On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors

TL;DR

This paper advances the probabilistic analysis of growth-fragmentation semigroups using the Feynman-Kac approach, establishing necessary conditions for Malthusian behavior and providing criteria for exponential convergence speed.

Contribution

It proves the necessity of the Malthusian condition and introduces a simple criterion for exponential convergence, extending previous results in the field.

Findings

Necessary condition for Malthusian behavior established

Criterion for exponential convergence speed provided

Broader cases covered than in previous literature

Abstract

This work develops further a probabilist approach to the asymptotic behavior of growth-fragmentation semigroups via the Feynman-Kac formula, which was introduced in a joint article with A.R. Watson [4]. Here, it is first shown that the sufficient condition for a Malthusian behavior which was established in [4], is also necessary. We then provide a simple criterion to ensure exponential speed of convergence, which enables us to treat cases than were not covered previously in the literature.