학술논문
Spectral gap for the growth-fragmentation equation via Harris's theorem.
Document Type
Journal
Author
Cañizo, José A. (E-GRAN-AM) AMS Author Profile; Gabriel, Pierre (F-VER-LM) AMS Author Profile; Yoldaş, Havva (F-LYON-ICJ) AMS Author Profile
Source
Subject
45 Integral equations -- 45K Integro-partial differential equations
45K05Integro-partial differential equations
47Operator theory -- 47D Groups and semigroups of linear operators, their generalizations and applications
47D06One-parameter semigroups and linear evolution equations
92Biology and other natural sciences -- 92D Genetics and population dynamics
92D25Population dynamics
45K05
47
47D06
92
92D25
Language
English
ISSN
10957154
Abstract
Summary: ``We study the long-time behavior of the growth-fragmentationequation, a nonlocal linear evolution equation describing a wide rangeof phenomena in structured population dynamics. We show the existenceof a spectral gap under conditions that generalize those in theliterature by using a method based on Harris's theorem, a result comingfrom the study of equilibration of Markov processes. The difficultyposed by the nonconservativeness of the equation is overcome byperforming an $h$-transform, after solving the dual Perron eigenvalueproblem. The existence of the direct Perron eigenvector is then aconsequence of our methods, which prove exponential contraction of theevolution equation. Moreover the rate of convergence is explicitlyquantifiable in terms of the dual eigenfunction and the coefficients ofthe equation.''