Spectral Gap for the Growth-Fragmentation Equation via Harris's Theorem

Abstract

We study the long-time behavior of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalize those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the nonconservativeness of the equation is overcome by performing an h-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.

On spectral gaps of growth-fragmentation semigroups in higher moment spaces

“…We note also that most of the known literature on spectral gaps deals with Assumption (7), see however [13]. Our paper is close in spirit to [10] even if our statements are not the same and our constructions are different and more systematic; see below.…”

“…Some results have been established by probabilistic methods, see e.g., [11][12] but we are focused on operator-theoretic results for which we refer to the recent works [23][9][10][13] [7]. In particular, quantitative estimates of the gap are obtained by means of Harris's theorem, [13], while [7] contains a comprehensive theory for the discrete case written in the spirit of this paper. A special mention should be given to [16], where the Perron eigenvector and eigenvalue were found and analysed for (1) with fairly general coefficients.…”

On spectral gaps of growth-fragmentation semigroups with mass loss or death

“…Without pretense to completeness, we refer e.g. to [18,14,21,15,22,6,7,8] where some results combine relative entropy techniques; see also [9] for a probabilistic approach. We refer to the introductions of [22,15,7,8] for a comprehensive review of the existing tools and results.…”

“…to [18,14,21,15,22,6,7,8] where some results combine relative entropy techniques; see also [9] for a probabilistic approach. We refer to the introductions of [22,15,7,8] for a comprehensive review of the existing tools and results. In particular, we point out that asymptotic stability need not be uniform with respect to initial data and, at least in suitable weighted spaces, we cannot expect the existence of a spectral gap for bounded total fragmentation rates a(.)…”

“…18) for a suitable threshold α * ≥ 1 (see also [22,8]). In this paper, we deal with the asynchronous exponential growth (1.17) in the physical spaces (mass loss or death) which does not occur in most of the literature on the subject even if mass loss occurs in some physical contexts, see e.g.…”