynamics of the slow component under very general assumptions. Depending on the heavy-tail behavior of the perturbation and its interplay with dissipation, one can observe asymptotic regimes where the limit process is either a diffusion (diffusion approximation regime) or a discontinuous Lévy-type process (non-linear Lévy filter regime). To perform such asymptotic analysis, it was necessary to establish estimates for the moments of the fast component on the infinite time horizon. To this end, we adapted the method of Lyapunov functions, well-known for Markov processes, to a very general semimartingale setting and obtained novel individual moment bounds for dissipative semimartingales with heavy-tail jumps. As the main mathematical tool for the analysis of a totally coupled slow-fast system, we developed new mathematical techniques based on the ”long-step semimartingale” regression scheme. Under very general assumptions, we determined the diffusion limit of the slow component. We also studied numerically and analytically a decoupled non-linear Lévy filter. A general theory for fully coupled slow-fast systems in the non-linear Lévy filter regime is yet to be developed. We believe that the ”balance condition” between the strength of dissipation and the heaviness of the tails of the perturbation will play an eminent role in this theory. We hope that the results obtained in the project will contribute substantially to the general asymptotic theory of multi-scale stochastic systems and will advance the understanding of the non-linear effects in realistic stochastic models of physics, biology, and climate sciences.
Publications
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Moment bounds for dissipative semimartingales with heavy jumps. Stochastic Processes and their Applications, 141, 274-308.
Kulik, Alexei & Pavlyukevich, Ilya
