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.

arXiv comment30 pages, accepted version with minor changes

References in corpus (18)

• Eigenelements of a General Aggregation-Fragmentation Model
• On the long time behavior of the TCP window size process
• Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions
• Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
• Measure solutions for some models in population dynamics
• A non-conservative Harris ergodic theorem
• Asymptotic behaviour of neuron population models structured by elapsed-time
• Measure solutions to the conservative renewal equation
• A probabilistic approach to spectral analysis of growth-fragmentation equations
• Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
• Probabilistic aspects of critical growth-fragmentation equations
• On an irreducibility type condition for the ergodicity of nonconservative semigroups
• Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate
• Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
• Exponential relaxation to self-similarity for the superquadratic fragmentation equation
• Global stability for the prion equation with general incidence
• Convergence Rates for Semistochastic Processes
• A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

Cited by in corpus (8)

• Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
• A multiple time renewal equation for neural assemblies with elapsed time model
• On quantitative hypocoercivity estimates based on Harris-type theorems
• Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics
• On spectral gaps of growth-fragmentation semigroups in higher moment spaces
• Exponential ergodicity of a degenerate age-size piecewise deterministic process
• Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes
• On the Krein-Rutman theorem and beyond