Spectral gap for the growth-fragmentation equation via Harris's Theorem

Abstract

We study the long-time behaviour 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 generalise 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 non-conservativeness 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.

30 pages, accepted version with minor changes

References in corpus (2)

• Convergence Rates for Semistochastic Processes
• A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

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