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

The subject of this paper is a fragmentation equation with non-conservative 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 non-increasing self-similar Markov processes that reach continuously 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on non-extinction and is then used for the solutions to the fragmentation equation. We notice that two parameters influence significantly 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 to the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a non-trivial limit which is related to the quasi-stationary solutions to the equation. Besides, these quasi-stationary solutions, or equivalently the quasi-stationary distributions of the self-similar Markov processes, are entirely described.ou