Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar Markov processes

TL;DR

This paper analyzes the long-term behavior of solutions to a nonconservative fragmentation equation with shattering, using probabilistic methods involving self-similar Markov processes, revealing convergence to nontrivial limits influenced by key parameters.

Contribution

It introduces a probabilistic framework to describe the asymptotic behavior of fragmentation solutions with mass loss, including the characterization of quasi-stationary solutions.

Findings

Solutions converge to nontrivial limits under certain conditions.

The asymptotic behavior depends on the rate of formation of nearly-1 relative masses.

Quasi-stationary solutions are fully characterized.

Abstract

The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the large time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via nonincreasing self-similar Markov processes that continuously reach 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on nonextinction and is then used for the solutions to the fragmentation equation. We note that two parameters significantly influence these large time behaviors: the rate of formation of "nearly-1 relative masses" (this rate is related to the behavior near 0 of the L\'evy measure associated with the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a nontrivial limit which is related to the quasi-stationary solutions of the equation. Besides, these quasi-stationary solutions, or, equivalently, the quasi-stationary distributions of the self-similar Markov processes, are fully described.